Nonlinear physics (solitons, chaos, discrete breathers)

Transcription

1 Nonlinear physics (solitons, chaos, discrete breathers) N. Theodorakopoulos Konstanz, June 2006

2 Contents Foreword vi 1 Background: Hamiltonian mechanics Lagrangian formulation of dynamics Hamiltonian dynamics Canonical momenta Poisson brackets Equations of motion Canonical transformations Point transformations Hamilton-Jacobi theory Hamilton-Jacobi equation Relationship to action Conservative systems Separation of variables Periodic motion. Action-angle variables Complete integrability Symmetries and conservation laws Homogeneity of time Homogeneity of space Galilei invariance Isotropy of space (rotational symmetry of Lagrangian) Continuum field theories Lagrangian field theories in 1+1 dimensions Symmetries and conservation laws Perturbations of integrable systems Background: Statistical mechanics Scope Formulation Phase space Liouville s theorem Averaging over time Ensemble averaging Equivalence of ensembles Ergodicity The FPU paradox The harmonic crystal: dynamics The harmonic crystal: thermodynamics The FPU numerical experiment i

3 Contents 4 The Korteweg - de Vries equation Shallow water waves Background: hydrodynamics Statement of the problem; boundary conditions Satisfying the bottom boundary condition Euler equation at top boundary A solitary wave Is the solitary wave a physical solution? KdV as a limiting case of anharmonic lattice dynamics KdV as a field theory KdV Lagrangian Symmetries and conserved quantities KdV as a Hamiltonian field theory Solving KdV by inverse scattering Isospectral property Lax pairs Inverse scattering transform: the idea The inverse scattering transform The direct problem Time evolution of scattering data Reconstructing the potential from scattering data (inverse scattering problem) IST summary Application of the IST: reflectionless potentials A single bound state Multiple bound states Integrals of motion Lemma: a useful representation of a(k) Asymptotic expansions of a(k) IST as a canonical transformation to action-angle variables Solitons in anharmonic lattice dynamics: the Toda lattice The model The dual lattice A pulse soliton Complete integrability Thermodynamics Chaos in low dimensional systems Visualization of simple dynamical systems Two dimensional phase space dimensional phase space dimensional phase space; nonautonomous systems with one degree of freedom Small denominators revisited: KAM theorem Chaos in area preserving maps Twist maps Local stability properties Poincaré-Birkhoff theorem Chaos diagnostics The standard map The Arnold cat map The baker map; Bernoulli shifts ii

4 Contents The circle map. Frequency locking Topology of chaos: stable and unstable manifolds, homoclinic points Solitons in scalar field theories Definitions and notation Lagrangian, continuum field equations Static localized solutions (general KG class) General properties Specific potentials Intrinsic Properties of kinks Linear stability of kinks Special properties of the SG field The Sine-Gordon breather Complete Integrability Atoms on substrates: the Frenkel-Kontorova model The Commensurate-Incommensurate transition The continuum approximation The special case ɛ = 0: kinks and antikinks The general case ɛ > 0: the soliton lattice Breaking of analyticity FK ground state as minimizing periodic orbit of the standard map Small amplitude motion Free end boundary conditions Metastable states: spatial chaos as a model of glassy structure Solitons in magnetic chains Introduction Classical spin dynamics Spin Poisson brackets An alternative representation Solitons in ferromagnetic chains The continuum approximation The classical, isotropic, ferromagnetic chain The easy-plane ferromagnetic chain in an external field Solitons in antiferromagnets Continuum dynamics The isotropic antiferromagnetic chain Easy axis anisotropy Easy plane anisotropy Easy plane anisotropy and symmetry-breaking field Solitons in conducting polymers Peierls instability Electrons decoupled from the lattice Electron-phonon coupling; dimerization Solitons and polarons in (CH) x A continuum approximation Dimerization The soliton The polaron iii

5 Contents 12 Solitons in nonlinear optics Background: Interaction of light with matter, Maxwell-Bloch equations Semiclassical theoretical framework and notation Dynamics Propagation at resonance. Self-induced transparency Slow modulation of the optical wave Further simplifications: Self-induced transparency Self-focusing off-resonance Off-resonance limit of the MB equations Nonlinear terms Space-time dependence of the modulation: the nonlinear Schrödinger equation Soliton solutions Solitons in Bose-Einstein Condensates The Gross-Pitaevskii equation Propagating solutions. Dark solitons Unbinding the double helix A nonlinear lattice dynamics approach Mesoscopic modeling of DNA Thermodynamics Nonlinear structures (domain walls) and DNA melting Local equilibria Thermodynamics of domain walls Pulse propagation in nerve cells: the Hodgkin-Huxley model Background The Hodgkin-Huxley model The axon membrane as an array of electrical circuit elements Ion transport via distinct ionic channels Voltage clamping Ionic channels controlled by gates Membrane activation is a threshold phenomenon A qualitative picture of ion transport during nerve activation Pulse propagation Localization and transport of energy in proteins: The Davydov soliton Background. Model Hamiltonian Energy storage in C=O stretching modes. Excitonic Hamiltonian Coupling to lattice vibrations. Analogy to polaron Born-Oppenheimer dynamics Quantum (excitonic) dynamics Lattice motion Coupled exciton-phonon dynamics The Davydov soliton The heavy ion limit. Static Solitons Moving solitons iv

6 Contents 17 Nonlinear localization in translationally invariant systems: discrete breathers The Sievers-Takeno conjecture Numerical evidence of localization Diagnostics of energy localization Internal dynamics Towards exact discrete breathers A Impurities, disorder and localization 164 A.1 Definitions A.1.1 Electrons A.1.2 Phonons A.2 A single impurity A.2.1 An exact result A.2.2 Numerical results A.3 Disorder A.3.1 Electrons in disordered one-dimensional media A.3.2 Vibrational spectra of one-dimensional disordered lattices Bibliography 173 v

7 Foreword The fact that most fundamental laws of physics, notably those of electrodynamics and quantum mechanics, have been formulated in mathematical language as linear partial differential equations has resulted historically in a preferred mode of thought within the physics community - a linear theoretical bias. The Fourier decomposition - an admittedly powerful procedure of describing an arbitrary function in terms of sines and cosines, but nonetheless a mathematical tool - has been firmly embedded in the conceptual framework of theoretical physics. Photons, phonons, magnons are prime examples of how successive generations of physicists have learned to describe properties of light, lattice vibrations, or the dynamics of magnetic crystals, respectively, during the last 100 years. This conceptual bias notwithstanding, engineers or physicists facing specific problems in classical mechanics, hydrodynamics or quantum mechanics were never shy of making particular approximations which led to nonlinear ordinary, or partial differential equations. Therefore, by the 1960 s, significant expertise had been accumulated in the field of nonlinear differential and/or integral equations; in addition, major breakthroughs had occurred on some fundamental issues related to chaos in classical mechanics (Poincaré, Birkhoff, KAM theorems). Due to the underlying linear bias however, this substantial progress took unusually long to find its way to the core of physical theory. This changed rapidly with the advent of electronic computation and the new possibilities of numerical visualization which accompanied it. Computer simulations became instrumental in catalyzing the birth of nonlinear science. This set of lectures does not even attempt to cover all areas where nonlinearity has proved to be of importance in modern physics. I will however try to describe some of the basic concepts mainly from the angle of condensed matter / statistical mechanics, an area which provided an impressive list of nonlinearly governed phenomena over the last fifty years - starting with the Fermi-Pasta-Ulam numerical experiment and its subsequent interpretation by Zabusky and Kruskal in terms of solitons ( paradox turned discovery, in the words of J. Ford). There is widespread agreement that both solitons and chaos have achieved the status of theoretical paradigm. The third concept introduced here, localization in the absence of disorder, is a relatively recent breakthrough related to the discovery of independent (nonlinear) localized modes (ILMs), a.k.a. discrete breathers. Since neither the development of the field nor its present state can be described in terms of a unique linear narrative, both the exact choice of topics and the digressions necessary to describe the wider context are to a large extent arbitrary. The latter are however necessary in order to provide a self-contained presentation which will be useful for the non-expert, i.e. typically the advanced undergraduate student with an elementary knowledge of quantum mechanics and statistical physics. Konstanz, June 2006 vi

8 1 Background: Hamiltonian mechanics Consider a mechanical system with s degrees of freedom. The state of the mechanical system at any instant of time is described by the coordinates {Q i (t), i = 1, 2,, s} and the corresponding velocities { Q i (t)}. In many applications that I will deal with, this may be a set of N point particles which are free to move in one spatial dimension. In that particular case s = N and the coordinates are the particle displacements. The rules for temporal evolution, i.e. for the determination of particle trajectories, are described in terms of Newton s law - or, in the more general Lagrangian and Hamiltonian formulations. The more general formulations are necessary in order to develop and/or make contact with fundamental notions of statistical and/or quantum mechanics. 1.1 Lagrangian formulation of dynamics The Lagrangian is given as the difference between kinetic and potential energies. particle system interacting by velocity-independent forces For a L({Q i, Q i }) = T V (1.1) T = 1 s m i Q 2 i 2 i=1 V = V ({Q i }, t). where an explicit dependence of the potential energy on time has been allowed. Lagrangian dynamics derives particle trajectories by determining the conditions for which the action integral t S(t, t 0 ) = dτl({q i, Q i, τ}) (1.2) t 0 has an extremum. The result is d L dt Q = L (1.3) i Q i which for Lagrangians of the type (1.2) becomes i.e. Newton s law. m i Qi = V Q i (1.4) 1.2 Hamiltonian dynamics Canonical momenta Hamiltonian mechanics, uses instead of velocities, the canonical momenta conjugate to the coordinates {Q i }, defined as P i = L Q i. (1.5) 1

9 1 Background: Hamiltonian mechanics In the case of (1.2) it is straightforward to express the Hamiltonian function (the total energy) H = T + V in terms of P s and Q s. The result is H({P i, Q i }) = s I=1 P 2 i 2m i + V ({Q i }). (1.6) Poisson brackets Hamiltonian dynamics is described in terms of Poisson brackets {A, B} = s i=1 { A B A } B Q i P i P i Q i (1.7) where A, B are any functions of the coordinates and momenta. The momenta are canonically conjugate to the coordinates because they satisfy the relationships Equations of motion According to Hamiltonian dynamics, the time evolution of any function A({P i, Q i }, t) is determined by the linear differential equations Ȧ da dt = {A, H} + A t. (1.8) where the second term denotes any explicit dependence of A on the time t. Application of (1.8) to the cases A = P i and A = Q i respectively leads to P i = {P i, H} Q i = {Q i, H} (1.9) which can be shown to be equivalent to (1.4). The time evolution of the Hamiltonian itself is governed by dh dt = H ( = V ). (1.10) t t Canonical transformations Hamiltonian formalism important because the symplectic structure of equations of motion (from Greek συµπλɛκω = crosslink - of momenta & coordinate variables -) remains invariant under a class of transformations obtained by a suitable generating function ( canonical transformations). Example, transformation from old coordinates & momenta {P, Q} to new ones {p, q}, via a generating function F 1 (Q, q, t) which depends on old and new coordinates (but not on old and new momenta - NB there are three more forms of generating functions - ): P i = F 1(q, Q, t) Q i p i = F 1(q, Q, t) q i K = H + F 1 t (1.11) 2

10 1 Background: Hamiltonian mechanics new coordinates are obtained by solving the first of the above eqs., and new momenta by introducing the solution in the second. It is straightforward to verify that the dynamics remains form-invariant in the new coordinate system, i.e. ṗ i = {p i, K} q i = {q i, K} (1.12) and dk(p, q, t) K(p, q, t) =. (1.13) dt t Note that if there is no explicit dependence of F 1 on time, the new Hamiltonian K is equal to the old H Point transformations A special case of canonical transformations are point transformations, generated by F 2 (Q, p, t) = i f i (Q, t)p i ; (1.14) New coordinates depend only on old coordinates - not on old momenta; in general new momenta depend on both old coordinates and momenta. A special case of point transformations are orthogonal transformations, generated by F 2 (Q, p) = i,k a ik Q k p i (1.15) where a is an orthogonal matrix. It follows that q i = k p i = k a ik Q k a ik P k. (1.16) Note that, in the case of orthogonal transformations, coordinates transform among themselves; so do the momenta. Normal mode expansion is an example of (1.16). 1.3 Hamilton-Jacobi theory Hamilton-Jacobi equation Hamiltonian dynamics consists of a system of 2N coupled first-order linear differential equations. In general, a complete integration would involve 2N constants (e.g. the initial values of coordinates and momenta). Canonical transformations enable us to play the following game: 1 Look for a transformation to a new set of canonical coordinates where the new Hamiltonian is zero and hence all new coordinates and momenta are constants of the motion. 2 Let (p, q) be the set of original momenta and coordinates in eqs of previous section, 1 Hamilton-Jacobi theory is not a recipe for integration of the coupled ODEs; nor does it in general lead to a more tractable mathematical problem. However, it provides fresh insight to the general problem, including important links to quantum mechanics and practical applications on how to deal with mechanical perturbations of a known, solved system. 2 Does this seem like too many constants? We will later explore what independent constants mean in mechanics, but at this stage let us just note that the original mathematical problem of integrating the 2N Hamiltonian equations does indeed involve 2N constants. 3

11 1 Background: Hamiltonian mechanics (α, β) the set of new constant momenta and coordinates generated by the generating function F 2 (q, α, t) which depends on the original coordinates and the new momenta. The choice of K 0 in (1.11) means that F 2 t + H(q 1, q s ; F 2 q 1,, F 2 q s ; t) = 0. (1.17) Suppose now that you can [miraculously] obtain a solution of the first-order -in general nonlinear-pde (1.17), F 2 = S(q, α, t). Note that the solution in general involves s constants {α i, i = 1,, s}. The s + 1st constant involved in the problem is a trivial one, because if S is a solution, so is S + A, where A is an arbitrary constant. It is now possible to use the defining equation of the generating function F 2 β i = S(q, α, t) α i (1.18) to obtain the new [constant] coordinates {β i, i = 1,, s}; finally, turning inside out (1.18) yields the trajectories q j = q j (α, β, t). (1.19) In other words, a solution of the Hamilton-Jacobi equation (1.17) provides a solution of the original dynamical problem Relationship to action It can be easily shown that the solution of the Hamilton-Jacobi equation satisfies or S(q, α, t) S(q, α, t 0 ) = ds dt = L, (1.20) t t 0 dτ L(q, q, τ) (1.21) where the r.h.s involves the actual particle trajectories; this shows that the solution of the Hamilton-Jacobi equation is indeed the extremum of the action function used in Lagrangian mechanics Conservative systems If the Hamiltonian does not depend explicitly on time, it is possible to separate out the time variable, i.e. S(q, α, t) = W (q, α) λ 0 t (1.22) where now the time-independent function W (q) (Hamilton s characteristic function) satisfies ( H q 1, q s ; W,, W ) = λ 0, (1.23) q 1 q s and involves s 1 independent constants, more precisely, the s constants α 1, α s depend on λ 0. 4

12 1.3.4 Separation of variables 1 Background: Hamiltonian mechanics The previous example separated out the time coordinate from the rest of the variables of the HJ function. Suppose q 1 and W q 1 enter the Hamiltonian only in the combination φ 1 (q 1, W q 1 ). The Ansatz W = W 1 (q 1 ) + W (q 2,, q s ) (1.24) in (1.23) yields H (q 2, q s ; W,, W ; φ 1 q 2 q s ( q 1, W ) ) 1 = λ 0 ; (1.25) q 1 since (1.25) must hold identically for all q, we have H ( φ 1 q 1, W ) 1 q 1 ) (q 2, q s ; W,, W ; λ 1 q 2 q s = λ 1 = λ 0. (1.26) The process can be applied recursively if the separation condition holds. Note that cyclic coordinates lead to a special case of separability; if q 1 is cyclic, then φ 1 = W q 1 = W 1 q 1, and hence W 1 (q 1 ) = λ 1 q 1. This is exactly how the time coordinate separates off in conservative systems (1.23). Complete separability occurs if we can write Hamilton s characteristic function - in some set of canonical variables - in the form W (q, α) = i W i (q i, α 1,, α s ). (1.27) Periodic motion. Action-angle variables Consider a completely separable system in the sense of (1.27). The equation p i = S q i = W i(q i, α 1,, α s ) q i (1.28) provides the phase space orbit in the subspace (q i, p i ). Now suppose that the motion in all subspaces {(q i, p i ), i = 1,, s} is periodic - not necessarily with the same period. Note that this may mean either a strict periodicity of p i, q i as a function of time (such as occurs in the bounded motion of a harmonic oscillator), or a motion of the freely rotating pendulum type, where the angle coordinate is physically significant only mod 2π. The action variables are defined as J i = 1 2π p i dq i = 1 2π dq i W i (q i, α 1,, α s ) q i (1.29) and therefore depend only on the integration constants, i.e. they are constants of the motion. If we can turn inside out (1.29), we can express W as a function of the J s instead of the α s. Then we can use the function W as a generating function of a canonical transformation to a new set of variables with the J s as new momenta, and new angle coordinates θ i = W J i = W i(q i, J 1,, J s ) J i. (1.30) 5

13 1 Background: Hamiltonian mechanics In the new set of canonical variables, Hamilton s equations of motion are J i = 0 θ i = H(J) J i ω i (J). (1.31) Note that the Hamiltonian cannot depend on the angle coordinates, since the action coordinates, the J s, are - by construction - all constants of the motion. In the set of action-angle coordinates, the motion is as trivial as it can get: J i = const θ i = ω i (J) t + const. (1.32) Complete integrability A system is called completely integrable in the sense of Liouville if it can be shown to have s independent conserved quantities in involution (this means that their Poisson brackets, taken in pairs, vanish identically). If this is the case, one can always perform a canonical transformation to action-angle variables. 1.4 Symmetries and conservation laws A change of coordinates, if it reflects an underlying symmetry of physical laws, will leave the form of the equations of motion invariant. Because Lagrangian dynamics is derived from an action principle, any such infinitesimal change which changes the particle coordinates and adds a total time derivative to the Lagrangian, i.e. q i q i = q i + ɛf i (q, t) q i q i = q i + ɛf i (q, t) (1.33) L = L + ɛ df dt, (1.34) will leave the equations of motion invariant. On the other hand, the transformed Lagrangian will generally be equal to and therefore the quantity will be conserved. L ({q i, q i}) = L({q i, q i}) = L({q i, q i }) + = L({q i, q i }) + = L({q i, q i }) + s i=1 s [ L ɛf i + L ] ɛf q i q i i s [ ( ) d L ɛf i + L ] ɛf dt q i q i i s ( ) d L f i dt q i i=1 i=1 i=1 Such underlying symmetries of classical mechanics are: L q i f i F (1.35) 6

14 1.4.1 Homogeneity of time 1 Background: Hamiltonian mechanics L = L(t + ɛ) = L(t) + ɛdl/dt, i.e. F = L; furthermore, q i = q i(t + ɛ) = q i + ɛ q i, i.e. f i = q i. As a result, the quantity s L H = q i L (1.36) q i (Hamiltonian) is conserved. i= Homogeneity of space The transformation q i q i + ɛ (hence f i = 1) leaves the Lagrangian invariant (F = 0). The conserved quantity is s L P = (1.37) q i (total momentum). i= Galilei invariance The transformation q i q i ɛt (hence f i = t) does not generally change the potential energy (if it depends only on relative particle positions). It adds to the kinetic energy a term ɛp, i.e. F = m i q i. The conserved quantity is (uniform motion of the center of mass). s m i q i P t (1.38) i= Isotropy of space (rotational symmetry of Lagrangian) Let the position of the ith particle in space be represented by the vector coordinate q i. Rotation around an axis parallel to the unit vector ˆn is represented by the transformation q i q i + ɛ f i where f i = ˆn q i. The change in kinetic energy is ɛ i q i f i = 0. If the potential energy is a function of the interparticle distances only, it too remains invariant under a rotation. Since the Lagrangian is invariant, the conserved quantity (1.35) is s i=1 L q i f i = s m i qi (ˆn q i ) = ˆn I, i=1 where is the total angular momentum. s I = m i ( q i q i ) (1.39) i=1 7

15 1.5 Continuum field theories 1 Background: Hamiltonian mechanics Lagrangian field theories in 1+1 dimensions Given a Lagrangian in 1+1 dimensions, L = dxl(φ, φ x, φ t ) (1.40) where the Lagrangian density L depends only on the field φ and first space and time derivatives, the equations of motion can be derived by minimizing the total action S = dtdxl (1.41) and have the form d dt ( L φ t ) + d ( ) L L dx φ x φ = 0. (1.42) Symmetries and conservation laws The form (1.42) remains invariant under a transformation which adds to the Lagrangian density a term of the form ɛ µ J µ (1.43) where the implied summation is over µ = 0, 1, because this adds only surface boundary terms to the action integral. If the transformation changes the field by δφ, and the derivatives by δφ x, δφ t, the same argument as in discrete systems leads us to conclude that L L δφ + δφ x + L ( dj0 δφ t = ɛ φ φ x φ t dt + dj ) 1 (1.44) dx which can be transformed, using the equations of motion, to ( ) d L δφ + L δφ t + d ( ) L δφ + L ( dj0 δφ x = ɛ dt φ t φ t dx φ x φ x dt + dj ) 1 dx (1.45) Examples: 1. homogeneity of space (translational invariance) x x + ɛ δφ = φ(x + ɛ) φ(x) = φ x ɛ δφ t = φ t (x + ɛ) φ t (x) = φ xt ɛ δφ x = φ x (x + ɛ) φ x (x) = φ xx ɛ δl = dl dl δx = dx dx ɛ J 1 = L, J 0 = 0. (1.46) Eq. (1.45) becomes ( ) d L φ x + L φ xt + d ( ) L φ x + L φ xx = dl dt φ t φ t dx φ x φ x dx or (1.47) ( ) d L φ x + d ( ) L φ x L = 0 ; (1.48) dt φ t dx φ x 8

16 1 Background: Hamiltonian mechanics integrating over all space, this gives dx L φ x P φ t (1.49) i.e. the total momentum is a constant. 2. homogeneity of time t t + ɛ δφ = φ(t + ɛ) φ(t) = φ t ɛ δφ t = φ t (t + ɛ) φ t (t) = φ tt ɛ δφ x = φ x (t + ɛ) φ x (t) = φ xt ɛ δl = dl dl δt = dt dt ɛ J 0 = L, J 1 = 0. (1.50) Eq. (1.45) becomes ( ) d L φ t + L φ tt + d ( ) L φ t + L φ tx = dl dt φ t φ t dx φ x φ x dt or (1.51) ( ) d L φ t L + d ( ) L φ t = 0 ; (1.52) dt φ t dx φ x integrating over all space, this gives [ ] L dx φ t L H (1.53) φ t i.e. the total energy is a constant. 3. Lorentz invariance 1.6 Perturbations of integrable systems Consider a conservative Hamiltonian system H 0 (J) which is completely integrable, i.e. it possesses s independent integrals of motion. Note that I use the action-angle coordinates, so that H 0 is a function of the (conserved) action coordinates J j. The angles θ j are cyclic variables, so they do not appear in H 0. Suppose now that the system is slightly perturbed, by a time-independent perturbation Hamiltonian µh 1 (µ 1) A sensible question to ask is: what exactly happens to the integrals of motion? We know of course that the energy of the perturbed system remains constant - since H 1 has been assumed to be time independent. But what exactly happens to the other s 1 constants of motion? The question was first addressed by Poincaré in connection with the stability of the planetary system. He succeeded in showing that there are no analytic invariants of the perturbed system, i.e. that it is not possible, starting from a constant Φ 0 of the unperturbed system, to construct quantities Φ = Φ 0 (J) + µφ 1 (J, θ) + µ 2 Φ 2 (J, θ), (1.54) where the Φ n s are analytic functions of J, θ, such that {Φ, H} = 0 (1.55) 9

17 1 Background: Hamiltonian mechanics holds, i.e. Φ is a constant of motion of the perturbed system. The proof of Poincaré s theorem is quite general. The only requirement on the unperturbed Hamiltonian is that it should have functionally independent frequencies ω j = H 0 / J j. Although the proof itself is lengthy and I will make no attempt to reproduce it, it is fairly straightforward to see where the problem with analytic invariants lies. To second order in µ, the requirement (1.55) implies {Φ 0 + µφ 1 + µ 2 Φ 2, H 0 + µh 1 } = 0 {Φ 0, H 0 } + µ ({Φ 1, H 0 } + {Φ 0, H 1 }) + µ 2 ({Φ 2, H 0 } + {Φ 1, H 1 }) = 0. The coefficients of all powers must vanish. Note that the zeroth order term vanishes by definition. The higher order terms will do so, provided {Φ 1, H 0 } = {Φ 0, H 1 } (1.56) {Φ 2, H 0 } = {Φ 1, H 1 }. The process can be continued iteratively to all orders, by requiring {Φ n, H 0 } = {Φ n+1, H 1 }. (1.57) Consider the lowest-order term generated by (1.57). Writing down the Poisson brackets gives s ( Φ1 H 0 Φ ) 1 H 0 s ( Φ0 H 1 = Φ ) 0 H 1. (1.58) θ i J i J i θ i θ i J i J i θ i j=1 The second term on the left hand side and the first term on the right-hand side vanish because the θ s are cyclic coordinates in the unperturbed system. The rest can be rewritten as s j=1 ω i (J) Φ 1 θ i = j=1 s j=1 Φ 0 J i H 1 θ i. (1.59) For notational simplicity, let me now restrict myself to the case of two degrees of freedom. The perturbed Hamiltonian can be written in a double Fourier series H 1 = n 1,n 2 A n1,n 2 (J 1, J 2 ) cos(n 1 θ 1 + n 2 θ 2 ). (1.60) Similarly, one can make a double Fourier series ansatz for Φ 1, Φ 1 = n 1,n 2 B n1,n 2 (J 1, J 2 ) cos(n 1 θ 1 + n 2 θ 2 ). (1.61) Now apply (1.59) to the case Φ 0 (J) = J 1. Using the double Fourier series I obtain B (J n 1) 1 n 1,n 2 = A n1,n n 1 ω 1 + n 2 ω 2, (1.62) 2 which in principle determines the first-order term in the µ expansion of the constant of motion J 1 which should replace J 1 in the new system. It is straightforward to show, using the same process for J 2, that the perturbed Hamiltonian can be written in terms of the new constants J 1 as H = H 0 (J 1, J 2) + O(µ 2 ). (1.63) Unfortunately, what looks like the beginning of a systematic expansion suffers from a fatal flaw. If the frequencies are functionally independent, the denominator in (1.62) will in general vanish on a denumerably infinite number of surfaces in phase space. This however means that Φ 1 cannot be an analytic function of J 1, J 2. Analytic invariants are not possible. All integrals of motion - other than the energy - are irrevocably destroyed by the perturbation. 10

18 2 Background: Statistical mechanics 2.1 Scope Classical statistical mechanics attempts to establish a systematic connection between microscopic theory which governs the dynamical motion of individual entities (atoms, molecules, local magnetic moments on a lattice) and the macroscopically observed behavior of matter. Microscopic motion is described - depending on the particular scale of the problem - either by classical or quantum mechanics. The rules of macroscopically observed behavior under conditions of thermal equilibrium have been codified in the study of thermodynamics. Thermodynamics will tell you which processes are macroscopically allowed, and can establish relationships between material properties. In principle, it can reduce everything - everything which can be observed under varying control parameters ( temperature, pressure or other external fields) to the equation of state which describes one of the relevant macroscopic observables as a function of the control parameters. Deriving the form of the equation of state is beyond thermodynamics. It needs a link to microscopic theory - i.e. to the underlying mechanics of the individual particles. This link is provided by equilibrium statistical mechanics. A more general theory of non-equilibrium statistical mechanics is necessary to establish a link between non-equilibrium macroscopic behavior (e.g. a steady state flow) and microscopic dynamics. Here I will only deal with equilibrium statistical mechanics. 2.2 Formulation A statistical description always involves some kind of averaging. Statistical mechanics is about systematically averaging over hopefully nonessential details. What are these details and how can we show that they are nonessential? In order to decide this you have to look first at a system in full detail and then decide what to throw out - and how to go about it consistently Phase space An Hamiltonian system with s degrees of freedom is fully described at any given time if we know all coordinates and momenta, i.e. a total of 2s quantities (=6N if we are dealing with point particles moving in three-dimensional space). The microscopic state of the system can be viewed as a point, a vector in 2s dimensional space. The dynamical evolution of the system in time can be viewed as a motion of this point in the 2s dimensional space (phase space). I will use the shorthand notation Γ (q i, p i, i = 1, s) to denote a point in phase space. More precisely, Γ(t) will denote a trajectory in phase space with the initial condition Γ(t 0 ) = Γ Note that trajectories in phase space do not cross. A history of a Hamiltonian system is determined by differential equations which are first-order in time, and is therefore reversible - and hence unique. 11

19 2.2.2 Liouville s theorem 2 Background: Statistical mechanics Consider an element of volume dσ 0 in phase space; the set of trajectories starting at time t 0 at some point Γ 0 dσ 0 lead, at time t to points Γ dσ. Liouville s theorem asserts that dσ = dσ 0. (invariance of phase space volume). The proof consists of showing that the Jacobi determinant (q, p) D(t, t 0 ) (q 0, p 0 (2.1) ) corresponding to the coordinate transformation (q 0, p 0 ) (q, p), is equal to unity. Using general properties of Jacobians (q, p) (q, p) (q 0, p 0 = ) (q 0, p) (q 0, p) (q 0, p 0 ) = (q) (p) (q 0 ) p=const (p 0 ) (2.2) q=const and D(t, t 0 t = t=t0 s i=1 ( qi + ṗ ) i = q i p i t=t0 s i=1 ( 2 H q i p i ) 2 H = 0, (2.3) p i q i and noting that D(t 0, t 0 ) = 1, it follows that D(t, t 0 ) = 1 at all times Averaging over time Consider a function A(Γ) of all coordinates and momenta. If you want to compute its longtime average under conditions of thermal equilibrium, you need to follow the state of the system over a long time, record it, evaluate the function A at each instant of time, and take a suitable average. Following the trajectory of the point in phase space allows us to define a long-time average 1 T Ā = lim dta[γ(t)]. (2.4) T T 0 Since the system is followed over infinite time this can then be regarded as a true equilibrium average. More on this later Ensemble averaging On the other hand, we could consider an ensemble of identically prepared systems and attempt a series of observations. One system could be in the state Γ 1, another in the state Γ 2. Then perhaps we could determine the distribution of states ρ(γ), i.e. the probability ρ(γ)δγ, that the state vector is in the neighborhood (Γ, Γ + δγ). The average of A in this case would be < A >= dγρ(γ)a(γ) (2.5) Note that since ρ is a probability distribution, its integral over all phase space should be normalized to unity: dγρ(γ) = 1 (2.6) A distribution in phase space must obey further restrictions. Liouville s theorem states that if we view the dynamics of a Hamiltonian system as a flow in phase space, elements of volume are invariant - in other words the fluid is incompressible: d dt ρ(γ, t) = {ρ, H} + ρ(γ, t) = 0. (2.7) t 12

20 2 Background: Statistical mechanics For a stationary distribution ρ(γ) - as one expects to obtain for a system at equilibrium - {ρ, H} = 0, (2.8) i.e. ρ can only depend on the energy 2. This is a very severe restriction on the forms of allowed distribution functions in phase space. Nonetheless it still allows for any functional dependence on the energy. A possible choice (Boltzmann) is to assume that any point on the phase space hypersurface defined by H(Γ) = E may occur with equal probability. This corresponds to ρ(γ) = 1 δ {H(Γ) E} (2.9) Ω(E) where Ω(E) = dγ δ {H(Γ) E} (2.10) is the volume of the hypersurface H(Γ) = E. This is the microcanonical ensemble. Other choices are possible - e.g. the canonical (Gibbs) ensemble defined as ρ(γ) = 1 Z(β) e βh(γ) (2.11) where the control parameter β can be identified with the inverse temperature and Z(β) = dγe βh(γ) (2.12) is the classical partition function Equivalence of ensembles The choice of ensemble, although it may appear arbitrary, is meant to reflect the actual experimental situation. For example, the Gibbs ensemble may be derived - in the sense that it can be shown to correspond to a small (but still macroscopic) system in contact with a much larger reservoir of energy - which in effect holds the smaller system at a fixed temperature T = 1/β. Ensembles must - and to some extent can - be shown to be equivalent, in the sense that the averages computed using two different ensembles coincide if the control parameters are appropriately chosen. For example a microcanonical average of a function A(Γ) over the energy surface H(Γ) = ɛ will be equal with the canonical average at a certain temperature T if we choose ɛ to be equal to the canonical average of the energy at that temperature, i.e. < A(Γ) > micro ɛ =< A(Γ) > canon T if ɛ =< H(Γ) > canon T. If ensembles can be shown to be equivalent to each other in this sense, we do not need to perform the actual experiment of waiting and observing the realization of a large number of identical systems as postulated in the previous section. We can simply use the most convenient ensemble for the problem at hand as a theoretical tool for calculating averages. In general one uses the canonical ensemble, which is designed for computing average quantities as functions of temperature Ergodicity The usage of ensemble averages - and therefore of the whole edifice of classical statistical mechanics - rests on the implicit assumption that they somehow coincide with the more physical time averages. Since the various ensembles can be shown to be equivalent (cf. 2 or - in principle - on other conserved quantities; in dealing with large systems it may well be necessary to account for other macroscopically conserved quantities in defining a proper distribution function. 13

21 2 Background: Statistical mechanics above), it would be sufficient to provide a microscopic foundation for the ensemble most directly accessible to Hamiltonian dynamics, i.e. the microcanonical ensemble. The ergodic hypothesis states that 1 T lim dta [Γ(t)] = 1 T T 0 Ω(E) dγ δ {H (Γ) E} A(Γ) (2.13) i.e. that time averages and microcanonical averages coincide. This requires that as a point Γ moves around phase space, it spends - on the average - equal times on equal areas of the energy hypersurface (recall that the phase point must stay on the energy hypersurface because H(Γ) is a constant of the motion. This seems like a strong & rather nonobvious assertion; Boltzmann had a rough time when he tried to sell it as a plausible basis for the emerging theory of statistical mechanics. One of the reasons why (2.13) appears implausible was a theorem proved by Poincaré which stated that if a Hamiltonian system is bounded, its trajectory in phase space - although not allowed to cross itself - will return arbitrarily close to any point already traveled, provided one waits long enough. Therefore, even statistically improbable microstates may recur. The catch is that Poincaré recurrence times for rare events in large systems are of order e N and may easily exceed the age of the universe[1]. In fact, ergodicity was later shown by Birkhoff to hold if the energy surface cannot be divided in two invariant regions of nonzero measure (i.e. regions such that the trajectories in phase space always remain in one of them). The energy surface is then called metrically indecomposable. One way this decomposition could occur might be if further integrals of motion are present. 14

22 3 The FPU paradox 3.1 The harmonic crystal: dynamics Consider a chain of N point particles, each of unit mass. Each of the particles is coupled to its nearest neighbor via a harmonic spring of unit strength; let Q i be the displacement of the ith particle; the Hamiltonian (1.6) is H(P, Q) = 1 2 N Pi i=1 N (Q i+1 Q i ) 2, (3.1) where the canonical momenta are P i = Q i and the end particles are held fixed, i.e. Q 0 = Q N+1 = 0 (NB: N degrees of freedom). The Fourier decomposition Q i = P i = 2 N N + 1 λ=1 2 N N + 1 λ=1 i=0 ( ) iπλ sin A λ N + 1 ( ) iπλ sin B λ (3.2) N + 1 is a canonical transformation (cf. above) to a new set of coordinate and momenta {A λ, B λ }. (NB: exercise, check properties, orthogonality, trigonometric sums, boundary conditions satisfied). In this new set of coordinates, the Hamiltonian can be written as where H = N H λ 1 2 λ=1 N ( B 2 λ + Ω 2 λa 2 λ) λ=1 { } πλ Ω 2 λ = 4 sin 2 2(N + 1) (3.3). (3.4) This is a case of a separable Hamiltonian, where Hamilton-Jacobi theory can be trivially applied, i.e. ( ) 2 1 Wλ A λ 2 Ω2 λa 2 λ = ɛ λ λ = 1,, N. (3.5) where each ɛ λ is a constant representing the energy stored in the λth normal mode. The substitution 2ɛλ A λ = sin Ω θ λ (3.6) λ transforms (3.5) to The corresponding action variable J λ = 1 2π W λ θ λ = 2ɛ λ Ω λ cos 2 θλ. (3.7) B λ da λ = 1 Wλ da λ (3.8) 2π A λ 15

23 3 The FPU paradox can now be evaluated as J λ = 1 2ɛ λ 2π Ω λ 2π 0 d θ λ cos 2 θλ = ɛ λ Ω λ (3.9) by integrating over a full cycle of the substitution variable θ λ. The Hamiltonian can be rewritten in terms of the action variables H = λ ɛ λ = λ Ω λ J λ (3.10) The angle variables conjugate to the action variables can be found from (1.30 It can be shown explicitly that θ j = θ j. The Hamiltonian equations in action-angle variables are θ λ = W λ(a λ, J λ ) J λ. (3.11) J λ = 0 θ λ = H J λ = Ω λ, (3.12) i.e. the Ω λ s are the natural frequencies of the normal modes. Note that we did not need the explicit form of the solution of the Hamilton-Jacobi equation to derive this. More explicitly, the time evolution of the normal mode coordinates is ( ) 1/2 2Jλ A λ (t) = sin ( Ω λ t + θλ 0 ) Ω λ, (3.13) with an analogous expression for the momenta B λ. In the action-angle representation, the 2N constants of integration are the N action variables {J λ } and the N initial phases {θ 0 λ }. 3.2 The harmonic crystal: thermodynamics The average energy of the harmonic chain at any given temperature T is given by the canonical average < H >= 1 dγe H(Γ)/T H(Γ), (3.14) Z where Z is the partition function Z(T ) = dγe H(Γ)/T. (3.15) It is possible to transform the integrals in both numerator and denominator of (3.14) to action-angle coordinates (cf. previous section). Because of the separability property of the Hamiltonian, the denominator splits into product over all N normal modes Z = N Z λ (3.16) λ=1 16

24 3 The FPU paradox where Z λ = 0 2π dj λ dθ λ e Ω λj λ /T 0 = 2πT Ω λ (3.17) whereas the numerator transforms to is a sum of the form N Z λ N λ (3.18) where It follows that < H >= N λ = λ=1 0 λ λ 2π dj λ dθ λ e Ω λj λ /T Ω λ J λ 0 = 2πT 2 Ω λ. (3.19) N < ɛ λ >= λ=1 N N λ /Z λ = λ=1 N T = NT, (3.20) i.e. each the average energy which corresponds to each degree of freedom is equal to T (equipartition property). The statistical mechanics of the harmonic chain has a fundamental flaw: although canonical averages are straightforward to obtain, there is obviously no basis for assuming ergodicity - in the presence of N integrals of motion. Now, this might not be a serious problem if one could argue that a tiny generic perturbation, as might arise from e.g. a small nonlinearity of the interactions, could drive the system away from complete integrability, and into an ergodic regime. If this turned out to be the case, one could still argue that the computed canonical averages reflect the intrinsic thermodynamic properties of the harmonic chain, in the programmatic sense of statistical mechanics. Fermi, Pasta and Ulam decided to put this implicit assumption to a numerical test. λ=1 3.3 The FPU numerical experiment Fermi, Pasta and Ulam (FPU[2]) investigated the Hamiltonian H(P, Q) = 1 2 N 1 i=1 P 2 i N 1 i=0 (Q i+1 Q i ) 2 + α 3 N 1 i=0 (Q i+1 Q i ) 3, (3.21) where the canonical momenta are P i = Q i and the end particles are held fixed, i.e. Q 0 = Q N = 0. Their work - undertaken as a suitable test problem for one of the very first electronic computers, the Los Alamos MANIAC - is considered as the first numerical experiment. In other words, it is the first case where physicists observed and analyzed the numerical output of Newton s equations, rather than the properties of a mechanical system described by these same equations. The dynamics of the Hamiltonian (3.21) was studied as an initial value problem; the initial configuration was a half-sine wave Q i = sin(iπ/n), with N = 32 and all particles at rest; the nonlinearity parameter was chosen as α = 1/4. Energy was thus pumped at the lowest 17

25 3 The FPU paradox Figure 3.1: The quantity plotted is the energy (kinetic plus potential in each of the first four modes). The time is given in thousands of computational cycles. Each cycle is 1/2 2 of the natural time unit. The initial form of the string was a single sine wave (mode 1). The energy of the higher modes never exceeded 6% of the total. (from [2]). Fourier mode, λ = 1, in the notation of (). The objective of the experiment was to study the energies stored in the first few Fourier modes, i.e. the quantities where A λ = H λ 1 2 (Ȧ2 λ + Ω 2 λa 2 λ 2 N i=1 ) (3.22) N ( ) iπλ sin Q i (3.23) N as a function of time, i.e. to test the onset of equipartition. Note that the decomposition of the total energy in Fourier modes is not exact - but as long as α stays small, H λ H λ will hold. Fig. 3.1 shows the time dependence of the energies of the first four modes. After an initial redistribution, all of the energy (within 3%) returns to the lowest mode. The energy residing in higher modes never exceeded 6 % of the total. Longer numerical studies have shown the return of the energy to the initial mode to be a periodic phenomenon; the period is about 157 times the period of the lowest mode. The phenomenon is known as FPU recurrence. The results of a more recent numerical study on FPU recurrence[3] are summarized in Fig The Hamiltonian (3.21) is fairly generic. In fact, the original FPU paper describes a further study with quartic, rather than cubic, anharmonicities which exhibits similar behavior. FPU recurrence has been shown to be a robust phenomenon. The upshot of those exhaustive numerical observations is that anharmonic corrections to the Hamiltonian, contrary to the original expectation which held them as agents that might help establish ergodicity, actually appear to generate new forms of approximately periodic behavior. The process of under- 18

26 3 The FPU paradox Figure 3.2: FPU recurrence time, divided by N 3 vs a scaling variable R = α(e/n) 1/2 N 2 where E/N [πb/(2n)] 2 is the energy density. Typical values used by FPU correspond to R 1. The asymptotic regime is well described by the relationship T r /N 3 = R 1/2 (from Ref. [3]). standing the source of this behavior - also known as the FPU paradox - and relating it to other manifestations of nonlinearity [4] has led to a profound change in theoretical physics. 19

27 4 The Korteweg - de Vries equation 4.1 Shallow water waves Original context: Wave motion in shallow channels, cf. Scott-Russell 1 Mathematical description due to Korteweg and devries (KdV [6]). The equation arises in wide variety of physical contexts (e.g. plasma physics, anharmonic lattice theory). Hence it counts as one of the canonical soliton equations. Long waves (typical length l) in a shallow channel l h. Small amplitude ( h) waves (weak nonlinearity) Two-dimensional fluid flow (motion in lateral dimension of channel neglected) x: horizontal direction, y: vertical direction Background: hydrodynamics Fluid velocity V uˆx + vŷ (4.1) Equations of (Eulerian) incompressible fluid dynamics continuity equation Euler equation where g = gŷ plus V = 0 (4.2) V t + ( V ) V = 1 p + g (4.3) ρ irrotational flow (no vortices) V = 0 V = Φ. (4.4) Using vector identity ( V ) V = 1 2 V 2 V ( V ) (4.5) in (4.3) (only first term survives due to (4.4) ), and (4.4) in (4.2) transforms hydrodynamics equations to 1 I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of translation. [5] 20

28 4 The Korteweg - de Vries equation 1. continuity 2. Euler Φ = 0, (4.6) Φ t ( Φ)2 + p + gy = 0. (4.7) ρ Statement of the problem; boundary conditions The above eqs (4.6) and (4.7) must now be solved subject to the boundary conditions 1. bottom: no vertical motion of the fluid v(x, y = 0) = 0 x (4.8) 2. top: free surface defined as y = h + η(x, t). (4.9) Velocity of free boundary coincides with fluid velocity, holds at the free surface. dy dt = η t + η dx x dt hence v = η t + η x u (4.10) The solution will involve two steps: first, find a general class of solutions of (4.6) which satisfy the bottom BC (4.8), and then use this general class to determine the height profile (4.9) by demanding that the Euler equation (4.7) be satisfied at the free surface, where p = 0 holds. The Euler equation can then be used to determine the pressure at any point Satisfying the bottom boundary condition Consider the general form of an expansion (the height O(h) is small in a sense which will be made precise below) of the type u = f(x) + f 1 (x)y + f 2 (x)y 2 + f 3 (x)y 3 + v = g 1 (x)y + g 2 (x)y 2 + g 3 (x)y 3 +. (4.11) The conditions u as and y = v x and u x = v y imposed by (4.6) can now be written, respectively, f 1 + 2f 2 y + 3f 3 y 2 = g 1x y + g 2x y 2 (4.12) f x + f 1 y + f 2 y 2 = g 1 2g 2 y 3g 3 y 2 (4.13) from which f 1 = 0 (4.14) 2f 2 = g 1x (4.15) 3f 3 = g 2x (4.16) 21

29 4 The Korteweg - de Vries equation and f x = g 1 (4.17) f 1x = 2g 2 (4.18) f 2x = 3g 3 (4.19) follow. Using the second set in the first, results in f 1 = 0, 2f 2 = f xx, 2f 3 = 1/2f 1xx (= 0); it follows that g 2 = 0 and g 3 = 1/3f 2x = 1/3!f xxx. Collecting terms, u = f 1 2 f xxy 2 + O(y 4 ) (4.20) v = f x y + 1 3! f xxxy 3. (4.21) Euler equation at top boundary Set p = 0 in (4.7) and differentiate with respect to x: u t x (u2 + v 2 ) + g η t = 0. (4.22) The problem is now to solve the system of coupled differential equations (4.22) and (4.10) using the expressions (4.20) and (4.21). Key: follow the scale of variation of the physical quantities involved. First note that if the water height is not much different from h (small nonlinearity), it will be useful to set η = ɛh η (4.23) Note ɛ is not a parameter of the problem. It simply serves as a tag to let us keep track of scales. At the end we will have to check the consistency of the assumptions and approximations made. According to our assumption, the length scale on which the fluid profile varies along the x direction is of the order l h. In order to incorporate this assumption in the approximation, I define a rescaled variable via x = l x. (4.24) Dimensional consideration determine a natural velocity scale c = gh. The motion should be slow with respect to that scale - in agreement with small amplitude variations of the profile. In other words, we expect u c. Note that from the leading orders of (4.20) and (4.21)it follows that v is typically of order h/l δ smaller than u. It is therefore reasonable to rescale Finally I use a rescaled time With these rescalings, keeping lowest order terms, i.e. equations (4.20) and (4.21) become - on the surface - f = ɛc f (4.25) u = ɛcū (4.26) v = δɛc v. (4.27) t = t l/c. (4.28) of O(ɛ) and O(δ 2 ), the rescaled ū = f 1 2 δ2 f x x (4.29) v = (1 + ɛ η) f x δ2 f x x x ; (4.30) 22

30 4 The Korteweg - de Vries equation accordingly, the top boundary condition (4.10) and the Euler equation (4.22) transform to f x + η t + ɛ( f η) x 1 6 δ2 f x x x = 0 (4.31) f t + η x + ɛ 2 ( f 2 ) x 1 2 δ2 f x x t = 0. (4.32) First we note that in the absence of nonlinearity (ɛ = 0) and dispersion (δ = 0), free wave propagation with unit velocity (in dimensionless units) occurs; in that (zeroth) order, f = η. But of course this is hypothetical because δ and ɛ are not parameters of the problem - they just help us keep track of things! However, the zeroth order approximation is useful in the sense that it suggests a coordinate transformation which absorbs the fastest time dependence; let Keeping terms to first order in ɛ and δ 2, we use the property (which holds for f as well) transform the system (4.32) to ξ = x t (4.33) τ = ɛ t. (4.34) η x = η ξ (4.35) η t = η ξ + η τ ɛ (4.36) f ξ η ξ + ɛ η τ + ɛ( η 2 ) ξ 1 6 δ2 η ξξξ = 0 (4.37) f ξ + η ξ + ɛ η τ + ɛ 2 ( η2 ) ξ δ2 η ξξξ = 0. (4.38) where we have used the property f = η in terms which contain ɛ or δ 2 factors. The sum of (4.38) is 2ɛ η τ ɛ( η2 ) ξ δ2 η ξξξ = 0. (4.39) The three terms in (4.39) will be of the same order if δ 2 = O(ɛ), i.e. if the nonlinearity balances the dispersion. We choose ɛ = δ 2 /6. Note that the choice must be tested at the end to check whether it satisfies the original requirements (small amplitude, long waves). With this choice and the substitution η = 4φ I arrive at the canonical KdV form, φ τ + 6φφ ξ + φ ξξξ = 0. (4.40) A solitary wave At this stage, without recourse to advanced mathematical techniques, it is possible to follow the path of KdV and look for special, exact, propagating solutions of (4.40) of the type φ(s), where s = ξ λτ. (4.40) becomes which has an obvious first integral λφ s + 3(φ 2 ) s + φ sss = 0 (4.41) λφ + 3φ 2 + φ ss = const. (4.42) If we are looking for solutions which vanish at infinity (lim s φ(s) = 0 and lim s φ s (s) = 0) the constant will be zero, i.e. φ ss = λφ 3φ 2 = d dφ (1 2 λφ2 φ 3 ) (4.43) 23

31 4 The Korteweg - de Vries equation Multiplying both sides by 2φ s we can integrate once more, obtaining φ 2 s = λφ 2 2φ 3 (4.44) where the integration constant must vanish once again (cf. above). Note that, if a solution exists, the parameter λ must be > 0 and φ < λ/2. Taking the square root of (4.44) and inverting the fractions I obtain dφ ds = ± φ (4.45) λ 2φ which can be integrated directly, resulting in φ(s) = [ λ 2 cosh 2 ] (4.46) λ 2 (s s 0) where s 0 is an arbitrary constant. (The plus sign in (4.45) has been chosen for s < s 0 and the minus for s > s 0 ). Note that the properties of the propagating solution (4.46) - except for its initial position, which is determined by s 0 - are all governed by a single parameter. If the velocity λ is given, the amplitude is fixed at λ/2 and the spatial extent at 2λ 1/2. In other words - in the canonical units of (4.40) - a slow pulse will also have a small amplitude and a large spatial extent Is the solitary wave a physical solution? Eq. (4.46 ) is an exact, propagating, pulse-like solution of (4.40). But is it an acceptable solution of the original problem? In other words, is the surface profile of low amplitude and is it a long wave? To do this, we have to go back to the original variables, and convince ourselves that (4.46) generates (some) acceptable solutions for the original problem (Exercise) 4.2 KdV as a limiting case of anharmonic lattice dynamics Consider the 1-d anharmonic chain; atomic displacements are denoted by {u n }; neighboring atoms of mass m interact via anharmonic potentials of the type V (r) = 1 2 kr kbr3 (4.47) where r is the distance between nearest neighbors. The equations of motion are m q n = q n [V (q n+1 q n ) + V (q n q n 1 ]) (4.48) = k(q n+1 + q n 1 2q n ) kb[ (q n+1 q n ) 2 + (q n q n 1 ) 2 ] = k(q n+1 + q n 1 2q n ) kb(q n+1 + q n 1 2q n )(q n+1 q n 1 ). If the displacements do not vary appreciably on the scale of the lattice constant a, we can use a continuum approximation; keeping terms of fourth order in the lattice constant, m q q tt = ka 2 q xx + ka 4 2 4! q xxxx + kba 2 q xx 2aq x, where x = na is the continuum space variable; defining c 2 = ka 2 /m, this can be written as 1 c 2 q tt q xx = 1 12 a2 q xxxx + 2αq x q xx, (4.49) 24

32 4 The Korteweg - de Vries equation where α = ab provides a dimensionless measure of the anharmonicity. I now look for solutions which vary smoothly in space, i.e. over a typical length of many lattice spacings, and where the main time dependence is contained in the wave equation part, i.e. of the form q(ξ, τ) q(ɛ x ct, δω 0 t), (4.50) a where ω 0 = c/a = k/m, ɛ 1 and δ ɛ; the exact dependence of δ on ɛ will be fixed later. The relevant derivatives transform according to q x = ɛ a q ξ q xx = q xxx = ( ɛ a ( ɛ a ) 2 qξξ ) 3 qξξξ q tt = ω0 2 ( ɛ 2 q ξξ 2ɛδq ξτ + O(δ 2 ) ). Using them in (4.49) gives 2δq ξτ ɛ3 q ξξξξ + 2αq ξ q ξξ = 0, (4.51) which, after a rescaling q ξ (= a ɛ q x) = ɛ aφ (4.52) 4α and setting 2 δ = 1 24 ɛ3 can be reduced to the canonical KdV form φ τ 6φφ ξ + φ ξξξ = 0. (4.53) Note that the rescaling of length, i.e. the value of the small parameter ɛ is still a matter of free choice, depending on the (initial) conditions of the problem. The above analysis shows that one may legitimately suspect that nonlinear propagating solitary waves will be generic in anharmonic lattices, at least for certain parameter ranges. Again, one has to make sure that the solutions found from solving the KdV equation (4.53) are appropriate for the original problem (4.49) (check consistency of approximations made). 4.3 KdV as a field theory KdV Lagrangian The KdV equation u t 3(u 2 x) x + u xxx = 0 (4.54) can be derived from the Lagrangian L = dxl(φ, φ t, φ x, φ xx ) (4.55) 2 note that this guarantees δ ɛ as demanded above. 25

33 4 The Korteweg - de Vries equation where L = 1 2 φ xφ t φ 3 x 1 2 φ2 xx. (4.56) Note that because the Lagrangian density depends on the second derivative of the field, (1.42) contain an extra term ( ) d2 L dx 2. (4.57) φ xx Minimization of the action leads to the field equations of motion which reduces to (4.54) upon the substitution φ xt 3(φ 2 x) x + φ xxxx = 0 (4.58) φ x = u. (4.59) Continuous symmetries of the Lagrangian will again give rise to an equation like (1.44), with an extra term L φ xx δφ xx (4.60) on the left-hand side. The above modifications generate an extra contribution to the left-hand side of (1.45). ( ) L δφ xx d2 L φ xx dx 2 δφ (4.61) φ xx Symmetries and conserved quantities For some infinitesimal transformations (cf. section ) one can verify explicitly that δφ xx = d 2 δφ/dx 2. If this is the case, the integral over all space of the extra contribution (4.61) can easily be seen to vanish (repeated integration by parts of either of the two terms). In this case, the standard symmetries are reflected in the same standard conservation (with the same densities of conserved quantities), as in section.... Translational invariance in space Conservation of the total momentum P = dx L φ t φ x = 1 2 dx φ 2 x = 1 2 dx u 2. (4.62) Translational invariance in time Conservation of the total energy H = dx ( ) L φ t L φ t = = ( ) 1 dx 2 φ2 xx + φ 3 x ( ) 1 dx 2 u2 x + u 3. (4.63) 26

34 4 The Korteweg - de Vries equation Conservation of mass The symmetry φ φ + ɛ generates δφ = ɛ, and all other variations are zero. From (1.45), conservation of M = dx L = 1 dx φ x = 1 dx u, (4.64) φ t 2 2 the total mass, is deduced. Galilei invariance The transformation x x ɛt, φ(x, t) φ(x ɛt) ɛx (or in terms of the u-field, u(x, t) u ɛ, generates (cf. section...) x x ɛt δφ = φ(x ɛt) φ(x) ɛx = ɛtφ x ɛx δφ t = φ t (x ɛt) φ t (x) = ɛtφ xt δφ x = φ x (x ɛt) φ x (x) ɛ = ɛtφ xx ɛ δl = dl δx = dl dx dx ɛt J 1 = tl, J 0 = 0. (4.65) Owing to δφ xx = (δφ) xx there are no extra terms in the conserved currents. Eq. (1.45) applies. Since δφ x = (δφ) x the two last terms in the left-hand side of (1.45) combine to form a total space derivative; similarly, because of δφ t = (δφ) t, the first two terms combine to form a total time derivative, i.e. the conserved density is L φ t δφ/ɛ = 1 2 φ x( tφ x x), (4.66) or, integrating over all space, and dividing by the total mass M, X = 1 M dx x u 2 = P M t + const. (4.67) which expresses the fact that the center of mass moves at a constant velocity KdV as a Hamiltonian field theory 27

35 5 Solving KdV by inverse scattering 5.1 Isospectral property Given the KdV equation u t 6uu x + u xxx = 0 (5.1) and a well behaved initial condition u(x, 0), which vanishes at infinity, it is possible to determine the time evolution u(x, t) in terms of a general scheme, which is known as inverse scattering theory. The scheme is based on the following particular property of (5.1): Given the linear operator L(t) = 2 xx + u(x, t) (5.2) whose parametric time dependence is governed by (5.1), and the associated eigenvalue equation L(t)ψ j (x, t) = λ j (t)ψ j (x, t), (5.3) it can be shown that 5.2 Lax pairs dλ j dt = 0. (5.4) The isospectral property can be formulated somewhat more generally: Suppose we can construct a linear, self-adjoint operator B = B, dependent on u and such that holds as an operator identity, i.e. il t i dl dt i lim L(t + ) L(t) = [L, B] (5.5) 0 il t f = [L, B]f f (5.1). (5.6) The operators L and B are then called a Lax pair. The time evolution of L is governed by where L(t) = U(t)L(0)U (5.7) U = e ibt. (5.8) Consider (5.3) at t = 0, and apply the operator U(t) to both sides from the left, i.e. U(t)L(0) U (t)u(t)ψ j (0) = λ j (0)U(t)ψ j (0) (5.9) where, in addition I have inserted a factor U U = 1. It can be recognized immediately that the l.h.s. of (5.9) and (5.3) are identical, provided ψ j (t) = U(t)ψ j (0), (5.10) 28

36 5 Solving KdV by inverse scattering and that, in order for the r.h.sides to coincide, I must have λ(t) = λ(0) t (5.11) (isospectral property). The form of the operator B in the KdV case is B = 4i xxx 3 3i (u x + x u) (5.12) (verify explicitly (5.6). 5.3 Inverse scattering transform: the idea The isospectral property tentatively suggests that it might possible to proceed as follows: solve the linear problem (5.3) at time t = 0, i.e. determine the eigenvalues {λ j } and the eigenfunctions {ψ j (x, 0)} from the known u(x, 0). determine the evolution of the eigenfunctions from (5.10) at a later time t. try to solve the inverse problem of determining the potential u(x, t) from the known spectra and eigenfunctions at the time t. In fact, the last step is the well known problem of inverse scattering theory in quantum mechanics, where physicists had tried to extract information on the nature of interparticle interactions from analyzing particle scattering data. The one-dimensional problem (corresponding to a spherically symmetric potentials in 3 dimensions) was completely solved in the 1950 s (Gel fand, Levitan & Marchenko). I will present the solution below, but before doing that, let me outline some broad features: Scattering data in the mathematical sense are the asymptotic properties of the solution of the associated linear problem, i.e. the properties far from the source of scattering, where the potential is effectively zero. What GLM have shown is that you can reconstruct the potential from the scattering data. Furthermore, it turns out that the operator B takes an especially simple form in the asymptotic limit, which allows us to write down an exact, analytic formula for the time evolution of scattering data. Evolution of the scattering data is the easy part of the game. But then if I only need scattering data at time t, and I know how these data evolve in time, all the input I need is the scattering data for the potential u(x, 0). This is exactly the program of the inverse scattering transform (IST). Because it is based only on the asymptotic part of the solution of the associated linear problem, it can be written down in closed form. I summarize the IST program schematically: 1. determine the scattering data S of the linear problem (5.3) at time t = 0, from the known u(x, 0). 2. determine the evolution of the scattering data S(t) at a later time t from the asymptotic from of the operator B. 3. do the inverse problem at time t, i.e. determine the potential u(x, t) from the known scattering data S(t). 5.4 The inverse scattering transform The direct problem This is just a summary of properties known from elementary quantum mechanics. 29

37 5 Solving KdV by inverse scattering Jost solutions The linear eigenvalue problem [ ] x 2 + u(x) ψ(x) = k 2 ψ(x) (5.13) has, in general, a discrete and a continuum spectrum, corresponding to imaginary and real values of k respectively. For real k there are in general two linearly independent solutions. Such a linearly independent set is provided by the Jost solutions: f 1 (x, k) e ikx x f 2 (x, k) e ikx x. (5.14) The Jost solutions of (5.13) satisfy the integral equations where f 1 (x, k) = e ikx f 2 (x, k) = e ikx + x x G(x, x ) = sin k(x x ) k dx G(x, x )f 1 (x, k) dx G(x, x )f 2 (x, k) (5.15) u(x ). (5.16) Eqs. (5.15) can be analytically continued to the upper half plane of complex k. Some information on the analytic properties can be obtained by considering the lowest iteration, where we substitute f 1 (x, k) = e ikx in the r.h.s. of the first equation. This gives f 1 (x, k) e ikx dx x) eik(x e ik(x x) u(x )e ikx x 2ik e ikx e ikx 1 dx {1 e 2ik(x x) }u(x ) (5.17) 2ik x which can be thought of as the beginning of a systematic expansion in inverse powers of k. Note that since x x > 0, the exponential will be convergent in the upper-half plane of k; therefore, if the potential vanishes sufficiently rapidly at infinity, I estimate where h vanishes as 1/k for high values of k. The property will be useful. g 1 (x, k) f 1 (x, k) e ikx e ikx h(x, k) (5.18) f 2 (x, k) = a(k)f 1 ( k, x) + b(k)f 1 (k, x). (5.19) For bound states, corresponding to k = iκ, the Jost solutions are degenerate. Asymptotic scattering data The asymptotic (scattering) data of (5.3) is defined as follows: discrete spectrum (bound states) λ n = κ 2 n n = 1,, N, (5.20) 30

38 5 Solving KdV by inverse scattering where κ n > 0; ψ n (x) = f 1 (x, k) e κ nx = C n f 2 (x, k) C n e κ nx x x. (5.21) I will also need the normalization integral of each bound state 1 α n = dxψ 2 n(x) = continuous spectrum (scattering states) dxf 2 1 (x, iκ n ) (5.22) λ(k) = k 2 < k <. (5.23) The physical scattering states corresponding to waves incident from the right, are ψ(x, k) e ikx + R(k)e ikx x T (k)e ikx x. (5.24) where R(k), T (k) are, respectively, the reflection and transmission coefficients, which satisfy R(k) 2 + T (k) 2 = 1. The Jost solutions are related to the physical solution (5.24) via ψ(x, k) = T (k)f 2 (x, k) = f 1 (x, k) + R(k)f 1 (x, k) x. (5.25) This identifies a(k) = 1/T (k) and b(k) = R(k)/T (k). The complete set of scattering data for any one dimensional potential of a Schroedinger-type equation is S [{κ n, C n, α n }, n = 1, N; T (k), R(k)]. (5.26) In fact, for the purposes of performing the inverse scattering transform I will only need the reduced set 1 S [{κ n, α n }, n = 1, N; R(k)] (5.27) Time evolution of scattering data I promised this will be the easy part. The operator B has the property lim = x ± B = 4i xxx 3. (5.28) Since t ψ j(x, t) = ibψ j (x, t) (5.29) holds for all eigenfunctions, we can apply in the asymptotic regime, where B B. In the case of a discrete eigenfunction, this gives ψ n (x) e κ nx+4κ 3 n t x C n e κ nx 4κ 3 n t x, (5.30) 1 Note that if scattering theory is to make sense, the potential must be vanishing at (±)infinity. I have not specified the minimal exact mathematical conditions which satisfy this demand. 31

39 5 Solving KdV by inverse scattering or, in keeping with the agreed normalization of the type (5.21), I multiply with a factor e 4κ3 n t, and obtain with In the case of Jost solutions I obtain ψ n (x) e κnx x C n (t)e κnx x, (5.31) C n (t) = C n (0)e 8κ3 n t. (5.32) f 1 (k, x) e ikx+4ik3 t f 2 (k, x) e ikx 4ik3 t the physical solution therefore evolves according to x ψ(k, x) e ikx 4ik3t + R(k)e ikx+4ik3 t T (k)e ikx 4ik3 t x ; (5.33) x, x or, multiplying both by a factor e 4ik3t, to keep the standard normalization of () where ψ(k, x) e ikx + R(k, t)e ikx x T (k)e ikx x, R(k, t) = R(k)e 8ik3 t. (5.34) The scattering data evolve according to (5.32) and (5.34). The transmission coefficient T (k) stays constant in time Reconstructing the potential from scattering data (inverse scattering problem) Reconstruction of the potential from scattering data is an old problem in quantum mechanics. A complete solution has been given in one dimension, subject to fairly general conditions, by Gelfand and Levitan [7] and Marchenko[8]. Reviews by Faddeyev[9] and Scott[10]. Definition of the problem: Given the eigenvalue equation ] [ d2 dx 2 + u(x) ψ(x) = k 2 ψ(x) (5.35) determine u(x) from scattering data in the form of eqs. (5.27). Fourier transforms of the g(x, k) functions ĝ j (x, y) = 1 dk e iky g j (x, k) 2π, (5.36) where j = 1, 2, with an inverse g j (x, k) = dy e iky ĝ j (x, y). (5.37) Note that, due to the analytic properties of f 1 (cf. (5.18), which allows to close the contour of (5.36) from above without finding any singularities) Similarly, ĝ 1 (x, y) = 0 if y < x. (5.38) ĝ 2 (x, y) = 0 if y > x. (5.39) 32

40 5 Solving KdV by inverse scattering Relating ĝ 1 (x, x) to the potential u(x) The starting point is to recognize that ( ) ( ) 2 2 x 2 + k2 u g 1 (x, k) = x 2 + k2 u (f 1 (x, k) 1) = u(x)e ikx ; (5.40) multiplying both sides by e iky /2π and integrating over all k, I obtain ( ) 2 x 2 2 y 2 u(x) ĝ 1 (x, y) = u(x) δ(x y) ; (5.41) defining new variables ζ = (x + y)/2, η = y x, I use x 2 y 2 = 2 η ζ, to transform (5.41) to ( 2 2 η ζ ĝ1 ζ η 2, ζ + η ) ( u ζ η ) ( ĝ 1 ζ η 2 2 2, ζ + η ) 2 = u(ζ η 2 )δ( η), which can be integrated over an interval of length ɛ around η = 0. The result is which in the limit ɛ 0 becomes where I have reverted to the original variables. 2 ζ ĝ1(ζ, ζ) ɛu(ζ)ĝ 1 (ζ, ζ) = u(ζ), (5.42) u(x) = 2 d dxĝ1(x, x) (5.43) Relating g 1 (x, y) to the scattering data Define Fourier transforms Rewrite (5.25) as ˆT (y) = ˆR(y) = dk 2π e iky [T (k) 1] (5.44) dk 2π eiky R(k). (5.45) T (k)f 2 (x, k) = f 1 (x, k) + R(k) [ f 1 (x, k) e ikx + e ikx] ; (5.46) adding f 2 (x, k) to both sides and adding and subtracting e ikx to the right hand side gives (T (k) 1) f 2 (x, k) = g 1 (x, k) g 2 (x, k) + R(k)g 1 (x, k) + R(k)e ikx ; (5.47) multiplying both sides by e iky /2π and integrating over all k produces dk 2π eiky [T (k) 1] f 2 (x, k) = ĝ 1 (x, y) ĝ 2 (x, y) + Lemma: T (k) has only simple poles dy ˆR(y + y ) ĝ 1 (x, y ) + ˆR(x + y) (5.48) 33

41 5 Solving KdV by inverse scattering It can be shown that the transmission coefficient T (k) is analytic in the upper half plane, including the real axis, except for simple poles which correspond to the bound states, i.e. at k = iκ n. In the neighborhood of such a pole T (k) i C nα n k iκ n, (5.49) where 1 = dx[f 1 (x, iκ n )] 2 (5.50) α n is the normalization integral of the bound state. Using the above lemma, it is possible to compute the integral in the left-hand side of (5.48) by closing the contour from above. The contribution from each bound state is 2πie κny i 1 2π C nα n f 2 (x, iκ n ) = α n e κny f 1 (x, iκ n ) = α n e κ ny g 1 (x, iκ n ) α n e κ n(x+y) where I have used the fact that Jost states are degenerate if k = κ n (cf. Eq. (5.21)). Moreover, I only need (5.48) for x y, since otherwise ĝ 1 (x, y) = 0. Defining a combined kernel which incorporates all scattering data, I finally obtain ĝ 1 (x, y) + ˆK(x + y) + ˆK(z) = ˆR(z) + (Gel fand, Levitan, Marchenko equation) IST summary N α n e κnz, (5.51) n=1 dy ˆK(y + y ) ĝ 1 (x, y ) = 0 if x y (5.52) ĝ 1 (x, y) = 0 if x > y (5.53) In order to solve the initial value problem of the KdV equation (5.1) we proceed as follows: Extract initial scattering data for the associated linear problem (5.13), from potential u(x) at time 0. Define the scattering kernel at time t: ˆK(z; t) = {κ n, α n ; n = 1, N}, {R(k), < k < } (5.54) dk 3t 2π eikz+8ik R(k) + N α n e κ nz+8κ 3 n t. (5.55) n=1 Solve the Gel fand, Levitan, Marchenko equation for x y, ĝ 1 (x, y; t) + ˆK(x + y; t) + x dy ˆK(y + y ; t) ĝ 1 (x, y ; t) = 0. (5.56) Extract the limit u(x, t) = 2 d dxĝ1(x, x + ; t). (5.57) Note that (i) I have explicitly included the parametric dependence on time, and (ii) the normalization integral α n has a time dependence such that the product C n α n stays constant (cf. time-independence of the transmission coefficient). 34

42 5 Solving KdV by inverse scattering 5.5 Application of the IST: reflectionless potentials Suppose the scattering kernel (5.51) contains only bound states, i.e. R(k) = 0. This would correspond to a reflectionless potential in the original quantum mechanical context. In this case it turns out that I can systematically derive a whole class of solutions to the original KdV equation by just solving the GLM equation (5.56) and taking the appropriate limit (5.57) A single bound state The scattering kernel has the form ˆK(z; t) = αe κz+8κ3 t. (5.58) I will look for solutions of (5.56) which are separable, i.e. with the above Ansatz, (5.56) tranforms to [ h(x, t) + αe 8κ3 t e κx + h(x, t) ĝ 1 (x, y; t) = e κy h(x, t) ; (5.59) x dy e 2ky ] = 0 ; inserting the expression for the integral, (2κ) 1 e 2κx, and setting α = 2κe 2δ, I obtain [ ] h(x, t) 1 + e 2δ e 2κx+8κ3 t = αe κx+8κ3 t t+2δ ĝ 1 (x, y; t) = 2κ e κ(x+y)+8κ3 1 + e [2κx 8κ3 t 2δ] the limiting form for y = x is it follows that 1 ĝ 1 (x, x; t) = 2κ 1 + e 2[κx 4κ3 t δ] u(x; t) = 2 d 2κ 2 dxĝ1(x, x; t) = cosh 2 [κ(x 4κ 2 t) δ] which is identical to the solitary wave (4.46), if we identify λ in (4.46) with 4κ 2 in (5.60). Comments: ; ; (5.60) the velocity of the wave corresponds (to within a factor of -4) to the eigenvalue of the associated problem. The velocity coincides with the ratio P/M (cf. symmetries conservation laws; show this (exercise)). Note that I made no attempt to guess the form of the wave. The form was imposed by the separation ansatz (5.59), i.e. it is built-in in the association of the KdV equation with the linear eigenvalue problem. This will be useful in the next subsection, where I will try to construct solutions that correspond to multiple bound states. It is of course possible to treat the proper initial value problem. Starting with any localized potential which may support a bound state, one can perform the IST steps. Exercise: do this (i) for an attractive delta function potential µδ(x), and (ii) for a potential of the type N(N + 1)sech 2 x, where N is an integer; (hint: in case (ii) the form of the solution is an example of the case discussed in the next subsection). 35

43 5 Solving KdV by inverse scattering Multiple bound states Again, I will restrict myself to the case where the reflection coefficient vanishes. The scattering kernel has the form N ˆK(z; t) = α i (t) e κ iz. (5.61) i=1 where α i (t) = α i e 8κ3 i t carries the time dependence. A generalized form of the separation ansatz (5.59) N ĝ 1 (x, y; t) = e κiy h i (x, t) (5.62) transforms the GLM equation (5.56) to i=1 N e κiy { h i (x, t) + α i (t)e κix + i=1 N j=1 α i (t) κ i + κ j e (κ i+κ j )x h j (x, t)} = 0, which must hold for all y > x; hence, in nonsymmetric matrix form, N A ij h j (x, t) = C i (x, t) (5.63) j=1 where and A ij (x, t) = δ ij + α i(t) κ i + κ j e (κi+κj)x (5.64) C j (x, t) = α i (t)e κix. (5.65) Thus h j = 1 det A det A 11 A 12 C 1 A 1 j+1 A 21 A 22 C 2 A 2 j+1 C 3. (5.66) where the jth column in the matrix A has been substituted by the vector C; it follows that A 11 A 12 C 1 e κjx A 1 j+1 ĝ 1 (x, x) = 1 A 21 A 22 C 2 e N κ jx A 2 j+1 det C 3 e κ jx det A j=1 ; (5.67) note however that, since da ij dx = α ie (κ i+κ j )x = C i e κ jx, this is equivalent to ĝ 1 (x, x) = d ln det A. (5.68) dx 36

44 5 Solving KdV by inverse scattering At this stage it is convenient to introduce the symmetrized form of the matrix A, obtained by  = DAD 1, where D ij = (α i ) 1/2 δ ij, i.e.  ij (x, t) = δ ij + (α iα j ) 1/2 κ i + κ j e (κ i+κ j )x, (5.69) whereupon u(x, t) = 2 d2 ln det  (5.70) dx2 where I have reintroduced the time dependence, with the understanding that it arises solely from the α i s. Application: N = 2, the two-soliton solution In the case N = 2 or, setting det  = 1 + α 1 2κ 1 e 2κ 1x + α 1 2κ 1 e 2κ 2x + α 1α 2 4κ 1 κ 2 ( ) 2 κ1 κ 2 e 2(κ 1+κ 2 )x κ 1 + κ 2 κ 1 κ 2 κ 1 + κ 2 e, α j 2κ j e 2θj+, (5.71) det  = 1 + e 2(κ 1x θ 1 2 ) + e 2(κ 2x θ 2 2 ) + e 2[(κ 1+κ 2 )x (θ 1 +θ 2 )] Note that now the time dependence is carried by the θ j s, i.e.,. (5.72) θ j θ j (t) = θ 0 j + 4κ 3 jt (5.73) In order to extract the asymptotic behavior of u(x, t) at early and late times, I proceed as follows: Assume κ 1 > κ 2 without loss of generality. Then as, t, at sufficiently early times, it is possible to satisfy the double inequality θ 1 κ 1 θ 2 κ 2. It is easy to see that, unless x θ1 κ 1 or x θ2 κ 2, the 2nd derivative of the expression (5.72) will be vanishingly small. This is true for x θ2 κ 2, because the last three terms vanish, leaving det  = 1 for x θ 1 κ 1, because, although the three last terms are all exponentially large, the last one will be dominant. This leaves ln det  x and the second derivative vanishes. for θ 1 κ 1 x θ 2 κ 2 the second term will be exponentially small, and the third term be much larger than the last. Again, ln det  x and the second derivative vanishes. This leaves the cases where x is appreciably near either θ 1 κ 1 or θ 2 κ 2. In the first case, the contributions to (5.72) come from the 3rd and 4th terms, i.e. det  e 2(κ2x θ2 2 ) [ 1 + e 2(κ1x θ1+ 2 )], or, ln det  2(κ 2x θ 2 2 ) (κ 1x θ ) + ln [ 2 cosh(κ 1 x θ ) ], 37

45 5 Solving KdV by inverse scattering κ 1 =2 κ 2 =1 θ 1 0 /κ 1 = -2. θ 2 0 /κ 2 = t t x Figure 5.1: The two-soliton solution [ u(x, t)] of the KdV equation as a function of space and time. Left panel: a 3-d plot shows the collision of the two solitons. Right panel: a contour plot of the same function; note the asymptotic motion of the local maxima and the phase shifts as a result of the interaction. x and hence Similarly, it can be shown that u(x, t) 2κ 2 1 sech 2 (κ 1 x θ ) if x θ 1/κ 1 u(x, t) 2κ 2 2 sech 2 (κ 2 x θ 2 2 ) if x θ 2/κ 2. Combining the above, and reintroducing the explicit time dependence, I can write that u(x, t) 2 2 κ 2 j sech 2 (κ j x 4κ 3 jt θj 0 ± ) if t, (5.74) 2 j=1 where the upper sign holds for j = 1, and the lower for j = 2. The above analysis can be repeated almost verbatim for asymptotically late times and leads to u(x, t) 2 2 κ 2 j sech 2 (κ j x 4κ 3 jt θj 0 ) if t. (5.75) 2 j=1 The above equations describe the soliton property in a mathematically exact fashion. As we follow the evolution from very early to very late times, we see the larger - and faster - local compression reach the smaller - and slower -, interact with it in an apparently intricate fashion, and then disengage itself and resume its motion with the same velocity. Both waves maintain shape, amplitude and speed. The interaction does however leave a signature. The center of mass of each wave becomes slightly displaced; the fastest by an amount of /κ 1 38

46 5 Solving KdV by inverse scattering (forwards), the slower by an amount of /κ 2 (backwards). Note that because the mass of each soliton is proportional to κ j, the center of mass of the combined two-soliton system moves at a constant speed before and after the two-soliton collision. This type of elastic, transparent interaction which leaves velocities unchanged and results only in spatial shifts 2 is characteristic of soliton bearing systems, and accounts for their remarkable dynamical properties. The analysis can be generalized to the N soliton solution. It can be shown that phase shifts are pairwise additive, i.e. the total phase shift of any soliton as a result of its interaction with the other N 1 solitons is the sum of the N 1 phase shifts resulting from the N 1 collisions. Fig. 5.1 exhibits graphically the dependence of the two-soliton solution on space and time. 5.6 Integrals of motion It is possible to deal with integrals of motion in a systematic fashion, by following the analytic structure of the scattering data. Recall that the transmission coefficient does not carry any time dependence under the IST, i.e. it can be treated as a constant of the motion! Lemma: a useful representation of a(k) Given the fact that a(k) (recall that a is the inverse of the transmission coefficient) has simple zeros in the upper half of the complex plane, the following identity holds: (cf. appendix...). ln a(k) = 1 dk ln a(k ) 2 2πi k k + ( ) N k k j ln k kj j=1 (5.76) Asymptotic expansions of a(k) The asymptotic expansion holds for k > max{ k j }. ln a(k) n=1 J n (2ik) n (5.77) Multiply both sides of (5.77) by k l 1 /(2πi) and integrate over a circle of radius R > max{ k j } centered at the origin of the complex k-plane. The only term which survives in the sum is that with j = l, hence (2i) l J l = 1 2πi dk k l 1 ln a(k) ; performing the dk integration in the first term of (5.76) generates a contribution k l 1. The second term can be integrated by parts and generates contributions from all poles. This results in J l (2i) l = 1 dk k l 1 ln a(k) 2 + 2πi 2 the term phase shifts is generically applicable. N j=1 1 ( k l j k l ) j l. (5.78) 39

47 5 Solving KdV by inverse scattering So far this has been general. Applying to the KdV equation, I set k j = iκ j ; note that the terms in the discrete sum vanish if l is even. Due to the reflection symmetry a(k) = a( k), the integrals vanish as well for even l. This leaves J 2m+1 = ( 1)m 22m+1 dk k 2m ln a(k) N j=1 κ 2m+1 j 2m + 1. (5.79) In what follows, I will relate the J l s - which are by definition constants of the motion, since they only depend on a(k) and bound state eigenvalues - to the family of conserved quantities generated in the previous section, independently of the IST. From the general property of the Jost functions I deduce that if Imk > 0 f 2 (x, k) = a(k)f 1 (x, k) + b(k)f 1 (x, k) (5.80) [ lim f2 (x, k)e ikx] = a(k). (5.81) x This is because in the limit of large positive x the first term in (5.80), f 1 (x, k) e ikx is exponentially large and the second f 1 (x, k) e ikx is exponentially small. On the other hand, [ lim f2 (x, k)e ikx] = 1. (5.82) x holds by definition. Knowledge of the two limits allows me to define with the property σ(x) = d [ { ln f2 (x, k)e ikx}] = f 2 + ik (5.83) dx f 2 dx σ(x) = ln a(k). (5.84) Now we can use the fact that f 2 is a solution of the associated linear problem, to derive a differential equation for σ in terms of u. To do this I multiply both sides of (5.83) and differentiate with respect to x. This gives or substituting f 2 = (σ ik)f 2 generates f 2 + ikf 2 = f 2σ + f 2 σ, uf 2 k 2 f 2 + ikf 2 = f 2σ + f 2 σ ; σ 2ikσ + σ 2 u = 0 (5.85) a nonlinear ordinary first order differential equation of the Ricatti type. I can now try to generate an asymptotic solution of the Ricatti equation (5.85), σ(x, k) n=1 where I note that, because of (5.77) and (5.84), σ n (x, k) (2ik) n (5.86) dx σ n (x) = J n. (5.87) 40

48 5 Solving KdV by inverse scattering Indeed, I note that the asymptotic ansatz (5.86) in (5.85) generates the recurrence relationships σ n+1 (x) = d n 1 dx σ n(x) + σ j (x)σ n j (x), n = 2, 3, (5.88) j=1 with σ 1 (x) = u(x). This generates a countable infinity of conserved densities. The first few are σ 2 = u x (5.89) σ 3 = u xx + u 2 (5.90) σ 4 = u xxx + 2(u 2 ) x (5.91) σ 5 = u xxxx + (u 2 ) xx + u 2 x + 2uu xx 2u 3. (5.92) Note that the even σ n s are total derivatives, i.e. they generate trivial, vanishing integrals; we know this, since the corresponding J n s vanish. The first few odd σ n s generate the mass, momentum, and energy integrals of section IST as a canonical transformation to action-angle variables It can be shown [] that the inverse scattering transform is a canonical transformation from the original field variables to action-angle variables. The scattering data of the IST are in essence action-angle variables. This demonstrates the KdV Hamiltonian system is completely integrable. 41

49 6 Solitons in anharmonic lattice dynamics: the Toda lattice The Toda lattice [11] is a unique example of a nonlinear discrete particle system which is completely integrable. Although the property of complete integrability is certainly a singular feature due to the peculiarity of the lattice potential, the model has been extremely useful as a theoretical laboratory for the exploration of a number of novel concepts and phenomena related to loss-free supersonic pulse propagation. 6.1 The model The Hamiltonian H = n { } p 2 n 2m + φ(q n q n 1 ) (6.1) where φ(r) = a b { e br + br 1 } (6.2) describes a chain of N particles with equal mass m, which interact via nearest-neighbor repulsive potential of exponential form. The range of the potential is given by 1/b and its strength by a/b. The linear term in the potential represents an external force which is necessary to achieve confinement. Important limiting cases of (6.2) are: the harmonic limit which leads to a, b 0, ab k φ(r) = 1 2 kr2 ; the hard-sphere limit b (i.e. the range approaches zero) with a finite, which leads to φ(r) = 0 if r > 0 = if r < 0. I will, from now on, set a = 1, b = 1, m = 1. Units will be reintroduced when appropriate. The equations of motion are q n = p n ṗ n = e (qn qn 1) e (qn+1 qn) (6.3) 42

50 6 Solitons in anharmonic lattice dynamics: the Toda lattice 6.2 The dual lattice Consider the variables r n = q n q n 1 which describe the difference in displacements of neighboring sites. sides with respect to time gives Differentiating both ṙ 1 = q 1 q 0 ṙ 2 = q 2 q 1 ṙ j = q j q j 1. Summing left and right sides separately gives allows me to express the velocity coordinates as j q j = ṙ l, (6.4) l=1 where I have assumed q 0 = 0. The total kinetic energy is T = 1 2 ( N n ) 2 ṙ l. If I now define new momentum variables, conjugate to the r n coordinates, via the s j s will satisfy n=1 l=1 s j = T ṙ j = ṙ ṙ j + ṙ ṙ j+1 + and therefore I can rewrite the kinetic energy as + ṙ ṙ N, s j s j+1 = ṙ 1 + ṙ j = q j (6.5) T = 1 2 N (s j+1 s j ) 2 Since the total potential energy is clearly only a function of the r n s, j=1 φ(r n ), n this process has defined a new canonically conjugate set of variables. Following Toda [11] we view this set as describing a new lattice, dual to the original. The equations of motion are ṡ j = H r j = φ (r j ) ṙ j = H s j = 2s j s j+1 s j 1 (6.6) and describe - by construction - the same dynamics as the original equations of motion (6.3). 43

51 6 Solitons in anharmonic lattice dynamics: the Toda lattice α=1 δ/α= r n E r 1E-5 n 1E-7 1E n n 5 10 Figure 6.1: The local compression corresponding to a Toda soliton (6.13). The value of α is equal to 1. The three curves represent different choices of the phase δ. Inset: the dependence of r n vs n on a logarithmic scale for the case δ = 0. Now it is possible to eliminate either the s n s or the r n s from (6.6). In the first case I obtain r n = 2e r n e r n+1 e r n 1 (6.7) and in the second where Note that owing to (6.6), 1 + S n = e 2Sn+Sn+1+Sn 1 (6.8) S n = t 0 dt s n (t ). (6.9) q n = S n S n+1. (6.10) A pulse soliton A special solution of (6.8) is S n (t) = ln cosh(αn βt δ) (6.11) where α > 0, δ is an arbitrary constant, and β = sinh α. Differentiating with respect to time, I obtain s n (t) = β tanh(αn βt δ) (6.12) and therefore β 2 e rn = 1 + cosh 2 (αn βt δ). (6.13) This special solution corresponds to a supersonic, compressional pulse soliton moving with the velocity v = ± β α = ±sinh α. α The form of the pulse is shown in Fig

52 6 Solitons in anharmonic lattice dynamics: the Toda lattice Mass of the soliton The total mass carried by the soliton can be shown - using (6.11) - to be M = j r j = lim n (q n q n ) = 2α. (6.14) Momentum of the soliton The total lattice momentum carried by the soliton can be shown - using (6.12) - to be P = j q j = lim n (s n s n ) = 2β = Mv. (6.15) Energy of the soliton The total energy of the soliton is given by 1 2 (s n+1 s n ) 2 + n n ( e r n 1 ) + r n. The sum of the first two terms can be shown to be sinh 2α; the third sum we recognize as the soliton mass. Thus E = sinh 2α 2α (6.16) 6.3 Complete integrability Define new coordinates in terms of the original positions and momenta a n = 1 2 e 1 2 (qn qn 1) Using the original equations of motion, I obtain and b n = 1 2 p n. (6.17) ḃ n = 1 2ṗn = 2(a 2 n+1 a 2 n) (6.18) ln(2a n ) = 1 2 (q n q n 1 ) ȧ n a n = 1 2 (p n p n 1 ) ȧ n = a n (b n b n 1 ). (6.19) Note that decaying boundary conditions at (plus or minus) infinity correspond to a n 1/2, b n 0. This allows for a constant value of the displacement q (cf. the pulse solution of the previous section). Now one can directly verify that the set of equations is equivalent to the condition i dl = [B, L], (6.20) dt 45

53 6 Solitons in anharmonic lattice dynamics: the Toda lattice where and L = B = i b n 1 a n 0 0 a n b n a n a n+1 b n+1 a n a n+2 b n+2 0 a n a n 1 0 a n 0 0 a n 0 a n a n+1 0 (6.21) (6.22) are tridiagonal matrices which form a Lax pair. In other words, the Toda lattice with decaying boundary conditions can be completely integrated using the inverse scattering transform. Details can be found in [11]. This means that there are multisoliton solutions, and that Toda solitons have all the nice properties of exact solitons which we encountered in the KdV example (e.g. elastic scattering which only results in phase shifts etc). 6.4 Thermodynamics The partition function of the Toda chain ( ) N Z = dp i dq i e βh, i=1 where β is the inverse temperature, can be factorized into two contributions, Z K and Z P, coming from the kinetic and potential energy respectively. The integration over momentum variables gives a product of N identical integrals, ( Z P = N dp e /2) βp2 = ( ) N/2 2π β whereas the integration over position coordinates gives ( N ) Z K = dq i e β N φ(q i q i 1 ) i=1 = = = i=1 ( N ) dr i i=1 e β N i=1 φ(ri) ( ) N dr e βφ(r) ) N (e β dy y β 1 e βy = [ e β β β Γ(β) ] N 0 (6.23) where the substitution y = e r has been made. Combining terms I obtain the free energy per site f = 1 1 N β ln Z = 1 + ln β 1 1 ln Γ(β) + β 2β ln β (6.24) 2π 46

54 6 Solitons in anharmonic lattice dynamics: the Toda lattice At low temperatures, β 1, one can use the Stirling approximation to the gamma function Γ(z) e z z z 1/2 ( 2π ) 12z + (6.25) and obtain f 1 β ln β 2π 1 12β 2 + (6.26) where the first term is identified as the free energy per site of a harmonic chain, and the second is the leading term of a systematic asymptotic expansion in powers of the temperature. 47

55 7 Chaos in low dimensional systems 7.1 Visualization of simple dynamical systems Two dimensional phase space Linear stability analysis Consider the following general dynamical system consisting of two coupled differential equations. x = F ( x), (7.1) where F 1 (x 1, x 2 ), F 2 (x 1, x 2 ) are arbitrary, in general nonlinear functions of x 1, x 2. Note that (7.1) does not necessarily represent a mechanical system. It could for example represent a coupled system of prey-predator species with populations x 1 and x 2 respectively, for which F 1 (x 1, x 2 ) = rx 1 kx 1 x 2 F 2 (x 1, x 2 ) = sx 2 + k x 1 x 2 (7.2) (Lotka-Volterra equation). In the absence of interaction, the prey and predator populations will, respectively, grow and die off, at exponential rates (Malthus model of population biology). Interaction creates new possibilities. Note first that for x 1 = s/k, x 2 = r/k, the right-hand side of (7.2) vanishes. The two populations may coexist stably at these levels. Suppose however that you are dealing with fish populations, and some outside agent, without the power to modify the biological parameters, simply removes a part of one - or both - populations. If the perturbation is large, we would have to solve the full system (7.1) with the new set of initial conditions. For small perturbations however, it is possible to make some general statements about the system s behavior in terms of linear stability analysis. Consider a state of the system near the fixed point, i.e. x = x + δ x(t). (7.3) If δ x(t) is sufficiently small, we may expand (7.1) around the fixed point, and obtain where d δ x(t) = M δ x(t) (7.4) dt ( ) Fi M ij = x j x= x. The ansatz δ x(t) = exp(λt) f leads to the eigenvalue equation M f = λ f, (7.5) A perturbation which has a nonzero component along an eigenvector with positive eigenvalue will grow exponentially. On the other hand, if both eigenvalues are negative, the system will be stable in all directions around the fixed point. The various possibilities are summarized as follows: 48

56 7 Chaos in low dimensional systems λ 1, λ 2 real. If λ 1 λ 2 < 0 we have a saddle (stable in one direction, unstable in the other); in the special case λ 1 + λ 2 = 0, the saddle is called a hyperbolic fixed point. If λ 1 λ 2 > 0 we have a node. A node is stable if both eigenvalues are negative, and unstable if both eigenvalues are positive. If λ 1, λ 2 are complex conjugates we have a focus. A focus will be stable or unstable according to whether the real part of λ is negative or positive, respectively. If λ 1, λ 2 are pure imaginary we have an elliptic fixed point. The undamped harmonic oscillator q = p ṗ = ω 2 0q (7.6) Elliptic fixed point at p = 0, q = 0. Eigenvalues are λ 1,2 = ±iω 0. Because there is a conserved quantity (Hamiltonian), orbits in phase space are one dimensional (ellipses). The damped harmonic oscillator q = p ṗ = ω 2 0q γp (7.7) The fixed point at p = 0, q = 0 is either a stable focus (if γ < 2ω 0 ), or a stable node (if γ = 2ω 0 ). There is no conserved quantity; orbits in phase space have a spiral form. The pendulum H(p, q) = 1 2 p2 ω 2 0 cos q (7.8) q = p ṗ = ω 2 0 sin q (7.9) There are fixed points at p = 0, q = kπ, where k = 0, ±1, ±2,. The points at even k are elliptic, the ones at odd k are hyperbolic. Orbits in phase space are again one-dimensional, due to energy conservation. They are either bounded (near a fixed point), or unbounded. A special orbit (separatrix) separates the two types of motion. The separatrix connects two hyperbolic fixed points. The bistable potential H(p, q) = 1 2 p (1 q2 ) 2 (7.10) q = p ṗ = (1 q 2 )q (7.11) There are fixed points at p = 0, q = 0 (hyperbolic), and p = 0, q = ±1 (elliptic). Motion is bounded but has a different topology according to the value of the energy. The different types of motion are separated by a particular orbit (separatrix). 49

57 dimensional phase space 7 Chaos in low dimensional systems The dynamics of a Hamiltonian system with two degrees of freedom H = 1 2 (p2 x + p 2 y) + V (q 1, q 2 ) (7.12) is formulated in terms of a system of 4 coupled differential equations which are first order in time: q 1 = p 1 q 2 = p 2 ṗ 1 = V (q 1, q 2 ) q 1 ṗ 2 = V (q 1, q 2 ) q 2. (7.13) In a generic Hamiltonian system only the energy is conserved. If for some reason there is a further constant of motion, orbits will lie on a 2-dimensional torus. In the generic case, phase space orbits will be on the energy shell, i.e a 3-d hypersurface. It is possible to visualize this with the help of Poincaré surfaces of section, i.e. by projecting the energy hypersurface on the q 1 = 0 plane. A Poincaré surface of section - briefly, Poincaré cut -, consists of points (q 2, p 2 on a plane, taken at q 1 = 0, p 1 > 0. A cut of a 2-d torus would be a continuous curve. A cut of a generic 3-d hypersurface would fill a portion of the plane. Evidence of such filling in systems which are perturbed away from an integrable limit, is interpreted as breaking of the torus. This is what Hamiltonian chaos is all about. The Henon-Heiles Hamiltonian H = 1 2 (p2 x + p 2 y) (x2 + y 2 ) + x 2 y 1 3 y3, (7.14) originally proposed as a model for integrable behavior in galactic motion [12], was a milestone in the study of Hamiltonian chaos. The equipotential surfaces, shown in Fig. 7.1, suggest its usefulness as a model for triatomic molecules y y x x Figure 7.1: Left: equipotential surfaces of the Henon-Heiles Hamiltonian; right: details of the region of bounded motion, E = 1/30, 1/15, 1/10, 2/15, 1/6 (outer surface). Fig. 7.2 shows Poincaré cuts obtained at increasing energies. At E = 1/12 - which is not a small energy! - the motion is almost entirely regular (note however the seeds of irregularity 50

58 7 Chaos in low dimensional systems in the immediate vicinity of the separatrix). As the energy increases further, the various tori begin to disappear. Widespread chaos ensues. Note that the scattered points all belong to the same trajectory in phase space py py 0.0 py y y y Figure 7.2: Poincare cuts for the Henon-Heiles system. Left, E=1/12; center E=1/8; right, E=1/6. The percentage of area covered by such scattered points provides a measure of chaos dimensional phase space; nonautonomous systems with one degree of freedom A nonautonomous Hamiltonian system with one degree of freedom H(p, q, t) = 1 2 p + V (q, t) 2m (7.15) is described by the equations q = p = H = p m V (q, t) q V (q, t). t (7.16) Phase space is now in general 3-dimensional. There are no conserved quantities to reduce it. However, if the system is externally driven by a periodic force of period T, one may attempt to visualize its behavior by using stroboscobic plots, i.e. plotting pairs pn, qn obtained at times tn = nt. As an example, consider the plots obtained for the bistable oscillator with m = 2 in a periodic field V (q, t) = 2q 2 + q 4 + ²q cos ωt. (7.17) In the absence of a driving field, the trajectories in 2-dimensional phase space are shown in Fig. 7.3 (left panel). The equations of motion have 3 fixed points, two of them (at q = ±1) elliptically stable, and one (at q = 0) hyperbolically unstable. If the total energy is low (near -1), the particle performs low-amplitude oscillations at the bottom of the left or the right well. The limiting natural frequency of oscillation is ω0 = 2. 51

59 7 Chaos in low dimensional systems The other two panels of Fig. 7.3 show what happens when a periodically varying field is turned on. The frequency of the field ω = 1.92 is chosen to lie near ω 0. At low amplitudes of the driving field and reasonably low energies, a stroboscopic plot of motion is not fundamentally different from the corresponding plot at the conservative limit ɛ = 0; the particle stays confined near the top of the potential well. As the driving amplitude increases, the particle escapes the well and performs a chaotic motion in the vicinity of the separatrix of the conservative limit (right panel) p p 0-1 p ϖ=1.92 ε= ϖ=1.92 ε= x x x Figure 7.3: Stroboscobic plot of the dynamics of (7.17) for ω = 1.92 and 0 < t < 2000 (after [13]). The left panel shows the contours of phase space trajectories of the unperturbed, conservative system; note the separatrix at E = 0, which separates bounded from unbounded motion. Initial conditions were p = 0 and q = 0.24, corresponding to E = 0.112, an energy near the top of the potential well. The middle panel, at ɛ = 0.01, shows that the particle remains trapped in the well. The right panel, at ɛ = 0.1, illustrates the escape from the well, and the breaking of the torus which occurs near the separatrix. 7.2 Small denominators revisited: KAM theorem Recall there was a problem of small denominators; if you start with an integrable Hamiltonian H 0 (J 1, J 2 ) and functionally independent frequencies ω i = H 0 / J i and perturb it with a small perturbation µh 1 (J 1, J 2, θ 1, θ 2 ) then Poincaré showed that there are no analytic invariants of the perturbed system H 0 + µh 1. Is chaos inevitable? The answer is more or less yes. Is chaos imminent and overwhelming, even for a small perturbation? The answer is no - as we have seen from circumstantial evidence in the Henon-Heiles Hamiltonian. Kolmogorov, Arnold and Moser (KAM) showed that, if the Hessian of the unperturbed Hamiltonian is nondegenerate, i.e. ( 2 ) ( ) H 0 ωi det = det 0, (7.18) J i J j J j a torus of the H 0 Hamiltonian with frequencies ω i survives, slightly deformed, in the perturbed system, provided n 1 ω 1 + n 2 ω 2 K(ɛ) n 1 + n 2 α n 1, n 2 (7.19) where α > 2 and K(ɛ) depend on the particulars of the problem. Tori which do not fulfill this condition may break up. The destroyed tori constitute a dense set. Yet they have a very small measure. Most tori survive. It is possible to understand this using an analogy 52

60 7 Chaos in low dimensional systems with the length obtained by excluding from the line continuum a small neighborhood, say ɛ/n 3, around every rational number m/n (recall that the rationals form a dense set). The measure of the continuum deleted is n n=1 m=1 ɛ n 3 = n ɛ n 2 = π2 6 ɛ. (7.20) Although irrationals do not form a dense set, they make most of the measure of real numbers. In this sense, almost all tori survive the addition of a small (in practice: even a moderately large) perturbation. Eventually however, as the perturbation grows, chaos ensues. Note I have used the language of systems with two degrees of freedom just for simplicity. In fact, the KAM theorem holds for an arbitrary number of degrees of freedom, under the conditions described above. 7.3 Chaos in area preserving maps Twist maps The twist map allows direct visualization of a Hamiltonian system with two degrees of freedom, moving on a torus. Let J 1, J 2 be the action coordinates, and θ 1, θ 2 the corresponding angle coordinates. Make a Poincaré cut each time θ 2 = 0 mod 2π. This will by definition be every τ = 2π/ω 2 seconds, where ω 2 = H 0 / J 2. Then plot the coordinates ρ = 2J 1 and φ = θ 1 on a plane. The points will lie on a circle. I can express the successive values of the angle coordinate on the cut by the sequence φ n+1 = φ n + ω 1 τ or, more generally, in terms of the winding number w = ω 1 /ω 2 ρ n+1 = ρ n φ n+1 = φ n + 2πw(ρ n ) (7.21) where I have explicitly allowed all possible J 1 s and hence all possible radii. For a given energy this fixes J 2, so that the winding number is only a function of ρ. In shorthand notation this will be ( ) ( ) ρn+1 ρn = T φ 0, (7.22) n+1 φ n where T 0 stands for the unperturbed twist map. Now if the winding number can be expressed as a rational fraction r/s, the cut will be composed of s points (s-cycle). If not, we have quasiperiodic motion; the cut fills the circle densely. We would like to find out what happens under a perturbation. This is described below (Poincaré-Birkhoff theorem). For the moment, let me just describe what a perturbed map will look like - and how to get it. In general, ρ n+1 = ρ n + ɛf 1 (ρ n, φ n ) φ n+1 = φ n + 2πw(ρ n ) + ɛf 2 (ρ n, φ n ) (7.23) where I must choose the functions f 1 and f 2 such that the map represents a Hamiltonian flow, i.e. it should be a canonical transformation. This can be achieved by using an appropriate 53

61 7 Chaos in low dimensional systems generating function F (φ 1, φ 2 ) such that ( ) F ρ n+1 = φ n φ n+1 ( ) F φ n+1 =. (7.24) φ n+1 φ n A class of such perturbed maps can be obtained by the generating function The maps have the form F (φ n, φ n+1 ) = 1 2 (φ n φ n+1 ) 2 + ɛv (φ n ). (7.25) ρ n+1 = ρ n + ɛv (φ n ) φ n+1 = φ n + ρ n+1 ; (7.26) The above map equations (7.26) can also be derived by demanding that the action W = m F (φ n, φ n+1 ) (7.27) n=0 should be an extremum with respect to any of the m internal coordinates φ 1,, φ m (i.e. the end coordinates are φ 0, φ m+1 are held fixed). F can thus be interpreted as a discrete Lagrangian. Later in the course I will show that this has important applications in an entirely different context - determining energy minima and studying prototypes of amorphous solids; in other words, spatial rather than temporal chaos Local stability properties The local stability properties of fixed points are governed by the tangent map (cf. continuous dynamics). Thus if X = (ρ, φ ) is a fixed point of the map T, X = T ( X ) (7.28) the tangent map of T is defined via a linearization procedure around the fixed point: where in general M( X n ) = X n = X + δ X n (7.29) δ X n+1 = M( X )δ X n (7.30) ( ρn+1 ρ n+1 ρ n φ n φ n+1 φ n+1 ρ n φ n ). (7.31) Since the map T is area preserving, the eigenvalues of M will satisfy the relationship λ 1 λ 2 = 1. There are two distinct cases both roots are imaginary; they must be of the form (elliptic fixed point), or both roots are real λ 1,2 = e ±iδ (7.32) λ 1 > 1 λ 2 < 1 (7.33) (hyperbolic fixed point if both positive, hyperbolic with reflection if both negative). 54

62 7 Chaos in low dimensional systems Periodic motion (s cycles in the form X 1, X 2,, X s ) is represented by fixed points of the T s map, X j = T s ( X j ) j = 1, 2,, s. (7.34) The stability of the s-cycle (7.34) is governed by the eigenvalues of the product matrix M (s) = M( X s )M( X s 1) M( X 1 ). Note that, since the determinant of each one of the terms in the above product is unity, det M (s) = 1. The classification of stability properties is therefore exactly the same (elliptic vs hyperbolic cycles) as in the case of fixed points (cf. above) Poincaré-Birkhoff theorem The unperturbed twist map with a rational winding number w = r/s will generate an s-cycle whose points lie on a circle C. This will happen no matter where one starts on the circle. In this sense, every point the circle will be a fixed point of the unperturbed T s o map, T s o C = C. (7.35) Note that this differs from the generic situation of an irrational winding number; the circle with a radius which corresponds to an irrational winding number maps onto itself - but its points are not fixed points of any finite repeated application of the map. What happens under the influence of a perturbation? In order to see this, consider two neighboring circles, C +, with a slightly larger, irrational winding number w +, and C with a slightly smaller, irrational winding number w. Under application of the same unperturbed twist map, C + will be slightly twisted - with respect to C -in the positive (counterclockwise) direction, since w + > w; similarly C will be slightly twisted in the negative (clockwise) direction, since w < w. These relative opposite twists of the circles survive under the perturbed twist map Tɛ s - although their form may be distorted. By a continuity argument it is possible to construct a zero twist curve R. If I now apply the map Tɛ s to R, the resulting curve will be distorted with respect to R only in the radial direction (zero twist). Because the map is area preserving, there should, in general, be an even number of intersections, 2ks (exceptions are possible in cases where the curve Tɛ s R might tangentially touch the curve R). These intersections are the only fixed points which survive from the original invariant circle C in the presence of a perturbation. Of the 2ks fixed points, half are elliptically stable and half hyperbolically unstable; elliptic and hyperbolic fixed points come in pairs and they alternate. This is the Poincaré-Birkhoff theorem Chaos diagnostics Power spectra Given a suitably averaged time-dependent quantity f(t), it is possible to define its power spectrum I(ω) = 1 dte iωt f(t). (7.36) 2π If the signal is periodic in time, i.e. if f(t) = f(t + T ), it is possible to express it as a Fourier series f(t) = α n e inωt (7.37) n= 55

63 7 Chaos in low dimensional systems Figure 7.4: Illustration of the Poincaré-Birkhoff theorem. (a) upper left: the unperturbed map: a circle C with a rational winding number w, along with neighboring circles C +, C with irrational winding numbers w + (positive twist), w (negative twist). (b) upper right: the perturbed map; outer and inner curve represent, respectively, the slightly deformed versions of C +, C. The intermediate curve R is a zero-twist curve obtained by the requirement of continuity. (c) lower right: tr(continuous curve) and its T s ɛ map (dashed curve). In this case s = 2. There is no twist under the action of the map, just pulling and pushing along the radial direction. There is a total of 4 intersections, corresponding to a stable and an unstable 2-cycle. Following the arrows, it is possible to determine which points are elliptic and which are hyperbolic. Note that the small arrows outside R are all pointing in the outward direction (positive twist), and those inside R in the negative direction (negative twist). (d) a more abstract view of the elliptic and hyperbolic 2-cycles. where Ω = 2π/T. It follows that the spectrum I(ω) = n= α n δ(ω nω) (7.38) will be composed of a series of δ-peaks situated at the fundamental frequency and its higher harmonics. One can generalize this to the case of a multiply periodic motion - which would be more apt to describe motion on on a torus. In this case of a doubly periodic motion f(t) is 56

64 7 Chaos in low dimensional systems described by a double Fourier expansion f(t) = α n1,n 2 e i(n 1Ω 1 +n 2 Ω 2 )t (7.39) n 1 = n 2 = and the spectrum I(ω) = α n1,n 2 δ(ω n 1 Ω 1 n 2 Ω 2 ), (7.40) n 1= n 2= forms peaks at all sum and difference frequencies. Under ideal conditions (cf. Fig. 7.5) it 1x10-5 1x10-4 1x10-5 1x Power spectra 10-6 Power spectra 10-6 Power spectra ϖ/2π ϖ/2π ϖ/2π Figure 7.5: Power spectra of p y(t) for quasiperiodic (left and center panels) and chaotic (right panel) trajectories of the Henon-Heiles system at energy E = 1/8. In the case of quasiperiodic motion (left) it is possible to make a detailed identification of the five peaks in terms of two fundamental torus frequencies at f 1 = 0.16 and f 2 = 0.12, their second harmonics, and the difference f 1 f 2 = A similar assignment can be made in the case of the center panel. Chaotic spectra (right panel) are characterized by broader, noisier features. should of course be possible to distinguish regularity from chaos by its spectral signatures. In the former case the spectrum is periodic or quasiperiodic, in the latter case there is a lot of noise, perhaps accompanied by broad peaks. In practice however, the intrinsic limitations of obtaining useful power spectra from finite numerical (or experimental) data, renders spectral information somewhat limited as a sole criterion of deciding whether a given process is chaotic or not. Lyapunov exponents Lyapunov exponents quantify the usual defining property of deterministic chaos, which is the sensitive dependence on initial conditions. Consider a certain trajectory of the - not necessarily area-preserving - N-dimensional map T X j+1 = T ( X j ) j = 0, n 1, (7.41) and a neighboring trajectory, which starts at X 0 + δ X 0. The difference between the two trajectories after the first iteration can be expressed in terms of the tangent map: δ X 1 = M( X 0 )δ X 0 ; 57

65 7 Chaos in low dimensional systems after the second iteration it will be and after n iterations δ X 2 = M( X 1 )δ X 1 = M( X 1 )M( X 0 )δ X 0, δ X n = M( X n 1 )M( X n 2 ) M( X 0 )δ X 0 = Λ n ( X 0,, X n 1 )δ X 0, (7.42) where the N N matrix Λ is the nth root of the product of all n tangent maps involved in the trajectory; in general, Λ will have N eigenvalues λ α (n), α = 1, N, which will depend on the order of iteration n. The Lyapunov exponents are defined as σ i = lim n ln λ α(n) α = 1,, N. (7.43) Note that in general there are as many Lyapunov exponents as the dimensionality of the map. If the map is area preserving, they come in pairs, i.e. for each positive exponent, a negative exponent with the same magnitude must occur. This corresponds to expanding and shrinking directions; It is obvious from (7.42) that, if we order Lyapunov exponents in decreasing order σ 1 > σ 2 > σ N (7.44) the largest (positive) exponent will eventually dominate the right hand side of (7.42). This will happen even if there is a vanishingly small component of δx 0 in the direction of the eigenvector of Λ which corresponds to σ 1. The norm δx n will grow exponentially as e σ1n. This is exactly the physical content of sensitive dependence on initial conditions. Lyapunov exponents provide a measure of just how sensitive this dependence is. Note: here I have defined Lyapunov exponents in the context of maps. If time permits, I will present the definitions - and computational procedures - for dynamical systems governed by differential equations, i.e. Hamiltonian or dissipative dynamics The standard map kicked pendulum, kicked rotator Consider the nonautonomous Hamiltonian system defined by a kicked pendulum, where gravity acts in bursts, every τ seconds: H = p2 2 K (2π) 2 cos(2πq) n= δ(t nτ) (7.45) where p is the angular momentum and 2πq the angle, referred to the perpendicular direction (cf. H-atom in electric field.) The equations of motion are ṗ = H q = K 2π sin(2πq) q = H p. n= δ(t nτ) The first equation implies that p is constant, except at times t = nτ, when it changes by a discrete step. Defining p n = lim ɛ 0 p(nτ ɛ), 58

66 7 Chaos in low dimensional systems I can integrate in the neighborhood of t = nτ, set τ = 1 and obtain what is known as the standard map p n+1 = p n K 2π sin(2πq n) q n+1 = q n + p n+1. (7.46) Eqs belong to the general class (7.26) of area preserving twist maps. In the following, the coordinates p n, q n will be understood as mod1, unless stated otherwise. Fixed points The map (7.46) has two fixed points: p = 0, q = 0, which is elliptic, and p = 0, q = 1/2, which is hyperbolic. (NB: the published literature has adopted a variety of conventions; one of them has a different sign in the Hamiltonian; this amounts to a shift of q by 1/2) Summary of results At small values of K (cf. Fig. 7.6) there is no sign of chaos. We observe the tori which surround the elliptic fixed point, which extend up to the separatrix which leaves off the hyperbolic fixed point. Furthermore, we observe a large number of horizontal tori - meaning that they run all the way from left to right; these tori are the slightly deformed versions of the original irrational tori of the unperturbed system, which have survived the perturbation. Finally, the structure which emanates from the period 2 cycle, around the center of the picture, is visible. This resonant torus is broken; according to the Poincaré-Birkhoff theory, we observe a period 2 island chain, and the hyperbolic fixed points nested between them. 1.0 K= K= p p q q Figure 7.6: Trajectories of the standard map at K = 0.5 (left panel), K = 0.8 (right panel). As the perturbation increases, more and more near-resonant horizontal tori break up. Chaos develops around the separatrices of the leading resonances (hyperbolic fixed point 59

67 7 Chaos in low dimensional systems K= K= p p q q Figure 7.7: Trajectories of the standard map at K = Kc = (left panel), K = 1.17 (right panel). and in the crossings between period 2 island chain). The survival of a torus depends on how irrational - its winding number is. In order to see what this means, look at the continued fraction representation of an irrational number w = a0 + 1 a1 + 1 a2 + {a0 ; a1, a2, }, (7.47) where the integers {ai } satisfy a0 0 and ai > 1 i 1. An n-th order approximation w = rn /sn can be generated by the sequence rn sn = an rn 1 + rn 2 = an sn 1 + sn 2 (7.48) with r 2 = 0, r 1 = 1, s 2 = 1, s 1 = 0. Eq. (7.48) implies that sn+1 > an+1 sn It follows that w. rn 1 1 < < sn sn sn+1 an+1 s2n (7.49). (7.50) Thus, if an+1 is large, the nth approximation is a good one. An example is π = {3; 7, 15, 1, 292, } which leads to π = r3 /s3 = 355/113 = , good to 7 digits. Conversely, the representation 2 = {1; 2, 2, 2, 2, } leads to 2 = r3 /s3 = 17/12 = , which has an error in the 4th digit. In this sense, the golden mean 5+1 = {1; 1, 1, 1, 1, } (7.51) 2 and its inverse 5 1 = {0; 1, 1, 1, 1, } 2 60 (7.52)

68 7 Chaos in low dimensional systems can be considered as the most irrational numbers. Therefore, the non-resonant torus with a winding number equal to the inverse golden mean, is expected to be the last to break. The disappearance of the last, golden mean torus at K = K c = (cf. Fig. 7.7, left panel)is a key event in the nonlinear scenario. It signals the transition from local to widespread chaos. The following aspects deserve special attention: breaking of analyticity: As K approaches the critical value K c, the deformation of the torus increases dramatically. The following procedure [14] makes it possible to follow the torus shape and detailed properties. First observe, following Greene [15], that an instability of a torus with irrational winding number w can be associated with the instability of an s n cycle, where s n is defined in terms of the sequence r n /s n used to approximate w. Thus, rather that try to construct a torus directly, it is possible to determine successive cycles and their thresholds of instability. It useful to introduce the Moser representation (parametrization) [14] q j = t j + u(t j ), (7.53) appropriate to any cycle with a rational winding number w; here t j = jw = jr/s. The property q j = q j+s implies u(t j ) = u(t j + 1) mod 1. (7.54) Note that the periodic function u(t) - which can be shown to be odd - is only defined on a rational set t = t j = jr/s, but this set becomes more and populous as s is increased. Fig. 7.8 shows the dependence of u(t), evaluated for an s = 4181 cycle which approximates the torus with a golden mean winding number, as a function of K. Note how the function becomes less and less smooth as K c is approached. self similarity: Fig. 7.8 shows the shape of the KAM golden-mean torus at two noncritical K s and at K = K c. Note the detailed view of the non-smooth function. The detailed numerics [14] allows the conjecture of self-similarity; in other words, the valleys and hills of the curve repeat themselves at all possible scales of numerical observation. In this sense, KAM torus disappearance resembles a critical phenomenon. Self-similarity is very well demonstrated in the frequency spectra. The odd function u(t) can be represented in a Fourier series u(t) = A f sin 2πft f=1 The product fa f is shown in Fig. 7.9 as a function of f, for the same values of K as in Fig Note the presence of more and more peaks as the critical value of K is approached. At K = K c self similar behavior occurs, with primary peaks occurring at the Fibonacci numbers. Arnold diffusion: A picture of the widespread chaos which occurs at higher values of the nonlinear parameter K > K c is given in Fig. 7.7) (right panel). Unless a point starts in the immediate neighborhood of the elliptic fixed point, or the very few islands, it will typically generate a chaotic orbit which may diffuse over a large two-dimensional area of phase space. This diffusive behavior can be quantitatively characterized as follows. Suppose we relax the mod 1 condition on the momentum p in (7.46). In other words we allow the phase space to be a cylinder of perimeter 1 and look at the quantity [ ] (p j+n p n ) 2 D = lim, (7.55) n 2n 61

69 7 Chaos in low dimensional systems p u p q q t Figure 7.8: The torus with a winding number approximately equal to the inverse of the golden mean W = ( 5 1)/2, at K = 0.5, 0.9, K c. The curves shown are actually sets of discrete points belonging to cycles of order s = 4181 with a rational winding number w = r/s = 2584/4181 which differs from W by less than I. Left panel: the torus in the (p, q) plane. II. Center panel: a detailed view of the same torus in the cases K =.9 (right y-scale) and K = K c (left y-scale). III. Right panel: the function u(t) which describes the torus of the standard map in parametric form. Again, the curve shown is actually obtained from an 4181-cycle which approximates the irrational winding number W. Note how the function changes from smooth at K = 0.5, to somewhat bumpy at K = 0.9, to very bumpy at K = K c. The inset shows a detailed view of the critical curve, which suggests self-similar behavior (After [14]) f A(f) f A(f) f A(f) f f Figure 7.9: Fourier coefficients of the function u(t), which describes parametrically the torus with with a golden mean winding number, at K = 0.5, 0.9, K c. The quantity plotted is f A f. The curve at K = K c (right panel), with primary contributions at the Fibonacci numbers, suggests self-similar behavior (after [14]). f which describes the coefficient of diffusion in momentum space. As long as K < K c, the existence of even a single torus presents a topological barrier to diffusion 1. D should vanish. 1 this is no more the case in higher dimensions! Arnold diffusion is generic in higher dimensionality because tori can be bypassed. 62

70 7 Chaos in low dimensional systems In the opposite limit K K c, we can estimate the diffusion coefficient as follows. From (7.46) and hence (p j+n p j ) 2 = p j+n = p j K 2π j+n 1 l=j sin(2πq l ) (7.56) ( ) 2 j+n 1 K sin(2πq l ) sin(2πq l ). (7.57) 2π l,l =j Now, since q j+1 = q j + p j+1 mod 1, if p j+1 is large, q j+1 is essentially random, i.e. uncorrelated to q j. The only correlations which survive are from terms l = l. On the average, the double sum will therefore be equal to n/2 (the 1/2 factor from the average value of sin 2 ). Therefore D ( ) 2 K if K K c. (7.58) 4π In the case where K slightly exceeds K c, Chirikov has estimated D (K K c ) (7.59) Cantori: (7.59) makes clear that, even beyond the stochasticity threshold, diffusion does not proceed uninhibited. At values slightly above K c the diffusion constant is in fact very close to zero. It appears that some barriers to diffusion persist after all KAM tori have been broken. Resistance to diffusion can be related to a particular class of orbits with irrational winding numbers, which do not fully cover a one-dimensional curve (Fig. 7.10), but leave a countable set of open intervals empty - i.e. they form a Cantor set. Because of this property they were named cantori by Percival. The existence of cantori as isolated regular orbits embedded in a sea of chaos is remarkable. We will deal with them again in Chapter 9, in the context of solid state theory. An excellent review of the transport properties of Hamiltonian maps has been written by Meiss [16] The Arnold cat map The area preserving map x n+1 = x n + y n mod 1 y n+1 = x n + 2y n mod 1 (7.60) has a tangent map M = ( which does not depend on the coordinates. The eigenvalues of the map are λ 1,2 = 1 2 ) ( 3 ± ) 5 (7.61) (7.62) and the Lyapunov exponents σ 1 = ln λ 1 = σ 2. There is a single, hyperbolic fixed point at x = y = 0. Neighboring trajectories everywhere diverge exponentially. All cycles are unstable. What happens to a cat thus mapped is shown in Fig

71 7 Chaos in low dimensional systems p p q q Figure 7.10: Standard map cantori with a golden-mean winding number, obtained at K Kc = 0.01 (left panel) and K Kc = 0.3 (right panel) y y y x x x Figure 7.11: The fate of a cat under two iterations of the map (7.60) The baker map; Bernoulli shifts The map xn+1 yn+1 xn+1 yn+1 = = = = 2xn 1 2 yn 2xn yn + 2 ¾ 1 2 if 0 xn < if 1 2 xn < 1, ¾ (7.63) because of its action, which is to shrink (halve) in the vertical and stretch (double) in the horizontal direction (cf. Fig ), has been named the baker s map. 64

72 7 Chaos in low dimensional systems y y y x x x Figure 7.12: Evolution of the cat of Fig under three successive iterations of the baker s map (7.63). the map is fully reversible. just like the cat map, the baker s map has a single, hyperbolic fixed point at x = y = 0. the tangent map is the same for any trajectory: ( ) 2 0 M = 0 1/2 (7.64) does not depend on the coordinates. The eigenvalues of the map are λ 1 = 2; λ 2 = 1/2 and the corresponding Lyapunov exponents σ 1 = ln 2 = σ 2. mixing: The map has the mixing property. Bernoulli shifts: Let x 0, y 0 be represented in binary notation as x 0 =.a 1 a 2 a i y 0 =.b 1 b 2 b i (7.65) where a i, b i = 0 or 1. A symbolic back-to-back representation of both coordinates can be written as X 0 = (x 0, y 0 ) = b i b 2 b 1.a 1 a 2 a i. (7.66) The first iteration, by doubling the x and halving the y produces X 1 = (x 1, y 1 ) = b i b 2 b 1 a 1.a 2 a i. (7.67) i.e. shifts the decimal point by one position to the right. 2 Bernoulli shift. This process is called a Now look at a coarse-grained description of the sequence {X n mod 1}, where the only information retained is the first digit after the decimal point. For typical (i.e. irrational) x 0, y 0 this will be an aperiodic sequence of zeros and ones, i.e. a sequence which is essentially equivalent to the tossing of a coin. Note that this totally random behavior has been obtained by coarse-graining of an entirely deterministic, reversible map. I will return to Bernoulli shifts in the next section, because they are a general feature of deterministic chaos. 2 Convince yourselves that this is so by looking separately at the cases a 1 = 0 and a 1 = 1! 65

73 7 Chaos in low dimensional systems The circle map. Frequency locking The circle map θ n+1 = θ n + Ω K sin θ n (7.68) is a one-dimensional non-area-preserving map; I introduce it here in order to describe the principle behind frequency locking. In addition, the map exhibits KAM characteristics, breaking of rational tori etc (Arnold). I will look at the winding number R = 1 ( ) 2π lim θn θ 0 n n. (7.69) In the integrable limit, K = 0, θ n θ 0 = nw and therefore R = Ω. For K 0 2 θ n+1 θ n = 1 K cos θ n 1 (7.70) 1 A'' Ω/K A A' θ Figure 7.13: Tangent bifurcation scheme for the fixed point of Eq. (7.71). Consider first the fixed point, θ n = θ n, corresponding to R = 0. From (7.68), this will happen if Ω K sin θ = 0. (7.71) Eq has no solutions if Ω > K, two solutions if Ω < K (one corresponding to a stable and the other to an unstable fixed point), and one solution if Ω = K. This behavior corresponds to a tangent bifurcation scenario (cf. Fig ). Note that the winding number R will now be equal to zero not just for Ω = 0, but for any Ω < K. An analogous situation occurs for R = 1/2, i.e. for a 2-cycle. In that case, the 2-cycle remains stable within a band Ω 1/2 < Ω < Ω+ 1/2, where Ω± 1/2 = π ± K2 /4, i.e. the band width is Ω 1/2 = K 2 /2. More generally, a rational winding number R = P/Q will lock in to that value for any Ω within a band of bandwidth Ω P/Q K Q. (7.72) This is the phenomenon of frequency locking. The total length of frequency locked intervals tends to zero as K 0. Note the analogy with the breaking of KAM-tori (Arnold); irrational winding numbers occupy most of available phase space in the slightly perturbed system. Stable frequency-locked intervals in the Ω K plane are shown in Fig For values K < 1 the winding number R will therefore exhibit frequency-locking steps at various rational values of R; between those steps, there will be intervals where R will 66

74 7 Chaos in low dimensional systems Figure 7.14: Frequency locking in the circle map. For any value of K > 0 there is a small band of Ω values, of width Ω P/Q, for which the winding number R locks to the rational value P/Q. As long as K < 1, the total measure of such intervals is less than 1. At K = 1, the total measure of locked-in frequency intervals is equal to unity. At values K > 1, bands corresponding to different rational ratios begin to overlap (resonance overlap). This is indicated by the dotted lines (from [17]). Figure 7.15: The complete devil s staircase at K = 1. Frequency locking takes place at all rationals.the inset shows that the staircase exhibits self-similarity at all scales (from [17]). depend linearly on Ω. This kind of behavior is represented pictorially by the incomplete devil s staircase. At K = 1 there are steps at all rational numbers and no portions with a finite slope. Frequency-locked intervals now have measure 1. This is the complete devil s staircase (Fig. 7.15). For values K > 1 chaos occurs as the resonance regions depicted in Fig begin to overlap. More details in [17]. 7.4 Topology of chaos: stable and unstable manifolds, homoclinic points The stable manifold of a cycle 3 is the set of points such if the forward map is started from one of them, further iterates will approach the cycle. Similarly, in the case of an invertible map, the unstable manifold of a cycle is the set of points such if the inverse (backward) map is started from one of them, further iterates will approach the cycle. 3 A fixed point of a map is a cycle of period 1; for systems with continuous dynamics it is straightforward to generalize statements on cycles and apply them to periodic orbits. 67

75 7 Chaos in low dimensional systems Stable and unstable manifolds cannot intersect themselves; however, they can intersect each other. If the manifolds belong to the same hyperbolic fixed point, their intersections are called homoclinic points. If the manifolds belong to different hyperbolic fixed points, the intersections are called heteroclinic points. Fig shows what happens at a homoclinic point X 0. Let X 1, X 2 be two successive iterates of the X 0 along the stable manifold (s). Consider now the neighboring points Y 0 on the unstable manifold (u). Because it is a predecessors of X 0 (follow the arrows!), its iterate Y 1 must find a place on the unstable manifold prior to X 1. In order to accommodate this requirement, the unstable manifold must fold. Now, as the hyperbolic fixed point P is approached, the distance between iterates along the stable manifold decreases. The folds of the unstable manifold must lie closer and closer to each other. Because of the area preserving property (the crossed areas in the figure should be equal), this makes the folds larger and larger in amplitude. Thus, if a single homoclinic point exists, an infinite sequence is generated. Fig (right panel) illustrates this in the case of the hyperbolic fixed point of the standard map. The complexity of the intersecting stable and unstable manifolds near hyperbolic fixed points lies at the heart of chaos in conservative systems. It was aptly recognized by its discoverer, Poincaré, with the fitting response the complexity of this figure will be striking, and I shall not even try to draw it p K= Figure 7.16: Homoclinic points in the vicinity of a hyperbolic fixed point. Left: a schematic view (cf. text); Right: the stable (red, thicker points) and unstable (black, thinner points) manifolds of the hyperbolic fixed point of the standard map at a low value of the nonlinearity parameter K = 0.5. For more details consult the excellent textbooks available, e.g. by Ott [18] or Tabor [19]. q 68

76 8 Solitons in scalar field theories 8.1 Definitions and notation Lagrangian, continuum field equations Starting point: classical discrete Lagrangian L = N i=1 { 1 2 I φ 2 i mgl(1 cos φ i ) 1 } 2 K(φ i+1 φ i ) 2, (8.1) Physical realization, e.g. coupled torsion pendula. Disks of radius l with moment of inertia I and an extra mass m on the periphery. Terms represent, respectively: rotational kinetic energy potential energy of mass in gravitational field potential energy of coupling Other physical realizations: arrays of Josephson junctions, one-dimensional magnets,... Equations of motion Continuum approximation I φ i = K(φ i+1 + φ i 1 2φ i ) mgl sin φ i (8.2) φ i±1 = φ(x i±1 ) φ(x i ) ± aφ (x i ) a2 φ (x i ) where a is the distance between neighboring disks (lattice constant). x i x (continuum space variable) leads to where c 2 0 = Ka 2 /I, ω 2 0 = mgl/i. c 2 2 φ 0 x 2 2 φ t 2 = ω2 0 sin φ (8.3) The Klein-Gordon class Eq. 8.3 is a member of the wider class of Klein-Gordon(KG) field equations c 2 0φ xx φ tt = ω0v 2 (φ) (8.4) where the on-site potential can be (examples) 69

77 8 Solitons in scalar field theories V (φ) = 1 2 φ2 original Klein-Gordon (linear, QM ca 1930) V (φ) = 1 8 (1 φ2 ) 2 known as φ 4 (displacive phase transitions, continuum version of Ising model) V (φ) = 1 cos φ known as Sine-Gordon (misnomer 1970, rhymes with Klein-Gordon); proposed earlier by Frenkel-Kontorova in context of dislocations, Skyrme in the 60s as a nonlinear field model for nucleons. Of interest for characterization of on-site potential: vacuum state, defined by V (φ 0 ) = 0 V (φ 0 ) > 0 (8.5) As defined (dimensionless) all examples have V (φ 0 ) = 0 and V (φ 0 ) = 1. Hence for small displacements from φ 0 V (φ φ 0 ) 1 2 (φ φ 0) 2. Note further that the 3 examples defined above have, respectively, 1. a single vacuum 2. two degenerate vacua 3. an infinite number of degenerate vacua. The field equations (8.4) can also be directly derived from the continuous Lagrangian L = A dxl where L(φ, φ x, φ t ) = 1 2 ( ) 2 φ 1 t 2 c2 0 ( ) 2 φ ω 2 x 0V (φ) is the Lagrangian density, and A defines the energy scale. In the SG example, A = I/a. Symmetries and Conservation laws Symmetries and conservation laws have been discussed in Section 1.4. In particular, the invariance of the Lagrangian density with respect to space and time leads, respectively, to the conservation of total momentum P and energy E. Furthermore, Lorentz invariance leads to the conservation of angular momentum, which in 1+1 dimensional systems is simply EX P t. 70

78 8 Solitons in scalar field theories In the special case of P = 0 (localized field configurations with vanishing total momentum), this implies that the center of energy remains fixed. This is the relativistic analog of the center-of-mass theorem of Newtonian dynamics. It turns out to be quite useful in soliton dynamics. Furthermore, Lorentz invariance implies that if φ(x) is a solution of (8.4), so is φ(γ(x vt)), where γ = (1 v 2 /c 2 0) 1/2 and v < c 0. In other words, any static solution can be Lorentzboosted. 8.2 Static localized solutions (general KG class) General properties The vacuum Note that the vacuum (or vacua) is always a solution of the equations of motion (8.4). Other solutions I look for static, localized solutions - which may then be Lorentz-boosted. point is c 2 0φ xx = ω 2 0V (φ) The starting or, in dimensionless form, d 2 φ dξ 2 = dv dφ (8.6) where d = c 0 /ω 0 and ξ = x/d. Multiplying both sides of (8.6) by dφ/dξ, I obtain [ (dφ ) ] 2 1 d 2 dξ dξ = dv dξ which has a first integral 1 2 ( ) 2 dφ = V + const. dξ For solutions which are localized, i.e. dφ lim ξ± dξ = 0 (8.7) and lim V = V (φ 0) = 0 ξ± the integration constant vanishes. A second integral can then be formally written as φ ξ ξ 0 = ± dφ 1 φ 0 2V (φ ). (8.8) 71

79 8.2.2 Specific potentials The linear KG case 8 Solitons in scalar field theories In the linear KG case, V (φ) = φ 2 /2 (8.8) becomes ξ ξ 0 = φ ± dφ 1 φ = ± ln φ or φ = e ±(ξ ξ 0) which, although it formally satisfies the original field equation, is not a localized solution in the sense of (8.7). Therefore it is not a physical solution. The φ 4 kink In the φ 4 case (8.8) becomes ξ ξ 0 = φ ± dφ (1 φ 2 ) = ±2 arctanh φ or ( ) x x0 φ K (x) = ± tanh 2d, (8.9) where x 0 = ξ 0 d. The upper sign corresponds to a kink, the lower to an antikink. Both solutions interpolate between the two degenerate vacua. The SG kink In the SG case (8.8) becomes or φ ξ ξ 0 = ± dφ 1 2(1 cos φ ) φ = ± dφ 1 2 sin φ 2 = ln tan φ 4 φ K (x) = 4 arctan exp{± x c 0t x 0 }. (8.10) d The solution with the upper sign interpolates between φ 0 = 0 and φ 0 = 2π, the one with the lower sign conversely. 72

80 8.2.3 Intrinsic Properties of kinks Topological charge 8 Solitons in scalar field theories Kinks (and antikinks) interpolate between distinct, degenerate vacua. They are known as topological solitons. The conserved quantity Q = dx dφ = φ( ) φ( ) (8.11) dx is known as topological charge. A φ 4 kink has topological charge 1, an antikink 1. A SG kink has topological charge 2π, an antikink 2π. Rest energy of a kink The total energy can be obtained from the Hamiltonian density { ( ) 2 1 φ H = A + 1 ( ) } 2 φ 2 t 2 c2 0 + ω 2 x 0V (φ). (8.12) For a static kink, the first term is zero, and the second and third terms are equal (cf. above). Thus ( ) 2 φ E K Mc 2 0 = Ac 2 0 dx x = Ac 2 1 ( ) 2 φ 0 dξ d ξ or M = A d = Ac d φ2 φ2 φ 1 dφ φ ξ φ 1 dφ 2V (φ). (8.13) Note that we do not need the explicit form of the kink solution in order to calculate the rest mass. In the case of the φ 4 field, this gives M = 4A/3d. In the case of the SG field, M = 8A/d. Energy and momentum of a moving kink: classical wave-particle duality The energy of the moving kink where φ K (γ(x vt)) 1 γ = 1 v2 c 2 0 can be directly computed from the full Hamiltonian density. It is The momentum can be computed from (1.49) and is equal to E(v) = Mc 2 0γ. (8.14) P (v) = Mγv. (8.15) 73

81 8 Solitons in scalar field theories The energy-momentum relationship is also satisfied. E 2 = P 2 c M 2 c 4 0 The fact that kink and antikink solutions satisfy the usual relativistic kinematic relations which ordinarily hold for mass points suggests that these classical localized fields may, for many practical purposes, be effectively treated like particles. The remarkable property of particle-wave duality at a classical level suggests that soliton-bearing classical Lagrangians might be good candidates for the construction of nonlinear quantum field theories Linear stability of kinks Consider small displacements around a static kink solution of the KG class. The total spaceand time-dependent field is written as φ(x, t) = φ K (x) + χ(x, t) (8.16) where χ will be regarded as a small quantity. Keeping only linear terms in χ leads to Using a separation of variables ansatz c 2 0χ xx χ tt = ω 2 0 V (φ K ) χ. χ(x, t) = j α j e iωjt f j (x) (8.17) leads to the eigenvalue equation d2 f j dξ 2 + V (φ K )f j (ξ) = Ω 2 jf j (ξ) (8.18) where I have again introduced the dimensionless space variable ξ = x/d, and Ω j = ω j /ω 0. Eq. (8.18) is has the form of a Schrödinger equation. The effective potential is the second derivative of V, evaluated at φ = φ K. Because φ K asymptotically approaches the vacuum field values, the effective potential (Draw!) approaches V (φ 0 ) which with our conventions has the value 1. Possible eigenstates of (8.18) may then be localized (bound), with Ω 2 < 1 or extended (scattering) states, with Ω 2 > 1. Linear stability requires that Ω 2 j 0. (8.19) Bound states. The zero frequency (Goldstone) mode The function f 0 = dφ K dξ is always an eigenstate of (8.18), associated with the eigenvalue 0. noting that satisfying φ K,ξξξ + V (φ K )φ K,ξ = 0 (8.20) One can see this by 74

82 8 Solitons in scalar field theories is equivalent to satisfying d dξ [ φ K,ξξ + V (φ K )] = 0. But the brackets are identically zero for a kink solution. The zero-frequency (or translational) mode, named after Goldstone, reflects the invariance of the kink solution under translations. Note in this context that the integration constant ξ 0 (cf. above) does not enter the expression for the rest-energy. A kink (or antikink) can be translated in space at zero energy cost. A kink solution centered at zero has the same energy with a kink solution centered at α. If α is small, the latter can be obtained from the former by Taylor expansion φ K (ξ α) φ K (ξ) α dφ K dξ which is why dφ K /dξ must be an eigenstate of (8.18). The Goldstone mode must be the eigenstate with the lowest Ω 2 j value. One can see this from the fact that, since kinks are monotonic functions in space, dφ K /dξ has no nodes. Other bound states may or may not exist, depending on the details of the effective potential. For example, the SG kink has no further bound states. The φ 4 kink has a further localized mode, with an internal oscillation frequency... and an eigenfunction... Scattering states. Phase shifts In general, scattering states consist of an incident, a transmitted and a reflected wave. The effective potentials which correspond to both the SG and the φ 4 kink have the special property that they are reflectionless. In other words, the eigenfunctions corresponding to extended states with frequencies Ω 2 q = 1 + q 2 (8.21) have the asymptotic form lim f q(ξ) e iqξ±iδ(q)/2. (8.22) x± The total phase shift δ(q) describes the net effect of the interaction between an incident phonon plane wave and a static kink. Note that the above property is asymptotic. The exact form of extended eigenstates may be significantly distorted in the neighborhood of the kink. For example, in the SQ case from which follows. f q (ξ) = (iq + tanh ξ) e iqξ, δ(q) = 2 arctan 1 q 8.3 Special properties of the SG field The Sine-Gordon breather The SG field equations admit a family of special localized solutions with an internal oscillation [ ] sin ω(t t 0 ) φ br (x, t) = 4 arctan ρ (8.23) cosh[(x x 0 )/λ] 75

83 8 Solitons in scalar field theories 7 µ=π/ µ=π/ φ 1 0 φ x/d x/d Figure 8.1: Left: multiple snapshots of a SG breather with intermediate amplitude. Right: a very slow breather with µ = π/2.001 which looks like a bound kink-antikink pair (of course if you observe the very slow oscillation over an extremely long period of time you will note the periodic motion). The snapshots are taken at times which are very far apart from the point of view of the laboratory observer: ±π/(2ω 0 cos µ) and ±π/(16ω 0 cos µ). where ω = ω 0 cos µ, ρ = tan µ, λ = d/ sin µ and 0 < µ < π/2. The constants x 0, t 0 are arbitrary and can be shown to generate two Goldstone modes (cf. above), related, respectively, to spatial and temporal translations. The solution is known as a breather and the form shown is in its rest frame. It can be Lorentz-boosted by applying the Lorentz transformations. The rest energy of a breather is E 0 br = 2Mc 2 0 sin µ. (8.24) In the limit of µ 1 the breather reduces to a phonon. In the limit of µ π/2 the frequency of oscillation approaches zero, the energy approaches Mc 2 0, and the breather looks very much like a bound kink-antikink pair. The breather is a singular feature of the SG field theory. Continuum field equations with other potentials of the KG class do not exhibit time-periodic, spatially localized solutions Complete Integrability The SG field equation with decaying boundary conditions can be completely integrated using the inverse scattering transform. Details in... 76

84 9 Atoms on substrates: the Frenkel-Kontorova model The Frenkel-Kontorova (FK) model [20] is an attempt to describe structures of adsorbed layers which have to reflect two competing periodicities, that of the substrate and that of the adatoms. The total potential energy is assumed to be Φ = C ( (x n+1 x n a) 2 + V 0 1 cos 2πx ) n, (9.1) 2 b n where x n is the position of the nth atom, a, b are the natural periodicities of adatoms and substrate, respectively, and C, V 0 are material constants denoting the strength of the two potentials. The first term in (9.1) describes the harmonic interactions between the adatoms, whereas the second term models the periodic template provided by the substrate. I will use a dimensionless description of all relevant quantities. Let δ = (a b)/b be the mismatch between the two competing length scales; let further n x n = bn + bφ n (9.2) denote the breakup of the displacement of the nth atom into a part which follows the substrate and a rest. The dimensionless potential energy is then given by ˆΦ = Φ Cb 2 = 1 2 (φ n+1 φ n δ) (2πλ) 2 (1 cos 2πφ n ), (9.3) n where λ = ( Cb 2 /V 0 ) 1/2 /(2π) is the dimensionless coupling constant. The equilibria of (9.3), defined by are given in terms of the second-order recurrence equations n ˆΦ φ n = 0 n, (9.4) φ n+1 + φ n 1 2φ n = 1 2πλ 2 sin 2πφ n. (9.5) Eq. (9.5) is equivalent to the two-dimensional standard map discussed in Section This means that we should in general expect to find ground and metastable states of (9.1) which exhibit all the complexity encountered there - and discussed in the general context of a dynamical system - where the index n stood for a discrete time. In particular, we expect to find phase locking. i.e. adatoms and substrate locked into lattice periodicities whose ratios are rational numbers. Furthermore, we expect to find metastable, chaotic configurations at higher values of the nonlinearity (low values of the coupling constant λ). We begin by looking at the absolute minimum of (9.3) for weak nonlinearities and, more specifically, at the first transition between a commensurate and an incommensurate phase. In order to do this, we will always compare the energy of a local extremum defined by (9.5) with the energy ˆΦ 0 = 1 2 Nδ2 (9.6) 77

85 9 Atoms on substrates: the Frenkel-Kontorova model of the reference state φ n = 0 n where N is the total number of substrate atoms. 9.1 The Commensurate-Incommensurate transition The continuum approximation If the coupling is strong, λ 1, it is possible to make a continuum approximation φ n φ(n); in this case (9.5) becomes, to leading order, d 2 φ dn 2 = 1 sin 2πφ, (9.7) 2πλ2 which is known as the Sine-Gordon equation (cf. Chapter 8). Eq. 9.7 has a first integral ( ) 2 dφ = 1 dn 2π 2 ( cos 2πφ + const), λ2 which can be rewritten, by setting the constant equal to 1 + 2ɛ and taking the square root, as dn dφ = ± 1, (9.8) g(φ) where g(φ) = 1 sin 2 πφ + ɛ πλ. (9.9) Eq. 9.8 can be integrated again, in the form n ν = ±J(φ) = ± where ν is a further constant of integration 1. φ 0 d φ g( φ) (9.10) The total energy: an intermediate result The total energy associated with the solution (9.10) is { ( ) 2 1 dφ ˆΦ ɛ = dn + 1 (1 cos 2πφ) 2 dn (2πλ) 2 } δ dn dφ dn Nδ2 or, measured from the reference energy (9.6), ˆΦ ɛ = ˆΦ ɛ ˆΦ 0 = φ2 φ 1 dφg(φ) ɛ 2π 2 λ 2 N δ(φ 1 φ 2 ), (9.11) where φ 2,1 = lim n ± φ(n), and I have also used the fact that φ(n) is a monotonic function of n (cf. below). 1 A further substitution k 2 = (1 + ɛ) 1, χ = (φ 1/2)π and u = (n ν)/(kλ) transforms (9.10) to u = ±F (k, χ) where F (k, χ) is the elliptic integral of the second kind. The latter equation can be formally inverted as χ = am(k, ±u) where am is the elliptic Jacobian amplitude[21]. In these lectures I will follow [22] and present a noprerequisites description of the C-I transition. 78

86 9 Atoms on substrates: the Frenkel-Kontorova model The special case ɛ = 0: kinks and antikinks If ɛ = 0, (9.8) admits soliton solutions of the kink/antikink type, φ(n) = 2 π arctan e±(n ν)/λ. (9.12) The total energy of a kink (or antikink) is, according to (9.11), ˆΦ kink = 2 π 2 λ δ, (9.13) where I have used ɛ = 0 and g(φ) = (πλ) 1 sin πφ. Note that the energy is negative, i.e. the kink is formed spontaneously, if δ > δ c = 2 π 2 λ. (9.14) The general case ɛ > 0: the soliton lattice Let me now look at some general properties of (9.10). In the following, I will choose the upper sign; the analysis can be inverted for the lower sign. Note first that the integrand is positive, therefore J(φ) is a monotonic, and hence invertible function. This is formally expressed by φ(n) = J 1 (n ν). (9.15) Furthermore, since g(φ) = g(φ + 1), it follows that J(φ + 1) = φ 0 d φ g( φ) + 1 φ+1 φ d φ g( φ) d = J(φ) + φ 0 g( φ) = n ν + L (9.16) where L is defined as the value of the definite integral in the second line 2 1/2 d L = 2 φ 0 g( φ). (9.17) Consequently, φ(n) + 1 = J 1 (n ν + L) = φ (n + L) (9.18) i.e., each time the index n is increased by L, the field variable φ, which measures the deviation from the reference phase - the phase perfectly matched to the substrate -, increases by one. In the limit ɛ 1, the dominant contribution to L comes from φ near zero. It is possible to obtain the leading-order contribution to L by approximating g(φ) 1 λ max(φ, φ c) where φ c = ɛ 1/2 /π. This results in L λ ln ɛ A (9.19) 2 In the elliptic integral notation of the previous footnote L = 2λkK where K = F (k, π/2) is the complete elliptic integral of the first kind. 79

87 9 Atoms on substrates: the Frenkel-Kontorova model to leading order in ɛ. A is a numerical constant of order unity. A direct consequence of (9.18) is that φ can be written in the form φ(n) = n ν L + ψ(n ν) (9.20) where the first term denotes the average change in φ, and the second is a periodic function of n ψ(n + L) = ψ(n) (Fig. 9.1). The type of solution described by (9.20) is known as the soliton lattice. It corresponds to a regular sequence of domains of L sites commensurate with the substrate, interrupted by local discommensurations λ=6 ε=e -2 ψ φ n Figure 9.1: The soliton lattice. The continuous curve represents the monotonic function φ, which is a sum of a straight line with slope 1/L and a periodic function with periodicity L (cf. Eq. (9.20) ). Energy of the soliton lattice The total energy of the soliton lattice - always measured relative to the reference state of the commensurate state - consists, according to (9.11), of three terms. All contributions are of order N. I can use the fact that φ 1 = 0 and φ 2 N/L + O(1) (cf. (9.20) ) to write the energy in the form ˆΦ ɛ = N 1 dφ g(φ) N ɛ L 2π 2 λ 2 N L δ. (9.21) 0 The soliton lattice energy (9.21), regarded as a function of the - still undetermined - constant ɛ, has an extremum at a value determined by ˆΦ ɛ ɛ = N ( L L 2 δ ɛ 1 0 ) dφ g(φ) = 0. Since the first factor is nonzero at all ɛ > 0 - and in fact diverges as ɛ 0 - the above condition can only be satisfied if 1 0 dφ g(φ) = 1 πλ 1 0 dφ sin 2 πφ + ɛ = δ. (9.22) 80

88 9 Atoms on substrates: the Frenkel-Kontorova model The above condition can be used to determine the value of ɛ = ɛ(δ) which, for a given mismatch, gives an extremum of the soliton lattice energy. In order to determine the nature of the extremum we must first look at the second derivative 2 ˆΦɛ ɛ 2 = 1 N 2(πλ) 2 L ɛ= ɛ(δ) L ɛ. ɛ= ɛ(δ) In view of (9.19), the sign of the second derivative is positive. The local minimum thus determined will always have a lower energy than the reference state, by an amount (noting that the first and the third terms in (9.21) cancel out). ˆΦ ɛ = N ɛ 2π 2 λ 2 (9.23) It should however be noted that (9.22) has a solution for ɛ only for such δ > δ c, where the critical value of δ is the same as that derived in the context of the energetic stability of the single kink. In other words, at mismatches δ < δ c, the commensurate state will still be favored. Once this critical value is exceeded however, not only becomes the spontaneous creation of a single kink energetically possible, but the whole structure of a soliton lattice - a new, incommensurate phase -, acquires a macroscopic energetic advantage and is formed spontaneously. Relationship between ɛ and δ In order to to derive the explicit relationship between ɛ and δ, I must go back to (9.22). For notational simplicity let me from now on drop the bar from the ɛ. I first note that using the general expression dδ dɛ = 1 2π 2 λ 2 L(ɛ) ; δ δ c = and the leading-order result (9.19), I obtain or, in reduced dimensionless form, ɛ 0 dɛ dδ dɛ δ δ c = 1 2π 2 λ ɛ ln ɛ Ae δ δ c δ c Discommensurations repel each other = 1 4 ɛ ln ɛ Ae. (9.24) Using this relationship, it is possible to express the energy of the incommensurate phase per discommensuration as ˆΦ ɛ N/L = 1 2π 2 λ ɛ ln ɛ A = (δ δ c ) + δ c 4 ɛ = (δ δ c ) + 8 π 2 λ e L/λ. (9.25) The first term in the last line is exactly the energy (9.13) of an isolated kink. The second term, which is always positive, expresses the repulsive energy of interaction between neighboring discommensurations. 81

89 9 Atoms on substrates: the Frenkel-Kontorova model The mean interatomic spacing The mean interatomic spacing, defined by x n x 0 ā = lim n n can now be calculated for the incommensurate phase. It is equal to ( ā = b ), L which corresponds to a winding number, r = ā b 1 1 λ ln δ δc δ c (9.26) to leading order. Note that, as the mismatch approaches the critical value from above, the mean spacing approaches b continuously. A singularity at critical mismatch appears in the second order derivative of the energy with respect to the length L = Nā (check this!, exercise). The commensurate-incommensurate transition is a therefore a second order phase transition in the language of statistical mechanics. Free vs. fixed-end boundary conditions I have up to now considered free-end boundary conditions. In other words: given the material parameters (coupling constant λ and mismatch δ) we look for the energy minimum, which in turn determines the winding number (9.26). It is of course possible to consider fixed-end boundary conditions, in which the positions of the end atoms are held fixed. More precisely, the relevant quantity for a system of N atoms is the difference φ N φ 0. Holding it constant corresponds to fixing the winding number, i.e. the density of discommensurations L/N. The parameter ɛ is then determined by (9.19). The energy can be directly computed from (9.21) and the end result has exactly the form of the last line in (9.25). The interpretation is also the same: the soliton lattice has an energy which consists of contributions of the individual discommensurations and of an interaction part, arising from the mutual repulsion of neighboring discommensurations. Note however that since we are in effect fixing the soliton lattice, the expression for the energy is valid for any value of the misfit parameter. If δ < δ c this means that the extra positive energy must be supplied in order to maintain the fixed-end boundary conditions. Phasons The soliton lattice has a further important property which I have not discussed up to now. Its energy, (9.25), is independent of the integration constant ν, defined in 9.10). This means that the whole soliton lattice configuration can be translated by an arbitrary amount without an energy cost. As has been discussed in Section in the context of single kinks, this translational invariance implies the existence of a zero-frequency (Goldstone) mode in the spectrum of linearized excitations around the exact soliton lattice configuration. Let me examine this in some detail: Up to now we have only looked at equilibrium properties which are determined by the minima of the total potential energy (9.3). The dynamics of the FK model is governed by the equations of motion φ n = φ n+1 + φ n 1 2φ n 1 2πλ 2 sin 2πφ n, (9.27) 82

90 9 Atoms on substrates: the Frenkel-Kontorova model or, in the continuum approximation, where the time is measured in units of (m/c) 1/2. 2 φ t 2 2 φ n 2 = 1 2πλ 2 sin 2πφ n, (9.28) Linearization of (9.28) around the static soliton lattice configuration φ s (n ν) = (n ν)/l + ψ(n ν), φ(x, t) = φ s (n ν) + q e iωqt f q (n), (9.29) leads to a Schroedinger-like equation for the f q s d2 f q dn λ 2 cos(2πφ s) f q = ω 2 qf q. (9.30) The effective potential of (9.30) has the periodicity L of the soliton lattice. Consequently, the Bloch/Floquet theorem applies to the eigenfunctions: f q (n) = e iqn F q (n) where F q (n) = F q (n + L) (9.31) Now the eigenfunction corresponding to the Goldstone mode is dφ s dn = 1 L + ψ (n ν) and is therefore periodic in n with period L. By comparison with (9.31) we conclude that it must correspond to q = 0 and that F 0 (n) = dφ s /dn. Note the contrast with the situation encountered in the context of localized kinks; in that case the zero-frequency mode was a discrete state in the spectrum. Now it is part of a band. In fact, the spectrum ω q consists of (i) a low-q region, starting at zero and reaching out to π/l, with a linear dependence of ω q, and (ii) a high-q region, i.e. at wavelengths shorter than the the distance L between discommensurations, which is dominated by the short-range properties of the soliton lattice, which are effectively those of the commensurate phase. Region (i) gives rise to the so called phason branch of excitations, region (ii) to the optical phonon branch of the commensurate phase. The two branches are separated by a frequency gap. 9.2 Breaking of analyticity The treatment of the FK model up to now has been based on the continuum approximation, which will break down at values of λ 1. It is then necessary to treat the equilibria of the potential energy in terms of the second-order recurrence equation (9.5), i.e. to go back to the standard map of Section When applying results from that section, it should be noted that the correspondence is q n φ n 1/2 and K λ 2. The next section will mostly treat the case of fixed end points. The implications for the physically relevant case of free boundary conditions will be treated somewhat heuristically. 83

91 9 Atoms on substrates: the Frenkel-Kontorova model FK ground state as minimizing periodic orbit of the standard map Rational winding numbers If the ends are held so that the average interatomic distance is a rational multiple of the substrate periodicity (in the language of the standard map this corresponds to a rational winding number), i.e. φ N φ 0 = w = r, N s where r, s are integers, the energy minimum (ground state) will be an (r, s) orbit of the standard map, i.e. an s cycle such that φ s+1 φ 1 = r. This corresponds to a commensurate state and holds for any value of the nonlinearity. Irrational winding numbers The nontrivial part is of course what happens for irrational values of the winding number w. In this case, Aubry [23] has proved the following fundamental results: for K < K c (corresponding to λ > λ c = Kc 0.5 = ), the ground state is quasiperiodic, of the form φ n = t n α + u(t n α) (9.32) where t n = wn and the hull function is periodic with period 1, u(t) = u(t + 1) and analytic in t. The ground state of the FK chain thus corresponds to a torus trajectory of the standard map. This result effectively generalizes (9.20). As K approaches K c, and successive KAM tori break, the hull function becomes more and more bumpy (cf. Fig. 7.8). Aubry describes this behavior as a phase transition due to breaking of analyticity. Indeed, for K > K c (corresponding to λ < λ c = Kc 0.5 = ), the ground state is still quasiperiodic, given uniquely by (9.32). However, it now corresponds to a cantorus of the standard map (cf. Fig. 7.10). Accordingly, the corresponding hull function develops discontinuities (Fig. 9.2, left and center panels) u u+t 0.4 ω t t Figure 9.2: Left panel: the hull function, as represented by an 89-cycle rational approximation of a cantorus for K = K c Center panel: the function u(t) + t allows -within this finite approximation- a better view of the gaps. Right panel: the phonon spectra corresponding to this ground state. Note that the minimal frequency is nonzero (inset). j 84

92 9 Atoms on substrates: the Frenkel-Kontorova model Small amplitude motion Small amplitude motion around the ground state is described by linearizing the equations of motion (9.27) around the ground state, i.e. φ n (t) = φ GS n + q f n (j) e iω jt which leads to the eigenvalue equation where n h mn f (j) n h mn = = ω 2 j f (j) m (9.33) 2 ˆΦ φ n φ m (9.34) and the second derivatives are evaluated at the ground state positions defined by (9.32). The eigenvalues of the Hessian matrix correspond to the squares of the eigenfrequencies of local oscillations (phonon spectra). We have seen in section and in the beginning of the present chapter that all trajectories of the standard map correspond to local extrema of the potential energy function ˆΦ. Local extrema can be classified according to the properties of the eigenvalues of (9.33). If a single eigenvalue is negative, the extremum is unstable (maximum in at least one direction). If all eigenvalues are positive it is a local minimum, i.e. in general a metastable configuration. A zero eigenvalue (Goldstone mode) indicates an invariance of the energy with respect to a particular motion. We saw an example of this in the case of the soliton lattice - which was a stable configuration (a ground state of the FK chain) in the continuum approximation. In fact the spectra of torus-like ground states obtained below the threshold of analyticity breaking always include such a zero-frequency mode, indicating that the total energy is invariant with respect to a change of phase; the whole arrangement can slide freely. Above that threshold, the atomic arrangement of adatoms becomes pinned. Pinning is reflected in the phonon spectrum which develops a gap near zero frequency (cf. Fig. 9.2, right panel). Peyrard and Aubry [24] performed extensive numerical studies of the transition by breaking of analyticity and demonstrated that the gap frequency vanishes as the threshold is approached. More specifically, they determined that the minimal frequency scales as a power of K K c, ω min (K K c ] χ where 1 < χ < Free end boundary conditions The picture which emerges for the physically relevant case of free-end boundary conditions is the following: The average interparticle distance ā as a function of the misfit parameter is characterized by locked-in, flat regions, at rational winding numbers, which correspond to commensurate phases. As long as the nonlinearity is below threshold, these flat regions are interrupted by intervals of continuous variation of ā with δ, which correspond to irrational winding numbers (incommensurate phases); the function ā(δ) forms an incomplete devil s staircase. At the threshold of analyticity breakup, there are steps at every rational values of the ratio ā/b, separated by discontinuities. This is the complete devil s staircase (cf. the similar analysis of phase locking in the case of the circle map in section 7.3.8). Numerical results [25] are summarized in Fig

93 9 Atoms on substrates: the Frenkel-Kontorova model Figure 9.3: Left panel: Phase diagram (winding number vs. misfit parameter) for the FK model. The numbers represent values of the locked-in winding number. Unlabeled regions contain additional structure. Right panel: winding number as a function of misfit parameter for the FK model at K = 1, showing a devil s staircase structure (from [25]). 9.3 Metastable states: spatial chaos as a model of glassy structure Chaotic trajectories of the standard map - which proliferate beyond the threshold of analyticity breakup - have a special significance in the context of the FK model. A lot of them correspond to unstable extrema. On the other hand, a great number of them (of order e N ) represent local minima, i.e. metastable states. The number and the energy distribution of Figure 9.4: Left panel: The number of metastable equilibrium states vs. their energy per site (measured from the ground state energy). Bands are shown for K = 5 (6 upper segments) and K = 2 (5 lower segments); horizontal dashed lines show the border between energy bands. Right panel: Band energy spectrum of the same equilibria vs. K (upper scale) and the phonon gap (lower scale); bands are marked by filled areas corresponding to a given K value (from [26]). these metastable states have been recently computed [26]. They were found to bundle in energy bands. The left panel of Fig. 9.4 shows the bands and their populations for K = 2 and K = 5 and a golden mean winding number. The energies lie very close to the ground state. For example, in the case of K = 2, the lowest energy band has an energy per site of (in dimensionless units). The right panel shows the relationship of energy bands to the ground state. Since the ground state is a cantorus, it is characterized by a nonzero Lya- 86

94 9 Atoms on substrates: the Frenkel-Kontorova model pounov exponent - which is itself proportional to the minimal frequency of small oscillations found in the previous section. Small perturbations in the displacements of the boundary sites - while still compatible with the fixed winding number - generate an exponentially large number of extra trajectories branching out of the cantorus with energies exponentially close to that of the cantorus. The type of energy landscape described above is characteristic of disordered condensed matter systems, such as glasses or globular proteins. In this sense, the FK model provides considerable insight regarding the interplay of nonlinearity and disorder. 87

95 10 Solitons in magnetic chains 10.1 Introduction Anisotropic exchange interactions between localized magnetic moments cause many magnetic materials to assume an effectively one-dimensional character. Within a certain range of temperatures, interactions along chains of magnetic atoms can be far more significant than interactions across chains. It is then possible to describe material properties (statics and dynamics) in terms of a one-dimensional Hamiltonian H = J n S n S n+1 + A n (Sn) z 2 µ B S n. (10.1) Here, S n is the spin which resides at the n-th magnetic site, J the exchange interaction (positive for a ferromagnet, negative for an antiferromagnet), A the anisotropy ( easyplane - if A > 0 and easy-axis - if A < 0), and B the external magnetic field. The magnetic moment of the n-th spin is, following the standard notation, equal to µ S n. A typical example of a magnetic chain capable of supporting nonlinear excitations is CsNiF 3 in the paramagnetic regime T > T Neel = 2.65K, with parameters S = 1, J/k B = 23.6K, A/k B = 4.5K, µ = gµ B with a gyromagnetic ratio g = 2.28 (µ B = is the Bohr magneton) [27]. For some applications it is possible to neglect the explicit quantum nature of the spin operators. In this case, S n will be treated as a classical spin vector of length S (rather than the technically correct [S(S + 1)] 1/2. n 10.2 Classical spin dynamics Spin Poisson brackets Spin is an intrinsically quantum phenomenon. The way to deal with it at a classical level is by associating an appropriate Poisson bracket algebra directly with spin angular momentum vectors { I n } f g {f, g} = ɛ αβγ In α In γ (10.2) where ɛ αβγ is the antisymmetric Levi-Civita tensor (=1 if αβγ is a cyclic permutation, - 1 if anticyclic, and zero otherwise) and the Einstein summation convention over repeated symbols is implied. A special case of (10.2) is where δ mn = 1 if m = n and 0 otherwise. I β n {I α m, I β n} = ɛ αβγ δ mn I γ n (10.3) The Poisson bracket algebra defined above can be used to generate Hamiltonian dynamics I n α = {In α, H} = ɛ αβγ In β H I γ n, (10.4) 88

96 10 Solitons in magnetic chains or, in vector form, I n = I n H I n. I will mostly deal with dimensionless spin vectors defined as S n = I n / h; for these, the equations of motion take the form S n = 1 h S n H S. (10.5) n Note that the Poisson brackets (10.3) can be obtained from the standard spin commutation relations by using the correspondence principle. Note further that, independently of the details of the spin Hamiltonian, according to (10.5), the norms of all vectors { S n } remain constant in time. Introducing the one-dimensional Hamiltonian (10.1) into (10.5) results in S n = J h S n ( S n+1 + S n 1 ) 2 Ā h ( S n ẑ) S n ẑ γ B S n (10.6) where µ = hγ and ẑ is a unit vector in the z-direction. We will deal with various special cases of (10.6), which governs the nonlinear dynamics of a broad class of one-dimensional spin chains An alternative representation The polar form of the spin angular momentum vector I n of length I I x n = I sin θ n cos φ n In y = I sin θ n sin φ n In z = I cos θ n (10.7) can be used to provide an alternative representation of spin dynamics. The transformation P n = In z = I cos θ n ( ) I y q n = arctan n = φ n (10.8) I x n can be shown to be canonical, i.e. it preserves Poisson brackets: f g ɛ αβγ In α In β In γ = [ f g f ] g q n n P n P n q n (10.9) holds for any pair of functions f, g. The P s and q s are canonically conjugate sets of coordinates and momenta, in the sense that {q m, P n } = δ mn (10.10) (with any of the two expressions of Poisson brackets). The two representations are equivalent - and so are the resulting dynamics. In the polar representation the dynamics takes the usual symplectic form q n = {q n, H} = H P n P n = {P n, H} = H q n. (10.11) 89

97 10 Solitons in magnetic chains Again, I will use the dimensionless polar variable the equations of motion then take the form p n = P n / h = S cos θ n ; 10.3 Solitons in ferromagnetic chains The continuum approximation h q n = H p n hṗ n = H q n. (10.12) If the exchange constant J in (10.1) is positive, and in the absence of anisotropy and external fields, spins will tend to a parallel ordering. This is exactly true at zero temperature. It defines a ferromagnetic ground state Sn z = S, Sn x = Sn y = 0 1. At reasonably low temperatures, where thermal motion does not prevail, it is plausible to assume that spin orientations do not vary wildly from site to site. Thus, although the spin vector may be far from the reference state (0, 0, 1), it will still make sense to write down a continuum approximation. This approximates sites n by a continuous index variable, n x, and individual spins by a continuum field, S n S(x). Sums over n are approximated by integrals, with the rule dx a n where a is the lattice constant. Spins at neighboring sites can be obtained by Taylor expansion S n±1 S(x ± a) S(x) ± a S (x) a2 S (x) + where the primes denote derivatives with respect to x. According to the above rules, S n S n+1 S(x) 2 + a S(x) S (x) a2 S(x) S (x). The first term is the constant norm of the spin vector field. The second term is proportional to the derivative of the constant norm, therefore vanishes. The third term can be integrated by parts over all space (the contribution from the boundary vanishes identically). The resulting continuum version of the Hamiltonian (10.1) is H = 1 2 Ja dx ( S ) 2 + A x a dx Sz 2 µ B a The spin equations of motion (10.6) reduce, in the continuum limit, to S = Ja2 h S S 2Ā h dx S. (10.13) ( S ẑ ) ( S ẑ ) γ B S. (10.14) Comment: (10.14) may also be obtained by taking the continuum limit of (10.4). correspondence is δ S a n δs(x) 1 Note that the choice of the z direction is at this stage - with the full rotational invariance of the exchange interaction - entirely arbitrary. The 90

98 10 Solitons in magnetic chains where the right-hand side corresponds to a functional derivative. (10.4) becomes which, if we insert the functional derivative reproduces (10.14). δh S(x) = ā S(x) h δh δs(x) δ S(x) = Ja S + A a ( S ẑ ) ẑ µ a B, In what follows, I will also need the continuum limit of the alternative (polar) spin representation. p(x) = S z (x),, q(x) = arctan S y S x The Hamiltonian (10.13) can be transformed directly to the polar variables if we note that S x 2 + S 2 (pp ) 2 y = S 2 p 2 + (S2 p 2 )q 2. The result is H = 1 { } S 2 2 Ja dx S 2 p 2 p 2 + (S 2 p 2 ) q 2 + A dx p 2 µ a a B dx S 2 p 2 cos q, (10.15) where I have chosen to take the x-axis of the spin vector parallel to the magnetic field The classical, isotropic, ferromagnetic chain The isotropic ferromagnetic chain is described by the Hamiltonian (10.1) with A = 0. I will choose the z direction along the magnetic field. The classical spin dynamics in the continuum limit are described by (10.14). In polar canonical variables these transform to ṗ = Ja2 h q = Ja2 h [ (S 2 p 2 )q ] [ ] S 2 p S 2 p 2 + S2 p p 2 (S 2 p 2 ) 2 + p q 2 µb h. In the following I will use dimensionless units, i.e. measure lengths in units of the lattice constant and times in units of h/(js). The magnetic field will be denoted by the dimensionless quantity b = µb/(js). Finally, I set p Sp. The equations of motion in dimensionless form are ṗ = [ (1 p 2 )q ] q = p 1 p 2 p p 2 (1 p 2 ) 2 p q 2 b. (10.16) 91

99 10 Solitons in magnetic chains Soliton solutions I look for bounded propagating solutions of the type p = p(x vt) q = Ωt + ˆq(x vt) (10.17) where the extra term allows for an overall precession around the z axis. In addition to boundedness, the solution should decay at infinity, and approach the ferromagnetic ground state p 1 The equations of motion transform to the following system of coupled ODEs: vp = (1 p 2 )ˆq 2pp ˆq = [ (1 p 2 )ˆq ] Ω vˆq = p 1 p 2 p p 2 (1 p 2 ) 2 p (ˆq ) 2 b (10.18) We note that the upper equation has a first integral, which we write as ˆq = v(p p 0) 1 p 2 (10.19) - where p 0 is a constant to be chosen later - and use to eliminate ˆq from the lower equation. After rearranging some terms I obtain p 1 p 2 + p p 2 (1 p 2 ) 2 = v2 (p p 0)(1 p 0 p) (1 p 2 ) 2 ˆΩ where ˆΩ = Ω + b. Multiplying this by 2p produces a complete derivative on the left-hand side: ( ) p 2 1 p 2 = 2v 2 (p p 0)(1 p 0 p)p (1 p 2 ) 2 2ˆΩp [ ] p = 2v p 0 p + 1 2(1 p 2 2ˆΩp ) which can be integrated to give where p 1 is a new integration constant. p 2 1 p 2 = v2 p2 0 2p 0 p + 1 2(1 p 2 2ˆΩp + p 1 ) Now the requirement of boundedness, applied to the derivative ˆq as p 1, demands (cf. (10.19)) that p 0 = 1. As a consequence, [ (p ) 2 = (1 p) (1 + p)(p 1 2ˆΩp) v 2]. (10.20) An analytically favorable choice of the integration constant p 1 can be made by demanding that the brackets vanish at p = 1. This means taking p 1 = 2ˆΩ + v 2 and results in ( ) 2 dp [ = (1 p) 2 2ˆΩ(1 + p) v 2]. (10.21) dx 92

100 10 Solitons in magnetic chains Note that, in order for the right-hand side to be positive at least for some values of p, the conditions ˆΩ > 0 and v 2 < 4ˆΩ must hold. I therefore set v = 2ˆΩ 1/2 cos α 2 (10.22) and obtain 1/2 dx 2ˆΩ dp = ± 1 (1 p)(p cos α) 1/2 which can be formally integrated as 2ˆΩ 1/2 (x x 0 ) = p cos α = 21/2 sin α 2 1 d p (1 p)( p cos α) 1/2 [ ] (p cos α) tanh 1 1/2 2 1/2 sin α 2 or, after some rearrangement, p(x) = 1 2 sin 2 α {ˆΩ1/2 2 sech2 sin α } 2 (x vt x 0),. (10.23) Inserting (10.23) in (10.19) gives ˆq v/2 = 1 sin 2 {ˆΩ1/2 } α 2 sech2 sin α 2 (x x 0) which can be integrated to give q = q 0 + Ωt + v { 2 (x vt x 0) + tan 1 tan α 2 tanh[ˆω 1/2 sin α } 2 (x vt x 0)] (10.24) -where I have reverted to the original dynamical variable q. Eqs. (10.23) and (10.24) describe a soliton with an internal degree of freedom: in addition to its overall translational motion with velocity v, the soliton is characterized by a nonuniform internal precession of each spin with respect to the z-axis (Fig. ). The soliton solution contains two independent parameters, the internal precession frequency and α, which completely determine the soliton dynamics. In particular, the translational velocity is given by (10.22), the soliton spatial extent is (in units of the lattice constant a) Γ = 1 ˆΩ 1/2 sin α 2 and the amplitude - as defined by the maximum deviation of p from the ferromagnetically ordered state p = 1 - A = 2 sin 2 α. 2 In addition, the soliton solution contains the arbitrary constants x 0, q 0 which specify, respectively, the initial position and internal phase. Soliton magnetization The total magnetization carried by the soliton - measured, in units of h, with respect to the ferromagnetically ordered state - is M = n (S z n S) = S dx (p 1) = 4S sin α 2 ˆΩ 1/2. (10.25) 93

101 10 Solitons in magnetic chains In what follows, it will prove useful to express the soliton s translational velocity in terms of M, i.e. v = 4S M sin α = 4JS2 a h M where in the second line I have reintroduced the physical units. sin α, (10.26) Soliton energy I will restrict myself to the case of vanishing external field B = 0. In this case ˆΩ = Ω. The energy density is, from (10.15) and (10.17), { } 1 p 2 2 JS2 1 p 2 + (1 p2 )ˆq 2, or, using (10.21) and (10.19), which integrates to JS 2 Ω(1 p) E = 4JS 2 Ω 1/2 sin α 2 = 16JS 3 1 M sin2 2. (10.27) where in the second line I have eliminated the precession frequency in favor of the magnetization. It turns out that the set of dynamical variables M and P = 2 hsα/a (10.28) are a better choice for describing the dynamics of the soliton. This becomes clear by looking at the derivative ( ) E = 4JS2 a sin α = v, P h M M which indicates that P can be interpreted as the canonical momentum conjugate to the position of the soliton. Semiclassical quantization Systems which are classically integrable may be quantized according to the Bohr-Sommerfeld scheme, which demands that the total action along a closed orbit must be a multiple of Planck s constant J = p j dq j = nh (10.29) j where {p j, q j } is a set of canonically conjugate coordinates and momenta. The canonically conjugate polar spin coordinates defined in subsection may be used in the above quantization condition after an appropriate correction for their dimensions. Since polar spin coordinates are dimensionless an extra factor h must be added to the lefthand side of the action (cf. the extra factor h which appears in the equations of motion 94

102 10 Solitons in magnetic chains (10.12) ). Furthermore, since in this section we have made the substitution p Sp, the quantization condition over a motion which is periodic with period T reads S h j T 0 dt p j q j = nh or, going to the continuum limit (with the length measured in units of the lattice constant), N/2 N/2 T dx dt (p 1) q = 2πn 0 S. (10.30) Note that I have used p 1 rather than p, since I am interested in the properties of a localized soliton excitation, which approaches p 1 as x ±. There are two types of periodic motion associated with the soliton: The first is related to the translational motion. In other words, the soliton runs around the chain (which is in this case subjected to periodic boundary conditions) with a period T = N/v. This is best viewed in a coordinate system which rotates with angular velocity Ω around the z-axis. In this case and the left-hand side of (10.30) becomes q = vˆq = v p v 2 N/v 0 N/2 dt dx p 1 N/2 1 + p N/v = v 2 dt Ω 1/2 dξ 2 sin α 2 sech 2 ξ 2 2 sin 2 α 2 sech2 ξ 0 = v 2 Ω 1/2 sin α 2 N v = N Ω 1/2 2Ω 1/2 cos α 2 sin α 2 = 2Nα. 1 dρ cosh ρ + cos α 2α sin α In terms of P, the canonical momentum of the soliton given by (10.28), the quantization condition reads P hk = 2π Na h n (10.31) which is the usual quantization condition for the momentum of a free particle subject to periodic boundary conditions. The second type of periodic motion is related to the precession around the symmetry axis. In this case I consider a coordinate system moving with the soliton translational velocity v. Then q = Ω and the left-hand side of (10.30) becomes Ω p = 1 2 sin 2 α 2 sech2 [ Ω 1/2 sin α 2 x ] 2π/Ω 0 N/2 dt dx 2 sin 2 α [ N/2 2 sech2 Ω 1/2 sin α ] 2 x 95

103 10 Solitons in magnetic chains = Ω 2π Ω 2 sin α 2 Ω 1/2 2 = 2π M S (10.32) and the quantization condition simply expresses the fact that M = m. In order to complete the semiclassical quantization scheme, I rewrite the relation between energy and canonical momentum, or, as is more usual in condensed matter physics, the wavevector K. Equation (10.27) now reads E = 16JS 3 1 M sin2 ( ) Ka 2S. (10.33) In the special case S = 1/2, the above expression coincides with the exact quantum mechanical result found by Bethe, using the Bethe-Ansatz, for bound states of M magnons The easy-plane ferromagnetic chain in an external field Weak out-of-plane motion: the Sine-Gordon limit I will consider the case of strong anisotropy. The spin vectors are then approximately confined to the xy plane. The z component is small, so we can readily assume p S and set p 0 in (10.15). The continuum Hamiltonian becomes H = 1 2 JaS2 dx q 2 A + dx p 2 µ a a BS dx cos q. (10.34) Inserting the relevant functional derivatives δh δp(x) δh δq(x) into the equations of motion, I obtain = 2 A a p = JaS 2 q + µbs a q = ā δh h δp(x) = 2Ā h p ṗ = ā δh h δq(x) = Ja2 S 2 h µbs h sin q (10.35) sin q (10.36) from which I can eliminate p and get a differential equation which is of second order in space and time for q: 2 q t 2 c2 2 q x 2 = ω2 0 sin q (10.37) where and c = 2AJ as h 2AµBS ω 0 = h 2 From a more modern perspective, bound states of magnons should be appropriately called quantum solitons. 96

104 10 Solitons in magnetic chains We recognize (10.37) as the SG equation. We have derived it under the assumption of strong anisotropy. Moreover, in order for (10.37) to provide a meaningful approximation to the true dynamics of discrete spins, the length scale defined by (10.37) d = c ( ) 1/2 0 J = a ω 0 µbs should be considerably larger that the lattice constant. The inequality B J µs therefore defines the physical range of allowed magnetic fields consistent with contiuum SG dynamics. Making use of the first of the equations of motion (10.36) I can write the total energy as a function of the field q only: H = ɛ 0 dx { 1 2 ( ) 2 q + 1 t 2 c2 0 ( ) } 2 q + ω0 2 cos q x. (10.38) where ɛ 0 = JaS 2 /c 2 0. This allows me to effectively ignore the out-of-plane motion and deal with q as if it were a single scalar field with an effective Lagrangian density L = ɛ 0 { 1 2 ( ) 2 q 1 t 2 c2 0 ( ) } 2 q ω0 2 cos q x - which results in the SG dynamics and the total energy (10.38). Note however that this should not be misunderstood to imply a vanishing out-of-plane motion. Dynamical structure factor The quantity I αα (k, t) = 1 e ik(m n)a < Sα m (t)s n N α(0) >, m,n where the brackets denote a thermodynamic average over a canonical ensemble, measures the spatial Fourier transform of time-dependent correlations of the α-component of spins. Its temporal Fourier transform is the dynamical structure factor (DSF) dt I αα (k, ω) = 2π e iωt I αα (k, t) = 1 dt dx e i(kx ωt) < S α (x, t)s α (0, 0) >, (10.39) a 2π where in the second line I have made use of the system s translational invariance and, in addition, taken the continuum limit. The DSF can be experimentally deduced from inelastic neutron scattering experiments which detect k and hω, as the change in the neutron s momentum and energy, respectively. In the case of weak out-of-plane motion, the xx DSF can be written in terms of the q-field as I xx (k, ω) = S2 dt dx e i(kx ωt) < cos q(x, t) cos q(0, 0) >. a 2π 97

105 10 Solitons in magnetic chains DSF calculation for a dilute gas of solitons In the limit of weak out-of-plane motion, spin dynamics is effectively governed by the SG equation. Now the dynamics of the SG field equation, a completely integrable system, is truly exceptional. For decaying boundary conditions it implies that solitons have an essentially infinite lifetime. At a finite temperature (therefore finite energy density) the exact mathematics is somewhat more subtle, but there are good reasons to believe in the existence of a soliton gas. At low temperatures, such a kink (or antikink)-like soliton gas would consist of almost non-interacting particles of mass M = 8 ( S d ɛ 0 = 8J c 0 ) 2 a d, velocity-dependent energy E(v) = Mc 2 0γ Mc 2 0 (1 v2 c 2 0 ) 1/2 Mc Mv2 + (with the second expression valid at low velocities) and displacement field (cf. ) { γ(x vt x cos q(x, t) = 1 2 sech 2 0 } ) d where x 0 is a constant specifying the soliton position at time t = 0. The last equation implies that, far from the soliton position, spins are oriented along the ferromagnetic reference state. Only within a distance d from the soliton do spins deviate appreciably from that reference state. Now if the soliton gas is dilute, i.e. the density is much smaller than a/d, then it is very improbable that two solitons will be at the same place at the same time. We can then assume that cos q(x, t) 1 2 j where the sum runs over all solitons of the gas. The correlation function < cos q(x, t) cos q(0, 0) > = 1 2 j 2 j + 4 i,j sech 2 { γj (x v j t x 0 j ) d } { < sech 2 γj (x v j t x 0 j ) } > d } < sech 2 { γj x 0 j d < sech 2 { γj (x v j t x 0 j ) d > (10.40) } sech 2 { γj x 0 i d } > consists of four terms. The first three are constant in time and space and therefore generate contributions to the DSF only at zero momentum and energy transfer. The same holds for that part of the fourth term which comes from terms i j and can be factorized into spaceand time-independent averages. The only contribution to the DSF at nonzero k and ω will come from the i = j part of last term (incoherent scattering) and - after averaging - is the same for all solitons. The sum over j generates a factor equal to the total number of solitons N s. The averaging operation involves averaging over all initial positions and velocities of the jth soliton. The initial positions are uniformly distributed (remember that there is no 98

106 10 Solitons in magnetic chains energy cost involved in moving a SG kink from one position to another). The velocities are distributed according to the Boltzmann distribution appropriate for a one-dimensional gas. In the limit of low temperatures, relativistic corrections can be neglected and P (v) = Ce βmv2 2, (10.41) where β = 1/(k B T ) and C = (βm/2π) 1/2. Symbolically, the averaging operation can be denoted as < >= 1 dx 0 dv P (v). L The resulting DSF { N s dt L 2π dx0 dv dx e i(kx ωt) P (v) < sech 2 γj (x vt x 0 } { ) sech 2 γj x 0 } > d d is a quadruple integral. Setting ξ = x vt x 0 allows the integration over x 0 and ξ to be readily performed, resulting in where n s [ dt 2π dv ei(kvt ωt) P (v) f f(κ) = dx e iκx sech 2 x ( )] 2 kd γ is the soliton form factor and n s = N s /L the density of solitons. The time integral generates a delta function of v ω/k which then enables us to perform the integration over velocities. The resulting DSF is [ 1 I(k, ω) = n s f k ( )] 2 kd P ( ω γ k ) where γ = (1 (ω/c 0 k) 2 ) 1/2. At low temperatures k B T Mc 2 0, where the velocity distribution is well approximated by (10.41), this is well approximated by the Gaussian central peak (CP) π 1/2 I(k, ω) = n s [f(kd)] 2 e ω 2 /Γ 2 k with a width Γ k ( ) 1/2 2kB T Γ k = k. M The observed CP width in CsNiF 3 is consistent with the above predictions [27] Solitons in antiferromagnets Continuum dynamics The starting point is the spin Hamiltonian (10.1) and the resulting equations of motion (10.6) with J < 0. In the following, I will use A = 2δ J ; the dimensionless measure of the anisotropy will be assumed to be small. Consider first the case A = 0, B = 0. If zero-point classical fluctuations are neglected, the ground state of the isotropic antiferromagnet at zero external field is the Neél state, S n = ±( 1) n Sˆn 99

107 10 Solitons in magnetic chains where besides the rotational degeneracy (arbitrary direction of the unit vector ˆn), there is an even-odd degeneracy. If A denotes up and B denotes down, both ABABAB and BABABA are possible ground states with the same energy. The existence of two degenerate vacua makes antiferromagnets a priori good candidates as soliton bearing systems. I will define new vector fields which are well suited to describe the situation at low temperatures, i.e. not too far from the ground state. Note that this will not exclude large amplitude fluctuations; however, I will make the demand that the various field configurations should vary smoothly in space. Let φ n = 1 2S ( S 2n+1 S 2n ) ln = 1 2 ( S 2n+1 + S 2n ). (10.42) The new fields satisfy the properties and φ n l n = 0 (10.43) φ n S 2 l n 2 = 1. (10.44) If field configurations vary smoothly in space, it is possible to use a continuum field approximation φ n φ(x) with field values at neighboring sites (note that a neighboring site of the new field is at a distance 2a apart!) φ n±1 φ(x) ± 2a φ (x) (2a)2 φ (x), and a similar expansion for l(x). Note however that the two fields do not have the same status. At the Neel state, it is obvious that l = 0, whereas φ n = 1. In fact, a consistent field expansion treats l as a small quantity, of the order of φ. Therefore, terms of second order in l will be dropped. Under these conditions, the normalization condition (10.44) is exhausted by the φ field, which will henceforth be treated as a vector of unit length. It is a tedious but straightforward - and necessary - exercise to use the inverse relations S 2n = l n Sφ n S 2n+1 = l n + Sφ n (10.45) and express the total Hamiltonian in the form H = dxh where the Hamiltonian density is given in terms of the new vector fields: { H = 1 c 2 φ 2 2gc as l 2 + c 2 φ 2 2ω1δ 2 φ 2 γω 1 4 S } B l (10.46) The first two terms come from the isotropic exchange term; the third term comes from the anisotropy - note that φ = φ ( φ ẑ)ẑ is the transverse component of the φ-field; the fourth term comes from the interaction with the magnetic field. The new constants are related to the old as follows: g = 2/( hs), ω 1 = 2 J S/ h, c = ω 1 a. 100

108 10 Solitons in magnetic chains The total magnetization can be expressed (in units of h) as 3 M = n = S n dx 2a 2 l(x). (10.47) The next step is to obtain the dynamics of the coupled vector fields by differentiating both sides of (10.42), using the dynamics defined in (10.6), and rewriting the results in terms of the new fields: φ = ( cφ φ 2 ) as l γb φ (10.48) l = ( ) c l φ + cas φ φ ω 1 δs( φ ẑ) φ ẑ γb l (10.49) In deriving (10.48), I have further dropped anisotropy terms which are first order in l, if the anisotropy is small, they are negligible compared to the leading, first term in the right-hand side of (10.48). The above coupled first-order equations determine in principle the spin dynamics of the new variables 4. It is however possible to perform a further reduction by recognizing that l is completely determined by φ and its derivatives (i.e. it is a slave variable). This can be easily seen by forming the vector product of both sides of (10.48) with φ. After some rearrangements, 2c as l = φ [ ] φ + cφ + γ B ( B φ) φ, (10.50) which can be used to eliminate l from (10.49). The result is φ [ φ c 2 φ + 2ω1δ( 2 φ ẑ)ẑ + 2γB ] φ + γ 2 B( B φ) = 0. (10.51) In what follows, it will be useful to exploit (10.50) in order to eliminate the slave variable l from (10.46). The result is 5 H = 1 { φ 2 c 2 φ 2gc 2 2ω1δ 2 φ 2 + γ 2 ( B φ) 2} (10.52) I will now consider some special cases The isotropic antiferromagnetic chain If δ = 0 and B = 0, (10.51) is equivalent to the dynamics of the field theory defined by the Lagrangian density L 0 = 1 ( φ 2 c 2 φ 2gc 2) (10.53) subject to the constraint φ 2 = 1. This is the relativistically invariant nonlinear sigma model, which has been employed as a toy model in quantum chromodynamics. Furthermore, 3 Note that, strictly speaking, this holds for an even number of spins. An odd number of spins will generate a contribution from the boundary. 4 It should be noted that the equations preserve exactly both the normalization φ 2 = 1 and the orthogonality property φ l = 0. 5 Strictly speaking, the result in the brackets of (10.52) omits an irrelevant constant γ 2 B B and a total derivative term 2γ 2 B φ, which only generates contributions from the boundaries. 101

109 10 Solitons in magnetic chains a Wick rotation shows it to correspond to the Hamiltonian of the two-dimensional classical antiferromagnet. The Hamiltonian density obtained from the field theory (10.53) H = 1 2gc is the same as the sum of the first two terms in (10.52). ( φ 2 + c 2 φ 2) (10.54) Note: it possible to add to the Lagrangian density (10.53) a term L = 1 θ φ g 2πS ( φ φ ). (10.55) This so called topological term of the Lagrangian does not influence the classical equations of motion. This is because it generates a contribution to the action which depends only on general topological properties of the field; in the simplest of cases, one can see, using a polar representation p = φ z, q = arctan(φ y /φ x ) that Q = 1 dtdx φ 4π ( φ φ ) = 1 ( p q dtdx 4π t x q t = 1 (p, q) dtdx 4π (x, t) = 1 4π 1 1 dp 2π 0 dq ) p x = 1 ; (10.56) in general, the Pontryagin index Q of the vector field tells us how many times the vector sweeps the unit sphere as dxdt sweeps two-dimensional space-time. The resulting contribution to the action W = dx dt L, a constant, cannot modify the classical equations of motion, which are determined by the action derived from the Lagrangian density L 0. It may however be relevant for quantum phenomena. Noting in this context that 2/gS = h and θ = 2πS 6 we obtain W h = 2πSQ (10.57) which hints that whether or not the extra term is relevant for quantum mechanics may well depend on whether S is a half-integer or an integer, respectively Easy axis anisotropy Consider the case of easy axis anisotropy δ = 1 2 (ω 0/ω 1 ) 2. The dynamics of the vector field φ (10.51) is equivalent to the Lagrangian field theory L = 1 ( φ 2 c 2 φ 2gc 2 ω0 2 φ 2) (10.58) 6 The choice θ = 2πS is mandated by the requirement that the canonical momentum conjugate to φ, π = (L 0 + L )/ φ should form a triad with l and φ, i.e π = h/a l φ and h/a l = φ π; the choice θ = 2πS in (10.48) with B = 0 satisfies the first of these requirements; the second, which is a natural feature of a Hamiltonian theory, guaranteeing the right form for l, the generator of infinitesimal rotations, is then automatically satisfied. 102

110 10 Solitons in magnetic chains subject to the constraint φ 2 = 1. Note that the anisotropy term does not destroy Lorentz invariance. The simplest way to lift the constraint is to introduce polar coordinates α = arccos φ z β = arctan(φ y /φ x ) where 0 α π and 0 φ < 2π. The Lagrangian density (10.58) can then be written as L = 1 [ α 2 + sin 2 α 2gc β ( ) ] 2 c 2 α 2 + sin 2 α β 2 ω0 2 sin 2 α. (10.59) The corresponding energy density is given by (10.52) as H = 1 2gc [ α 2 + sin 2 α β 2 + c 2 ( α 2 + sin 2 α β 2 ) + ω 2 0 sin 2 α The resulting equations of motion are The vacua 1 c 2 t α 1 c 2 α = 1 ω0 2 β 2 2 c ) 2 ( sin 2 α β = x sin 2α ]. (10.60) ( sin 2 α β ). (10.61) I first determine the spatially and temporally uniform solutions. These are α = 0, π/2, π and β = β 0 (arbitrary). By inspection of (10.60) it can be seen that only α = 0, π correspond to - degenerate - energy minima (vacua), whereas α = π/2 corresponds to an energy maximum. Kinks and antikinks I next look for solutions which satisfy β = ω (uniform precession of the φ vector around the z-axis) and α = 0; the latter restriction can later be lifted because I can always Lorentz-boost a static solution. The second equation is satisfied identically; the first reduces to α = ω2 0 ω 2 2c 2 sin 2α, (10.62) which is a Sine-Gordon equation for the field 2α and a length scale R = c/ ω 2 0 ω2. It has a first integral R 2 (α ) 2 = 1 cos 2α + const 2 and can therefore support soliton solutions which interpolate from one vacuum (α = 0 to the other (α = π). This implies the choice of the constant equal to 1/2, therefore and or, Rα = ± sin α α = 2 arctan e ±(x x 0)/R, sin α = sech x x 0 R cos α = tanh x x 0 R where x 0 is an arbitrary constant., (10.63) The solitons are π-kinks, going from α = 0 at x to π at x (and the corresponding antikinks). 103

111 10 Solitons in magnetic chains The total magnetization of the π-kink Introducing (10.50) into (10.47), we obtain the following general expression for the magnetization, valid for B = 0: M = S ( dx φ φ + c ) φ 2c The z-component of the magnetization will then be M z = S ) ] dx [(φ x φy φ y φx + cφ z 2c = S dx sin 2 α 2c β + S 2 cos α = m S (10.64) where the limits of integration have been extended to infinity, since we assume that the spin configurations approach one of the two Neel states at infinity. The second term will then be equal to -S for a kink and S for an antikink. This is a contribution which is entirely independent of the structural details of the kink. The extra magnetization will be m = S 2c ω = S 2c ω 2R dx sech 2 x x 0 R ω = ω 2 0 ω S. (10.65) 2 The total energy of the π-kink Introducing the form of the kink solution into the expression for the energy density (10.60) gives H = which gives a total kink energy [ 1 sin 2 α ω 2 + c 2 sin2 α 2gc R 2 = ω2 0 x x 0 gc sech2 R ] + ω0 2 sin 2 α E = dx H = ω2 0 gc 2R = 2 g ω 2 0 ω 2 0 ω 2. (10.66) It is possible to express ω in terms of the kink magnetization m using (10.65). This gives the energy of the kink as a function of magnetization E = hω 0 S2 + m 2 (10.67) where I have made use of g = 2/( hs). The last form, along with (10.65), is eminently useful for developing a semiclassical quantization approach where m would take integer or half-integer values. 104

112 10 Solitons in magnetic chains Easy plane anisotropy A positive value of the anisotropy parameter δ favors spin orientations along the xy-plane. Setting ( ) 2 ω0 δ = 1 2 ω 1 implies a change of sign ω0 2 ω0 2 in the effective Lagrangian density (10.59), which now reads L = 1 [ α 2 + sin 2 α 2gc β ( ) ] 2 c 2 α 2 + sin 2 α β 2 + ω0 2 sin 2 α, (10.68) the energy density (10.60), and the equations of motion (10.61). Although the spatially and temporally uniform solutions are the same as in the case of the easy-axis anisotropy, their role is reversed. There is only one stable energy minimum at α = π/2 and maxima at α = 0, π. Looking for solutions with α = 0 and β = ω leads to α = 1 1 sin 2α 2 R2 where now c 2 R 2 = ω 2 + ω0 2. This is a SG equation for the field 2(π/2 α). We can write down the solution by making the substitution α π/2 α in (10.63): cos α = sech x x 0 R sin α = tanh x x 0 R, (10.69) where x 0 is again an arbitrary constant. Note that this type of solution does not interpolate between degenerate vacua. It drives the system out of the only available vacuum at x x 0, leads it to the energy maximum at x = x 0 and returns it to the vacuum as x x 0. The associated energy density is H = 1 [ ( ) ] c sin 2 α ω gc R 2 + ω2 0 cos 2 α and will therefore lead, from the first term, to a total energy proportional to the size of the system, as long as ω 0. If we restrict ourselves to finite-energy solutions ω must vanish, i.e. β = β 0. The characteristic length becomes R = c/ω 0 and the energy density which integrates to a total kink energy H = 1 [ 2gc 2ω2 0 sech 2 ω ] 0 c (x x 0), E = 2 g ω 0 = hω 0 S (10.70) Inspection of (10.64) shows that both the bulk and the boundary contributions to the magnetization vanish. 105

113 10 Solitons in magnetic chains Easy plane anisotropy and symmetry-breaking field In the case of easy-plane anisotropy ( ) 2 ω0 δ = 1 2 ω 1 and nonzero magnetic field, it is possible to satisfy the equations of motion (vanishing brackets in (10.51) ) by constructing a field theory based on the effective Lagrangian density L = L B = 1 { φ 2 c 2 φ 2gc 2 ω0 2 φ 2} + L B, where 1 { γ 2 ( B 2gc φ) 2 + 2γB ( φ } φ). (10.71) Two comments are in order here concerning the second term in the second line. First, this is the most general scalar that can be constructed from the magnetic field and the field vector φ and its derivatives, consistent with the property of time reversal invariance B B, φ φ. Second, that although the term is crucial for the dynamics -it generates the fourth term in the brackets in (10.51)-, it does not contribute to the energy, which is quadratic in the magnetic field (cf. (10.52) ). I will consider the case in which the magnetic field is in the x-direction, i.e. it serves to break the easy-plane symmetry. It is straightforward to derive the complete spin dynamics in polar coordinates. They correspond to those of the previous subsection, with additional terms which come from the field dependent part of the Lagrangian L B = 1 { γ 2 B 2 sin 2 α cos 2 β 2γB sin β α γb sin 2α cos β 2gc β } The equations of motion in polar coordinates are: α c 2 α 1 2 sin 2α( β 2 + ω 2 0) = 2γB cos β sin 2 α β 1 2 γ2 B 2 sin 2α cos 2 β t (sin2 α β) c 2 x (sin2 αβ ) = γb cos β α γ2 B 2 sin 2 α sin 2β (10.72) I will also need the total energy density (10.52), which in polar coordinates reads H = H exch + H anis + H B, where 1 [ H exch = α 2 + sin 2 α 2gc β ( )] 2 + c 2 α 2 + sin 2 α β 2 H anis = ω2 0 2gc sin2 α H B = (γb)2 2gc sin 2 α cos 2 β. (10.73). The vacua Spatially and temporally uniform solutions of (10.72) must satisfy the conditions sin 2α [ ω0 2 γ 2 B 2] = 0 sin 2 α sin 2β = 0. (10.74) 106

114 10 Solitons in magnetic chains If we look at the total energy of such a uniform state as a function of the field, E = L 1 2gc sin2 α ( ω 2 0 γ 2 B 2 cos 2 β ) we see that the energy minimum, which is at α = π/2 at zero magnetic field, shifts to α = 0, π if the field exceeds the critical value B c = ω 0 /γ. At moderate fields B < B c, the minimal energy configuration will be at α = π/2, β = ±π/2. The kinks If the anisotropy is nonvanishing, we can assume that the out-of-plane motion will be weak. As long as α π/2, the second equation of motion will reduce to It admits static (and Lorentz- which is a Sine-Gordon equation for the angle π 2β. boostable) kink and antikink solutions of the form 2 β t 2 c2 2 β x 2 = 1 2 γ2 B 2 sin 2β (10.75) π 2β = arctan e ±(x x 0)/d where d = c/(γb). Alternatively, we can write ( ) x x0 cos β = sech d ( ) x x0 sin β = ± tanh d which gives an energy density 1 2gc ( ) c 2 β 2 ω γ 2 B 2 cos 2 β and a total kink rest energy (measured from the vauum) Out-of-plane corrections E 0 = 2 c = hsγb = µsb. gd. (10.76) Corrections which arise from the weak out-of-plane motion may be estimated by considering the first of the equations of motion (10.72) which for α π/2 (so that the second space and time derivatives can be neglected) gives ( ) cos α β2 + ω0 2 γ 2 B 2 cos 2 β = 2γB cos β β For a static kink configuration, the first term on the left-hand-side vanishes 7 and the third is at most (in the vicinity of the kink) of order γ 2 B 2. At moderate fields B < B c this leaves the second term as the dominant one; hence α = π 2 + 2γB ω 2 0 cos β β which is the analog of the first of eqs (10.36) for the easy-plane ferromagnet and can be used to estimate e.g. out-of.plane corrections to the kink energy. 7 Note that this estimate is even better for moving kinks, since the first term then partially cancels the third. 107

115 10 Solitons in magnetic chains The dynamical structure factors The xx spectra < cos(x, t) cos(0, 0) > give rise to a DSF similar to that of the ferromagnet with easy-plane anisotropy and magnetic field in the x-direction - except for a slightly different form factor due to the sech rather than sech 2 function. The yy spectra can be approximated by < sin(x, t) sin(0, 0) > < σ(x, t)σ(0, 0) > where σ = ±1. The approximation suggests that they act as registers of a spin flip every time a soliton comes by. More precisely, we can approximate σ(x, t) = σ(x, 0)( 1) N1(t), where N 1 (t) is the number of times a soliton passes through the point x during the time interval (0, t), and σ(x, 0) = σ(0, 0)( 1) N 2(x), where N 2 (x) is the number of solitons which are in the segment (0, x) at any given time. The dynamical correlation can then be calculated, using the Poisson statistics which characterize the spin flips, to be < σ(x, t)σ(0, 0) >= e 2[N 1(t) + N 2 (x)]. The averages are straightforward to estimate: N 2 (x) is simply equal to 2n s x, where n s is the density of kink-type solitons (and n s = n s the density of antisolitons). The average number of solitons passing through a point during a given time interval of length t is given by n s ū t, where ū is the average thermal speed of solitons. If solitons can be thought of as forming an ideal (Boltzmann) gas (cf. the analogous discussion of the ferromagnetic case in section ) then ū = (2k B T/M) 1/2. This should of course be multiplied by a factor of 2, to account for the antisolitons. Putting everything together, and hence where and < σ(x, t)σ(0, 0) >= e 4n s x 4n s ū t π π I(k, ω) = k 2 + Γ 2 k ω 2 + Γ 2 ω Γ k = 4n s ( ) 1/2 2kB T Γ ω = 4 n s M (10.77) According to the above, the main temperature and magnetic field dependence of both widths comes from the soliton density which, in the appropriate temperature range, has a characteristic exponential dependence n s e E0/k BT. Since the kink rest energy is proportional to the magnetic field, we expect the leading B and T dependence of the width to scale with B/T. This is borne out by experiments [28] on TMMC (cf. Fig. 10.1). 108

116 10 Solitons in magnetic chains Figure 10.1: Energy and wavevector width of the central peak of TMMC. (from [28]). 109

117 11 Solitons in conducting polymers 11.1 Peierls instability This is not an attempt to cover the general subject of electronic transport properties of polymers. A good place to start learning about the physical properties, the chemistry and the technological significance of semiconducting and metallic polymers is Alan Heeger s Nobel lecture[29]. The theoretical and experimental background on solitons in conducting polymers has also been reviewed by Heeger, Kivelson, Schrieffer and Su [30]. Here I will only try to convey the main ideas about the Peierls instability, which turns a one-dimensional metallic chain into an insulator, and about how the resulting structures, if the proper conditions of energy degeneracy are met, can support solitons and polarons. The exemplary substance for the present discussion is trans-polyacetylene, (CH) x. The schematic structure shows that the backbone consists of carbon atom bonds which are neither single nor double. For large chains, where the end points are irrelevant, the picture of the electronic structure is relatively simple: each carbon atom contributes a single p z electron to the π- band. These electrons have a tendency to delocalize. One can express this in terms of a tight-binding model Hamiltonian H 0 = ( ) t n,,n+1 c n+1s c ns + c nsc n+1s, (11.1) ns where the c ns s denote creation and annihilation operators for electrons of spin s at the nth site, and the hopping parameters t n,n+1 correspond to the π-electron transfer integrals Electrons decoupled from the lattice In the absence of any coupling to the underlying lattice, the hopping parameters can be assumed to have a constant value, t 0. The Hamiltonian H 0 is then diagonal in Fourier space, i.e. H 0 = ɛ q c qsc qs, (11.2) qs where c qs = 1 N N n=1 e iqan c ns ; q = 2π a N is the number of sites and a the lattice constant, and n N, n = N 2 + 1,, N 2 1, N 2, (11.3) ɛ q = 2t 0 cos qa (11.4) The resulting band structure is shown in unfolded and folded form in Fig.. The point to note is that of the total N allowed values of q, N/2 lead to states of negative energy; since every state can be doubly occupied (spin up/down), and there is a total of N π-electrons, the negative energy states are all occupied, and the positive energy states are empty. Since there is no gap between them, the one-dimensional tight-binding electronic system is metallic. 110

118 11 Solitons in conducting polymers Electron-phonon coupling; dimerization Suppose that the carbon atoms in the backbone are slightly displaced from their reference positions. Let the displacement of the nth carbon atom be y n. It is reasonable to assume that if the distance from the nth to the n + 1st atom decrease, the probability amplitude for electron hopping should increase; for small relative displacements t n,n+1 = t 0 α(y n+1 y n ) (11.5) should hold. energy Furthermore, the atomic displacements contribute to a lattice deformation H L = 1 2 K n (y n+1 y n ) 2. (11.6) In principle, the lattice atoms also contribute a kinetic energy term. In the framework of the Born-Oppenheimer approximation, we will treat the slow motion of atoms as classical; in essence we want to compare the electronic energies of various lattice configurations. The kinetic energy of these configurations can be neglected in a first approximation. The possibility of dimerization I examine configurations with alternating bond lengths, i.e. y n = ( 1) n y 0. I define new creation and annihilation operators appropriate to the folded Brillouin zone, c ks = a ks if k < k F c ks = b k sgn(k)kf s if k > k F which restricts k-values to be smaller than k F = π/(2a). The tight-binding Hamiltonian - which now includes the interaction of the electrons with the lattice - can now be written as H 0 = qs { ɛq [ a qs a qs b qsb qs ] + i q [ b qs a qs a qsb qs ]} (11.7) with q = 0 sin qa and 0 = 4αy 0. In addition, the lattice deformation contributes to the total energy. H L = 1 2 NK4y2 0 Diagonalization of the electronic Hamiltonian The Hamiltonian (11.7) can be diagonalized via a Bogoliubov transformation a ks = αksa ks β ks B ks b ks = βksa ks + α ks B ks (11.8) where α ks, β ks are c-numbers, satisfying the relationship α ks 2 + β ks 2 = 1, and chosen such as to satisfy the anticommutation relations {A ks, A ks } = 1 and {B ks, B ks } = 1; all other 111

119 11 Solitons in conducting polymers anticommutators must vanish; furthermore, α ks can be chosen to be real and positive. The procedure leads to α ks = 1 [ 1 ɛ ] q 2 β ks = E q [ i 1 + ɛ ] q 2 E q, where E q = ɛ 2 q + 2 q = 2t 0 1 (1 z 2 ) sin 2 qa, (11.9) with z = 0 /2t 0, and a two-band diagonal Hamiltonian: H 0 = qs E q [ A qs A qs B qsb qs ]. (11.10) The energy bands now form a gap of width 2 0 at the Brillouin zone edge (cf. Fig. 11.1). In order to see whether this instability will materialize spontaneously, it is necessary to compute the total ground state energy and compare it with that of the undimerized state. The ground state energy of H 0 is 2 q E q = 2 Na π/2a 2π 2t 0 dq 1 (1 z 2 ) sin 2 qa π/2a = N 4 π/2 π t 0 dx 1 (1 z 2 ) sin 2 x 0 = N 4 π t 0 E(z) where E(z) is the complete elliptic integral of the second kind. In addition, the dimerized configuration includes a contribution from the lattice deformation, where H 0 = 2Nt 0 1 πλ z2, α 2 λ = 4 π Kt 0 is the dimensionless electron-phonon coupling constant. Collecting terms and using the small-z expansion of the elliptic integral, E(z) ( 2 z2 ln 4 z 1 ) 2 I obtain a total energy per site E 0 (z) = 4t 0 π { ( z2 ln 4 z )} λ. Subtracting the energy E 0 (0) of the undimerized state gives E 0 (z) = 2t ( 0 π z2 ln 4 z ) λ (11.11) 112

120 11 Solitons in conducting polymers 1 z= λ=0.4 energy E 0 /(4t 0 ) qa/π z Figure 11.1: Left panel: electronic spectra of the undimerized (dotted curve) and dimerized (dashed curve) cases of the SSH model; the energy is in units of 2t 0. The Peierls gap is formed at the edge of Brillouin zone. Right panel: the energy (11.11) of dimerization as a function of the dimensionless parameter z. as the energetic advantage of dimerization. Fig shows that the dimerization energy has a double well structure, with minima at z = ±z 0 = ±4e 1 1/λ. This corresponds to a band gap 2 0 at the BZ edge (Peierls gap). Expressed as a fraction of the total bandwidth, 2 0 4t 0 = z 0 = 4e 1 1/λ (11.12) Note the non-analytic dependence of the energy gap on the electron-phonon coupling constant, which bears a formal similarity to the Cooper pair condensation energy in superconductivity. Of course the effect here is the opposite. Switching on the electron-phonon interaction brings about a spontaneous lattice distortion 1 and turns a putative one-dimensional metal into an insulator (Peierls instability). The double minimum structure of the dimerization potential (11.11) makes plausible the existence of kink-like solitons, i.e. nonlinear configurations of the coupled electron-phonon system which interpolate between the two degenerate vacua. It turns out that these are not the only nonlinear configurations possible. The full arguments will be presented in the next section. 1 in the (CH) x case the lattice distortion corresponds to the formation of alternating single and double bonds. 113

121 11 Solitons in conducting polymers 11.2 Solitons and polarons in (CH) x A continuum approximation The original theoretical treatment of solitons in polyacetylene was given by Su, Schrieffer and Heeger [31], who wrote down the electron-phonon Hamiltonian of the previous section (SSH Hamiltonian). Here, I will present an alternative formulation, due to Takayama, Lin-Liu and Maki (TLM) [32], which has the advantage of being more tractable analytically. The objective of the theory is to look for exact, nonlinear configurations of the electronphonon theory. Assuming that any such configurations are obtainable as smooth spatial variations from the basic dimerization pattern (note the analogy with antiferromagnetic solitons!), it is reasonable to write down the Ansatz c n = e ik F naû n ie ik F naˆv n for the operator c n ; the idea is that we can approximate the operators û n, ˆv n by continuum field operators, i.e. û n aû(x); this translates and {c n, c n } = δ n,n {û(x), û (x )} = δ(x x ) û n+1 [ a û + a û ] x û n+1 ] a [û a û x The lattice distortion also forms a smooth variation with respect to the dimerization pattern, y n = ( 1) n 1 4α (x). Furthermore, it should be clear that only the electrons which are near the Fermi level may contribute to the physics; for k k F I can approximate the dispersion relation (measuring k from k F ) by where hv F = 2at 0. 2t 0 cos[(k ± k F )a] = ±2t 0 sin ka ±2t 0 ka ±v F k, Under these assumptions, the electronic part of the SSH Hamiltonian, including the electron-phonon interactions, transforms to [ ] H 0 = dx ˆΨ (x) i hv F σ 3 x + (x)σ 1 ˆΨ(x) (11.13) where are Pauli matrices, and The lattice part is ( ) 1 0 σ 3 = 0 1 ( ) 0 1 σ 1 = 1 0 ˆΨ(x) = ( û(x) ˆv(x) H L = ω2 Q 2g 2 ).. dx 2 (x) (11.14) 114

122 11 Solitons in conducting polymers where, conforming to standard field theoretic notation, ω 2 Q = 4K/M and g = 4α(a/M)1/2. I now look for the ground state of the Hamiltonian, allowing for any smooth deformation (x) of the lattice. Using standard second-quantized notation for the operators û(x) = l u l (x)a l ˆv(x) = l v l (x)b l I try to find the set of normalized one-electron states ( ) ul (x) Ψ l (x) = v l (x) and the deformation field (x) which minimizes the ground state energy < 0 H 0 >= [ ] dxψ l (x) i hv F σ 3 x + (x)σ 1 Ψ l (x) + ω2 Q 2g 2 l. dx 2 (x). This is a variational problem subject to the constraints imposed by the normalization of one-electron states dx Ψ l (x)ψ l (x) = 1 l ; it can be worked out by unrestricted minimization of < 0 H 0 > ɛ l dxψ l (x)ψ l (x) l and subsequent determination of the Lagrange multipliers to fit the normalization constraint. The procedure leads to the Bogoliubov-de Gennes equations, first derived in the context of the theory of superconductivity: ( ) i hv F σ 3 x + (x)σ 1 Ψ l (x) = ɛ l Ψ l (x) or, in component form, l l Ψ l (x)ψ l (x) + ω2 Q (x) = 0 (11.15) g2 i hv F x u l + (x)v l = ɛ l u l i hv F x v l + (x)u l = ɛ l v l [u l v l + u l vl ] + ω2 Q (x) = 0. (11.16) g2 The first of (11.15) comes from extremization with respect to the electronic wave function Ψ l (x), and the second from extremization with respect to the displacement field (x). Practically, one treats (x) as a parameter, describing a class of displacements, e.g. dimerization, soliton-like pattern, etc., and computes the total energy corresponding to the particular displacement class, E = dx Ψ l (x) ɛ l Ψ l (x) + ω2 Q 2g 2 dx 2 (x) l = ɛ l + ω2 Q 2g 2 dx 2 (x) ; (11.17) l 115

123 11 Solitons in conducting polymers note that the sum runs over all occupied states (factor 2 from spin implicit!). It will prove useful to recast the electronic part of the BdG equations by defining f (±) l = u l ± iv l, as i hv F f ( ) l + i f ( ) l = ɛ l f (+) l i hv F f (+) l i f (+) l = ɛ l f ( ) l. (11.18) If ɛ l 0, it is possible to use the first of these equations to express f (+) in terms of f ( ) ; inserting the result in the second equation gives [ h 2 vf 2 2 ] x 2 + ɛ2 l 2 (x) (x) hv F f ( ) l (x) = 0. (11.19) x Dimerization The continuum theory describes the Peierls instability. This can be seen by inserting a constant displacement field in (11.19). The solutions are plane waves with f q ( ) (x) = N q e iqx ɛ q = ± 2 + ( hv F q) 2, and N q a normalization constant to be determined. The total energy (electronic ground state plus deformation energy) is 2 q ɛ q + ω2 Q 2g 2 L 2 where the factor 2 comes from the spin states and L is the length of the chain. In order to obtain the excess energy due to dimerization we must subtract the energy of the uniform state. This gives a dimerization energy E( ) = 2 L Λ [ ] dq 2 + ( hv F q) 2π 2 hv F q + L 2 (11.20) Λ πλ hv F where, in the first term we have used q L 2π dq and introduced a cutoff to treat ultraviolet divergences, and in the second term we have used the dimensionless coupling constant λ = 2 g 2. Introducing the dimensionless variable z = hv F q/, we obtain πv F h ω 2 Q 2 zm [ ] dz 1 + z2 z + L 2 πλ hv F E( ) = 2L π hv F 0 = 2L 2 1 [ z m 1 + z 2 π hv F 2 m + ln 2L [ Λ 1 π W λ + ln W ] ( z m z 2 m ) ] zm 2 + L 2 πλ hv F (11.21) where z m = hv F Λ/ = W/(2 ) in terms of the bandwidth W = 2 hv F Λ which comes with the finite cutoff. The approximation should hold as long as the gap is small compared to the bandwidth. The dimerization energy (11.21) has a minimum at 0 = W e 1/λ, corresponding to a lowering of the total energy by an amount L 2 0/(2π hv F ). The ground state will therefore be dimerized - just as in the discrete version of the model -. The gap which opens at the Fermi level is 2 0. This is the energy scale - needed to create an electron-hole pair - with which other, nonlinear elementary excitations should be compared. 116

124 11 Solitons in conducting polymers energy 0 v u 0 q Figure 11.2: Electronic energy bands in the dimerized state of the TLM model (u, v); also shown are the bands in the undimerized case (dotted straight lines). The electronic energy spectra in the discrete lattice (SSH) case (dashed curves) are shown for comparison; note that the negative wavevectors of Fig have been translated by an amount 2π/a in order to bring the gap at zero wavevector The soliton The Bogoliubov-de Gennes equations turn out to be exactly solvable for the class of lattice deformations described by (x) = 0 tanh x. ξ The tanh Ansatz, introduced in (11.19) gives { 2 0ξ0 2 2 x 2 + ɛ2 l ( 1 ξ 0 ξ ) } sech 2 x f ( ) l (x) = 0, (11.22) ξ where ξ 0 = hv F / 0 is a length characteristic of the dimerized state. The above equation is analytically solvable in terms of hypergeometric functions. Here, I will only show the solution in the special case where the characteristic width of the kink ξ is equal to ξ 2 0. In this special case, the sech 2 term in (11.22) disappears, and the solutions are plane waves, with f q ( ) (x) = N q e iqx ɛ q = ± ξ 2 0 q2, i.e. the f q ( ) solutions are identical with those of the exactly dimerized state. However, in the case of the tanh deformation, the f q (+) solution derived from (11.18) is f (+) q = hv F q + i (x) ɛ q f ( ) q = ± qξ 0 + i tanh x ξ 0 N 1 + ξ 2 q e iqx (11.23) 0 q 2 2 It turns out [32] that this produces the soliton with the minimal energy 117

125 11 Solitons in conducting polymers where the ± refers to the sign of the energy. At x ξ 0 this is effectively a plane wave; however, there is a phase difference: where δ(q)/2 = arctan(1/qξ 0 ). lim f q (+) (x) e iqx iδ(q)/2 x lim f q (+) (x) e iqx+iδ(q)/2 (11.24) x + In addition, we can now investigate whether the BdG equations (11.18) admit zero-energy solutions (something that could be immediately excluded in the dimerized case). It can be readily seen that this is the case here, with f ( ) = cosh x ξ 0 f (+) = sech x ξ 0. The f ( ) state is not normalizable; but the f (+) is a legitimate localized state with energy exactly at midgap. I will return to its interpretation shortly. The modifications in the electronic spectrum, i.e. the phase shift δ(q) of the extended states and the appearance of a localized state at midgap, have important physical consequences. Phase shifts are important because, as I first discussed in the context of scalar field theories, they modify the density of states in q-space. Let us recall: if we demand periodic boundary conditions on a chain of length L, the phase of the wave function on the left end should differ from the phase on the right end by a multiple of 2π. Thus q n L = 2πn (n = 0, ±1, ) if = 0 (11.25) q nl + δ(q n) = 2πn (n = 0, ±1, ) if = 0 tanh x ξ 0 (11.26) For large L, this means that the wavevector q of an electronic state is shifted by an amount q n q n = δ(q n )/L. This is true with one exception. Eq. (11.26) has no solutions if n = 0. In other words the zero wavevector state does not exist in the presence of the soliton. How does this modify the energy of the coupled electron-phonon system, compared with the energy of the [dimerized] ground-state? The energy difference to be calculated is ( ɛq ɛ ) n q n + {0 ɛ0 } + 1 πλ hv F n 0 dx ( 2 0 tanh 2 x 2 ) 0 The first term comes from the shift in electronic states discussed above. Note that all contributions refer to occupied states, i.e. states of negative energy. The factor 2 comes from taking the spin into account. The factor 1/2 is there because electronic states are linear superpositions of f q ( ) and f q (+) and only the latter are shifted compared to the pure dimerized state. The sum does not include the n = 0 state. The term in curly brackets expresses the absence of the zero wavevector state from the deformed state - and its presence in the dimerized state. The second term is the change in the elastic energy due to the deformation. Finally, note that the midgap state does not appear in this calculation because it has zero energy. Transforming the sum into an integral, using a Taylor expansion ɛ q ɛ q ɛ q(q q) = ɛ qδ(q)/l, and noting that ɛ and δ are both odd functions of q, I obtain a soliton energy. E s = L 2π 2 Λ 0 + dq ɛ q δ(q) L ɛ 0 2 πλ 0 118

126 = 1 Λ π ɛ qδ(q) π + 0 π 11 Solitons in conducting polymers Λ [ Λ 2 ξ0 2 π Λξ 0 2 π π 0 ln(2λξ 0 ) 2 πλ 0 dq ɛ q δ (q) πλ 0 + ] + 2 Λ 0ξ 0 dq π q2 ξ πλ 0 = 2 π 0, (11.27) where the approximation signs mean leading terms in cutoff-dependent quantities. Within the general context of continuum theory, the expression obtained is exact. Let me summarize what has been derived: A lattice deformation of a tanh type (kink) can exist in the coupled electron-phonon system. It modifies the electronic spectrum taking half a state away from the top of the valence band (the q = 0 state corresponding to one of the two branches of solutions) and creating a localized state (localized around the center of the kink) at midgap. What remains in the Fermi sea consists of paired states, i.e. both spin up and spin down states are occupied. However, the localized state at midgap is unpaired. Therefore, the soliton excitation (which should be understood to consist of the lattice deformation, the localized state at midgap, and the small shifts in q-values of states in the Fermi sea, the so-called backflow ) has a spin 1/2 and a charge 0 (owing to overall electrical neutrality). The energy 2 0 /π needed to create a soliton is less than 0. Or, in terms of what really happens: The energy 4 0 /π needed to create a kink-antikink pair of solitons is less than the 2 0 needed to create an electron-hole pair. This is why solitons are of practical importance in determining the conductivity of polyacetylene. A further comment: it is in principle possible to feed an extra electron at the mid-gap state, thus obtaining a charged object with Q = e and spin 0. Or one can remove the electron from the midgap state, creating a soliton with positive charge and zero spin. This is the physical principle behind doping in polyacetylene and the unusual spin/charge relationships observed experimentally The polaron The soliton solution interpolates from the ABAB.. to the BABA... dimerization pattern. Is it possible to have a local deformation which starts off at the ABAB... dimerization pattern, make a possibly large change, perhaps go off to the BABA... pattern and return to the original ABAB... pattern? In other words, can we find deformation patterns of the type (x) = 0 C {tanh[κ(x + x 0 )] tanh[κ(x x 0 )]} which will solve the BdG equations? A way to achieve this would be to adjust the parameters so that the effective potential term in (11.19) should be a pure sech 2 term. Indeed, after some rearrangements, it turns out that the choice C = 0 tanh(2κx 0 ) leads to 2 + hv F = tanh(2κx 0 ) [tanh(2κx 0 ) + κξ 0 )] sech 2 [κ(x + x 0 )] 2 0 tanh(2κx 0 ) [tanh(2κx 0 ) κξ 0 )] sech 2 [κ(x x 0 )]. It is possible to make either one of the two sech 2 functions disappear; the choice which leads to an attractive effective potential is tanh(2κx 0 ) = κξ 0. (11.28) As long as κ does not exceed 1/ξ 0, this condition will specify an x 0 as a function of κ. Let me therefore denote acceptable parameter values as κ = ξ 1 0 sin θ. The effective potential 119

127 11 Solitons in conducting polymers now has a single-well form, 2 + hv F = 2 0 { 1 2κ 2 ξ 2 0 sech 2 [κ(x + x 0 )] } (11.29) with which I may recast (11.19), using the dimensionless variable y = κ(x + x 0 ) and the dimensionless eigenvalue r l = (ɛ 2 l 2 0)/( 0 κξ 0 ) 2, as ] [ 2 y 2 2 sech2 y f ( ) ( ) l (y) = r l f l (y). (11.30) Localized eigenstates of the BdG equations The recasting is useful in order to recognize the prefactor in the potential as n(n + 1) with n = 1, which gives a single localized eigenfunction at r l = 1, corresponding to ɛ b = ± 0 1 κ 2 ξ 2 0 = ± 0 cos θ. Note that the bound states - provided they exist, which we still have to establish by finding acceptable values of κ (or θ)- is not at midgap. Returning to the original units, I write the bound state eigenfunction as and from the BdG equation... f ( ) b (x) = N b sech[κ(x + x 0 )] (11.31) f (+) b (x) = ±in b sech[κ(x x 0 )] (11.32) where the ± sign matches the sign of the energy. The u v eigenstates corresponding to the two energies ± 0 cos θ are u ± b (x) = N b 2 [sechκ(x + x 0) ± i sechκ(x x 0 )] v ± b (x) = N b 2 [i sechκ(x + x 0) ± sechκ(x x 0 )]. (11.33) Their form shows that there is equal probability for the localized electron to be near x 0 or x 0. Extended eigenstates of the BdG equations The extended states of the BdG equations are f ( ) q (x) = N q [ iq + κ tanh κ(x + x 0 )] e iqx f q (+) qξ 0 + i (x) = ±N q [ iq + κ tanh κ(x x 1 + q2 ξ0 2 0 )] e iqx where again the ± sign matches the sign of the energy. The phase shift, in both cases, is δ(q) = 2 arccot q κ. (11.34) It is shown in Fig.. It runs, just like in the soliton case, from zero to π, makes a jump at q = 0 from π to π, and then drops off to zero as q approaches infinity. The similarity with the soliton case is deceptive. This phase shift is really the entire physical shift of the eigenfunction - not of half the eigenfunction. Both f s are phase-shifted by this amount (therefore the physical eigenstates u and v as well). As a result, the q = 0 state disappears entirely (not by a half!) from the valence band. There is, just like in the soliton case, a backflow in the Fermi sea, which redistributes q-vectors as a result of the phase shift. 120

128 11 Solitons in conducting polymers The total energy The states which appear in the gap can in principle be occupied singly, doubly, or not at all. But because the energies of the localized states are now nonzero, this will affect the total energy of the object. Let n +, n = 0, 1, 2 be, respectively, the populations of the ± 0 cos θ localized energy state. The total energy will again consist of an electronic part 2 ( ɛq ɛ ) n q n + (n+ n ) 0 sin θ ( 2 0 ) n 0 and a lattice deformation part. After some calculations analogous to the ones for the soliton, this sums up to E p = 4 0 π sin θ + 4 [ π π 2 θ + π 4 (n + n )] cos θ Considered as a function of θ, the total energy has a minimum at θ 0 = π 4 (n + n + 2). The energy of the polaron E p = 4 π 0 sin θ 0 (11.35) will therefore depend on the occupation of the gap states. distinguished: The following cases can be n n + = 2. We would expect this to be the lowest energy state of the polaron, where the two electrons excluded from the valence band end up, paired, in the lowest localized state (n = 2, n + = 0). This state has θ 0 = 0, i.e. κ = 0, E p = 0. The lowest localized state in reality has returned to the valence band. The deformation corresponds to a constant 0. This polaron is really nothing but the pure dimerized state. n + = n, θ 0 = π/2. The energy is 4 0 /π, the separation x 0 becomes infinite. This is really a kink/antikink pair with its components entirely separated. n n + = 1, θ 0 = π/4. It follows that κξ 0 = 2 1/2. The resulting separation x 0 is finite, x 0 /ξ 0 = arctanh(2 1/2 )/(2 1/2 ). The energy is equal to 2 2/π There are two ways to form this polaron: either by n = 2, n + = 1, i.e. the lower bound state is doubly, and the upper singly occupied, which makes it a charged excitation with an unpaired spin (Q = e, S = 1/2); or by n = 1, n + = 0, i.e. the lower bound state is singly occupied and the upper is empty; this would be a hole polaron, with Q = e, S = 1/2. Note that the spin/charge relationships of the polaron are the same as in ordinary electrons and holes. However, the energy to create a pair of polarons is about 10% lower than that required to create an electron/hole pair. There are no other possibilities. What the various combinations of the second case imply in terms of donor/acceptor character and spin/charge properties is summarized in Fig

129 12 Solitons in nonlinear optics 12.1 Background: Interaction of light with matter, Maxwell-Bloch equations Semiclassical theoretical framework and notation The propagation of electromagnetic waves through a material (gaseous) medium is modeled, at a semiclassical level, as follows: The medium is considered as an assembly of quantum-mechanical two-level systems (2LS) described by a Hamiltonian H 0 = 1 2 hω 0 ( ) with basis states ( 1 >= 0 ) ( 1, >= 0 ) and a general (mixed) state ( α Ψ >= β ) = α > +β >. The density of 2LS is n 0. Each 2LS carries an electric dipole moment. The corresponding dipole moment operator is ( 0 1 ˆp = q 1 0 The electromagnetic field is treated at a classical level. The electric field, propagating along the x-direction, satisfies Maxwell s wave equation ( ) 2 2 c2 t2 x 2 where the polarization is given by ). E(x, t) = 4π 2 t 2 P (x, t) (12.1) P = n 0 < Ψ ˆp Ψ >= n 0 q (α β + β α) } {{ }. (12.2) =2Re(α β) P + The 2LSs interact with electromagnetic radiation via a dipole interaction H 1 = p E. 122

130 12 Solitons in nonlinear optics Dynamics The quantum mechanical wavefunction evolves in time according to or, in component form, Let and and Then one can explicitly verify that or, taking real and imaginary parts, i h t Ψ >= (H 0 + H 1 ) Ψ >, i h α = hω 0 2 α ( q E)β (12.3) i h β = hω 0 2 β ( q E)α. (12.4) P = i (α β + β α) = 2Im(α β) Z = P + + ip = 2α β N = α 2 β 2. i hż = hω 0Z 2( q E)N, P + t P t = ω 0 P (12.5) = ω 0 P h ( q E)N. (12.6) Moreover, multiplying (12.3) by α, (12.4) by β, taking the real part of sum of the two equations, we obtain N t = 2 h ( q E)P (12.7) The triad of eqs (12.5), (12.6), (12.7) for the real functions P, P +, N is equivalent to the pair of eqs (12.3), (12.4) for the two complex amplitudes α, β. This is because the complex amplitudes have a constant normalization, which we choose as unity: α 2 + β 2 = 1 The system of equations (12.1) - with the right hand side given by P = n 0 q P + = n 0 ω 0 q P = n 0 ω 2 0 q P + 2 h n 0ω 0 ( q E) qn (12.8) -, (12.5), (12.6) and (12.7) are known as the Maxwell-Bloch (MB) equations. They were originally derived in the context of nuclear magnetic resonance (with the correspondence S x P +, S y P, S z N ) Propagation at resonance. Self-induced transparency Slow modulation of the optical wave At resonance, the carrier (optical) frequency of the electromagnetic wave coincides with the frequency of the 2LS: ω = ω 0 123

131 12 Solitons in nonlinear optics We look for solutions of the MB equations of the form E = h q E cos(kx ω 0t + φ) (12.9) } {{ } Ψ where k = ω 0 /c and E, φ are slowly varying functions of x, t; furthermore, we are taking the field to be polarized in the z-direction and q = qẑ. Using the transformation P + = Q cos Ψ + PsinΨ (12.10) P = P cos Ψ Q sin Ψ (12.11) one can rewrite (12.5), (12.6), respectively, as ( Q cos Ψ t + P φ ) ( ) P + sin Ψ t t Q φ t ( ) ( P Q cos Ψ t Q φ sin Ψ t t + P φ ) t = 0 = 2EN cos Ψ. The above equations can be further simplified if we (i) multiply the first by sin Ψ, the second by cos Ψ and then add them, or (ii) multiply the first by cos Ψ, the second by sin Ψ and then subtract them. The result is P t Q φ t Q t + P φ t = 2EN cos 2 Ψ = EN sin 2Ψ or, averaging over a period 2π/ω 0 of the (fast) carrier wave, Similarly, (12.7) can be rewritten as P t Q φ t Q t + P φ t = EN (12.12) = 0. (12.13) N t = 2E cos Ψ ( Q sin Ψ + P cos Ψ) which averages to N t = EP. (12.14) Finally, keeping only the first term in (12.8) - a valid approximation as long as ω 0 E -, transforms the field equation (12.1) to ( t + c ) ( x t c ) E cos Ψ = 2c α (Q cos Ψ + P sin Ψ), (12.15) x where α = 2πn 0q 2 ω 0 hc At resonance, the left hand side of (12.15) simplifies considerably. We recognize ( t c ) E cos Ψ 2ω 0 E sin Ψ x. 124

132 12 Solitons in nonlinear optics and ( t + c ) ( ) ( ) E φ E sin Ψ = x t + c E sin Ψ + cos Ψ x t + c φ x Combining these, and matching sine and cosine terms in (12.15) results in. E t + c E x = cα P (12.16) ( ) φ E t + c φ = cα Q. (12.17) x The set of 5 equations (12.12), (12.13), (12.14), (12.16), (12.16) describes the slow modulation of the coupled wave-medium system variables P, Q, N, E, φ Further simplifications: Self-induced transparency In the approximation of vanishing phase φ = 0 (12.17) implies Q = 0. This leaves a reduced set of three equations P t = EN N t = EP E t + c E x = cα P, where the first two have an obvious first integral, This suggests the parametrization N 2 + P 2 = const P = ± sin σ, N = ± cos σ, which in turn implies, from the first two equations, E = σ t ; the last equation, ( t + c ) σ x t = ±cα sin σ can be cast into a more useful form by introducing new space and time coordinates ξ = α x τ = t x c. The transformed version 2 σ ξ τ is a form of the Sine-Gordon(SG) equation. = ± sin σ (12.18) 125

133 12 Solitons in nonlinear optics Propagating solutions. Slowing down of light The SG equation is completely integrable. It is known to admit multisoliton solutions. Here I will restrict myself to some of the properties of single solitons. A property of (12.18) is that it admits solutions of the form σ(z), where z = aτ ξ ( a = a t x ) c α x a 2 (12.19) and a is an arbitrary constant (a 1 will be identified as the pulse duration). Introducing this type of solution Ansatz to (12.18) leads to d 2 σ = ± sin σ. (12.20) dz2 Before I discuss the properties of the solution, let me note that (12.19) can be further rewritten in the form ( z = a t x ) v with 1 v = 1 c + α a 2 (12.21) which implies that light will be slowed down. The 2π pulse The choice of the lower sign in (12.20) leads to the well-known kink/antikink solutions of the SG equation, σ = 4 arctan e ±(z z0). The resulting field is and satisfies the sum-rule The 2LS inversion E = σ τ = adσ dz = ± 2a cosh[a(t x v t 0)] dt E(x, t) = dτ σ τ 2 N = cos σ = 1 + cosh 2 [a(t x v t 0)] = σ( ) σ( ) = 2π. (12.22) starts off at 1 as t, increases up to 1 as the pulse reaches the 2LS, and then returns to 1 as t. Thus, the electromagnetic wave brings about a temporary inversion of the 2LS; as the pulse travels further however, the 2LS becomes deexcited and gives the energy back to the field. No net absorption of energy occurs. This is the phenomenon of self-induced transparency Self-focusing off-resonance Off-resonance limit of the MB equations Off-resonance propagation occurs when the carrier frequency of the optical wave is much lower than the eigenfrequency of the 2LS: ω ω 0 (12.23) 126

134 12 Solitons in nonlinear optics In this case, the field does not cause inversion, i.e. the MB equations hold with N 1. The dominant time dependence of the polarization vector is determined by the carrier frequency of the wave, hence P ω 2 P and P + ω 2 P +. On the other hand, (12.5) and (12.6) with N = 1 imply that i.e. ( ) 1 ω2 0 ω 2 P + = ω0p ω 0 q E h P + = 2ω 0 q E h and, making use of the off-resonance condition (12.23), P + = 2ω2 q hω E 0 (12.24) P + = 2 q hω E 0. (12.25) Inserting (12.24) in the field equation (12.1) yields ( ) 2 2 c2 E(x, t2 x 2 t) = 8πn 0 ω 2 ( q hω E) q. 0 Up to now, we implicitly assumed that all 2LS carry the same dipole moment. This is of course not quite true. Even if the magnitude of the dipole moment is the same (an assumption which we will make), the random orientation of 2LS in a medium implies a distribution of values for each component. If the field is polarized along the z-direction, what really enters the right-hand-side of the field equation is an orientational average of qz. 2 Therefore, ( ) 2 2 c2 t2 x 2 E = 8πn 0 < qz 2 > ω 2 E. (12.26) hω 0 Modulation of the carrier wave We will again look for solutions of the form E(x, t) = E c cos(kx ωt) + E s sin(kx ωt) (12.27) where ω = ck and E c, E s are slowly varying functions of x, t. In this context, the slowly varying modulation of the field responds only to averages over the fast phase. Thus, for example, the short-time average of the square of the field (over a period 2π/ω) will be E 2 = 1 2 ( E 2 c + E 2 s ). (12.28) Nonlinear terms The orientational average of the square of the z-component of the dipole moment < q 2 z >=< cos 2 θ > q 2 127

135 12 Solitons in nonlinear optics is defined as < cos 2 θ >= 1 1 d cos θ cos2 θ e βh d cos θ e βh 1 (12.29) where H 1 = p E = q z EP + = 2q2 z hω 0 E 2 = 2q2 hω 0 E 2 cos 2 θ, β is the inverse temperature, and I have made use of (12.25). Furthermore, since I am interested in slow wave modulation, it is legitimate to substitute the square of the field by its time average over a period of the carrier wave, using (12.28). Defining ρ = β q2 ( E 2 hω c + E 2 ) s 0 I can rewrite < cos 2 θ >= ln I(ρ) ρ where I(ρ) = 1 1 ( dx e ρx ρ ) 5 ρ2 + O(ρ 3 ) where the expansion is valid in the limit of low fields and/or high temperatures, ρ 1. In this limit ln I(ρ) ln ( 1 3 ρ ) ρ 2 18 < cos 2 θ > 1 ( ) 3 ρ. (12.30) and the wave equation (12.26) can be rewritten as, with ( ) 2 2 c2 t2 x 2 E = [ ( G 0 + G 2 E 2 c + Es 2 )] ω 2 E. (12.31) G 0 = 8π n 0 q 2 3 hω 0 G 2 = 4 βq 2 G hω Space-time dependence of the modulation: the nonlinear Schrödinger equation Consider the complex modulational field φ = E c + ie s. Then, if the following properties hold: F = φe i(kx ωt) E = ReF F 2 = φ 2 = E 2 c + E 2 s 128

136 12 Solitons in nonlinear optics Therefore, if F satisfies ( 2 2 c2 t2 x 2 ReF will satisfy the original field equation (12.31). solutions of (12.32). Now examine the left hand side of (12.32). First note that ) ( F = G 0 + G 2 F 2) ω 2 F, (12.32) ( t + c ) ( F = e i(kx ωt) x t + c ) φ x It is therefore sufficient to look for and therefore ( t c ) ( x t + c ) {( ) ( F = e i(kx ωt) 2 2 c2 x t2 x 2 φ + 2iω t + c ) } φ x. This allows me to rewrite (12.32) as ( ) ( 2 2 c2 t2 x 2 φ + 2iω t + c ) ( φ = G 0 + G 2 φ 2) ω 2 φ (12.33) x which involves only the modulating field φ. Eq. (12.33) is still exact. If we restrict ourselves to slow modulations, the second time derivative should be small and might be dropped. In this case, introducing new dimensionless variables ξ = kx ωt, τ = 1 2 ωt transforms (12.33) to ( iφ τ = φ ξξ + G 0 + G 2 φ 2) φ. Introducing φ = G 1/2 2 exp( ig 0 t) ˆφ eliminates the first term in the parentheses and rescales the rest, leading to i ˆφ τ = ˆφ ξξ + ˆφ 2 ˆφ, (12.34) the canonical form of the nonlinear Schrödinger (NLS) equation Soliton solutions The NLS equation can be integrated exactly, for arbitrary initial conditions (meaning: suitably vanishing at plus and minus infinity), by the inverse scattering transform. This means that it admits multisoliton pulses as exact solutions - a fact of potentially vast technological significance. The interested reader is referred to the specialized literature. Here, I will restrict myself to a heuristic derivation of the single pulse solution. I look for solutions of (12.34) of the form ˆφ = ue iθ with u, θ real. Real and imaginary terms lead, respectively, to u τ + 2u ξ θ ξ + uθ ξξ = 0 uθ τ + u ξξ uθ 2 ξ + u 3 =

137 12 Solitons in nonlinear optics I use the traveling wave cum linear phase Ansatz θ(ξ, τ) = µτ + ˆθ(z) u(ξ, τ) = u(z), where z = ξ λτ, to reduce the PDEs into ODEs: λu z + 2u z ˆθz + uˆθ zz = 0 (12.35) µu + λuˆθ z + u zz uˆθ z 2 + u 3 = 0 (12.36) Multiplying the first equation by 2u results in λ(u 2 ) z + 2(u 2 ) z ˆθz + 2u 2 ˆθzz = 0 which has a first integral u 2 (λ 2ˆθ z ) = constant. (12.37) An obvious choice for the value of the constant is zero. Introducing into the second equation yields u zz = ˆθ z = λ 2 ) (µ λ2 u u 3 = d ) 1 (µ λ2 u du u4 } {{ } V eff (u) (12.38) (12.39) which looks very much like the ODEs found in the context of scalar field theories of the µ-λ 2 / V eff (u) Figure 12.1: The effective potential (12.39) in the two cases µ > λ 2 /4 (upper curve) and µ < λ 2 /4 (lower curve) u Klein-Gordon class. The difference is that, whereas in the soliton-bearing KG class the effective potential had at least two degenerate stable minima, the effective potential here has either a single minimum (at u = 0) - and two maxima - if µ > λ 2 /4, or no minimum at all - and a single maximum at u = 0 - if µ λ 2 /4. Such a potential (Fig. 12.1) would 130

138 12 Solitons in nonlinear optics of course be entirely unphysical in the context of field theory, because of the instability at large displacements. Here there is no such physical restriction. A soliton-like solution can occur as long as there is a single, locally stable minimum; it will lead the system from the local minimum out to either one of the maxima and back to to the local minimum. Note that this implies that, in the first integral of (12.39), u 2 z = 2V eff (u) + const the constant vanishes. This leads to the formal second integral 1 z z 0 = ± du 2V (u) and the bounded solutions u(z) = ± κ sech κ(z z 0) 2 (12.40) where I have used the more appropriate constant ( ) κ = + 2 µ λ2 4. Note that since I have up to now introduced two arbitrary constants (in addition to the arbitrary phase z 0 ), I can take any value of κ > 0 and λ; µ will then have the fixed (and positive) value To conclude, I note that from (12.38) µ = κ2 2 + λ2 4. ˆθ = λ 2 z + θ 0 (12.41) where θ 0 is an arbitrary phase. Collecting terms, and returning to the original space-time variables, u(ξ, τ) = ± κ sech κ(ξ λτ z 0) (12.42) 2 θ(ξ, τ) = λ { ( ) } λ ξ 2 2 κ2 τ + θ 0 (12.43) λ Note that envelope amplitude and phase propagate with different velocities, v e = λ and v ph = λ/2 κ 2 /λ, respectively. 131

139 13 Solitons in Bose-Einstein Condensates 13.1 The Gross-Pitaevskii equation Starting point: Gross-Pitaevskii equation [33, 34] for the weakly interacting Bose gas (particles of mass m): i h } t Ψ 0 = { h2 2m 2 + V ext ( r) + g Ψ 0 2 Ψ 0 (13.1) where Ψ 0 ( r, t) is the condensate wave function; the condensate density is then g = 4π h 2 a/m the coupling constant, with n( r) = Ψ 0 2. a the s-wave scattering amplitude (low-energy characteristic of the effective potential). the external potential, typically of the form describes a cylindrically symmetric magnetic trap. V ext ( r) = α(x 2 + y 2 ) + λz 2 (13.2) In the limit λ α, I look for solutions which depend only on z; these must satisfy i h t Ψ 0 = { h2 2 } 2m z 2 + g Ψ 0 2 Ψ 0 (13.3) which I recognize as the nonlinear Schrödinger equation. A useful quantity is the characteristic length defined by ξ 2 = h2 2mgn = 1 (13.4) 8πan where n is the average condensate density Propagating solutions. Dark solitons I look for propagating solutions of the form where Ψ 0 = ne iµt/ h φ(ζ) ζ = z vt ξ and µ = gn is the chemical potential. I will treat the case g > 0 (repulsive interaction). 132

140 13 Solitons in Bose-Einstein Condensates Introducing the above ansatz in (13.3) reduces it to i 2 v c where c = gn/m is the sound velocity. dφ dζ = d2 φ ( dζ φ 2) φ (13.5) Figure 13.1: Absorption images of BEC s with kink-wise structures propagating in the direction of the long condensate axis, for different evolution times in the magnetic trap, t ev. The moving dark regions can be interpreted as a pair of gray solitons. (From [35]). I first rewrite (13.5) as a system of coupled ODEs for real and imaginary parts of φ = φ 1 + iφ 2 : v dφ 1 2 c dζ 2 v dφ 2 c dζ = d2 φ 2 dζ 2 + (1 φ2 1 φ 2 2)φ 2 = d2 φ 1 dζ 2 + (1 φ2 1 φ 2 2)φ 1. I now look for solutions with a constant imaginary part φ 2 = A. These must satisfy v dφ 1 2 c dζ A(1 A2 φ 2 1) = 0 d 2 φ 1 dζ 2 + (1 A2 φ 2 1)φ 1 = 0. (13.6) Muitiplying the first equation by φ 1 and the second by A, and taking the difference gives A d2 φ 1 dζ v c dφ 1 dζ φ 1 = 0 which has a first integral A dφ 1 2 dζ + v 2 c φ2 1 = C The latter, first order ODE must be however be identical with the first of (13.6). mandates A 2 = v 2 /c 2 and fixes the constant C. I then obtain the obvious solution 1 γ tanh ζ 2γ where γ = (1 v 2 /c 2 ) 1/2. Collecting terms, I obtain φ(x vt) = i v c + 1 γ This tanh x vt 2γξ (13.7) which has unit amplitude as x ±, and drops to v/c at x = vt (gray soliton). If v = 0 the soliton is dark, i.e. the amplitude vanishes at x = 0. Fig shows an experimental observation of a dark soliton in a BE condensate. 133

141 14 Unbinding the double helix 14.1 A nonlinear lattice dynamics approach Mesoscopic modeling of DNA Background: thermodynamic phase transitions Transitions between different states of matter (e.g. the transition from the paramagnetic to the ferromagnetic phase, or the liquid-gas transition) are reflected in singularities of the thermodynamic functions (free energy, entropy etc). The modern theory of critical phenomena developed by Fisher, Kadanoff and Wilson during the 1960 s and 70 s has demonstrated that the essential features of such mathematical singularities depend only on a few relevant degrees of freedom of the underlying Hamiltonian. As a consequence, substantial effort has been made by researchers in developing and studying appropriately reduced descriptions, minimal models of many complex phenomena related to transformations between different states of matter. Figure 14.1: Melting of poly(di)-poly(dc) (after [36]). Experiment suggests DNA denaturation is a sharp phase transition The thermal denaturation of DNA (also known as DNA melting) consists of the unbinding of the double helix into the two component strands. There is no breaking of covalent bonds along the chain, and the transition is in principle reversible. In the case of DNA chains which are long (of the order of N 10 4 base pairs) and homogeneous (i.e. all base pairs are identical), the transition, as observed by the difference in UV absorption spectra, can be very sharp (Fig. 14.1). It is then perfectly reasonable to assume that the underlying phenomenon would be an exact phase transition in the thermodynamic limit N and attempt to model it accordingly. 134

142 14 Unbinding the double helix Mesoscopic modeling: 1 degree of freedom per base pair A reduced, mesoscopic description of DNA consists of assigning a single continuous, transverse degree of freedom y n to the nth base pair, corresponding to the distance between the two bases which comprise the pair. The energy related to this degree of freedom has its physical origin in the hydrogen bonds which are responsible for pair binding. Accordingly, it is modeled by a Morse potential V (y) = D(e ay 1) 2 where D is an average measure of the binding energy and 1/a a length which characterizes the range of the hydrogen bonding. The tendency of successive base pairs to stack ( stacking interaction) can be modeled by assuming that they are bound together by springs. For simplicity, I will assume these springs to be harmonic. The total Hamiltonian will then be of the form [37] H = n [ p 2 n 2µ + 1 ] 2 µω2 0(y n+1 y n ) 2 + V (y n ), (14.1) where µ is the reduced mass corresponding to the base pair, p n = µẏ n is the canonical momentum conjugate to y n and µω 2 0 a measure of the strength of the stacking interaction. Note that this minimal modeling makes no reference to the helical structure of the molecule. Although generalizations to that effect have been formulated, it should be borne in mind that this type of modeling makes no attempt to describe structural details of the DNA molecule. Its scope begins and ends with capturing essential observed macroscopic features at and very near the denaturation point Thermodynamics The classical thermodynamics of H is described by the canonical partition function Z = N n=1 dp n dy n e βh. (14.2) which factorizes into a product of Gaussian integrals over the momentum variables, Z K = (2πµ/β) N/2, (14.3) and a nontrivial configurational part ( ) N Z P = dy n T (y 1, y 2 ) T (y N 1, y N ) T (y N, y N+1 ), (14.4) n=1 where [ ] T (x, y) = e β µω (y x)2 +V (x). (14.5) The transfer integral formalism: definitions and notation Consider the eigenvalue problem defined by the asymmetric kernel T (the kernel can be easily symmetrized but need not be so; in fact, working with the asymmetric kernel is technically 135

143 14 Unbinding the double helix advantageous in examining the validity of some approximations, cf. below): dy T (x, y) Φ R ν (y) = Λ ν Φ R ν (x) (14.6) dy T (y, x) Φ L ν (y) = Λ ν Φ L ν (x), (14.7) where left and right eigenstates have been assumed to be normalized; note that the normalization integral is dxφ L ν (x)φ R ν (x). Orthogonality and completeness dx Φ L ν (x) Φ R ν (x) = δ νν (14.8) Φ L ν (x) Φ R ν (y) = δ(x y) (14.9) relationships are assumed to hold. I will further use the notation (sensible as long as the eigenvalues are nonnegative). ν Λ ν = e βɛν (14.10) Relationship between Z and the spectrum of T The integrand of (14.4), as written down has a problem: it includes a reference to the displacement y N+1 of the N + 1st particle, which has not yet been defined. For a large system, this is best remedied by means of periodic boundary conditions (PBC), i.e. by demanding that y N+1 = y 1. Alternatively, the integration may be extended to one more variable, dy N+1, with the simultaneous introduction of a factor δ(y N+1 y 1 ) to take care of PBC. This however is the same as the sum in the left-hand-side of (14.9). I then obtain Z P = dy 1 dy N+1 Φ L ν (y 1 ) } {{ } T (y 1, y 2 ) T (y N, y N+1 )Φ R ν (y N+1 ) } {{ }. (14.11) ν The braces make clear that I can perform the integral over dy N+1 and obtain a factor Λ ν Φ R ν (y N+1 ), using the defining property of right-hand eigenfunctions. The process can be repeated N times, each time giving a further factor Λ ν and a right eigenfunction with an argument whose index is smaller by one. At the end, I am left with Z P = dy 1 Φ L ν (y 1 )Λ N ν Φ R ν (y 1 ) = Λ N ν. (14.12) ν ν In the thermodynamic limit, Z P is dominated by the largest eigenvalue Λ 0 or, equivalently, the lowest ɛ 0 : 1 lim N N ln Z P = ln Λ 0 = βɛ 0 (14.13) The order parameter < y i > = 1 Z P dy 1 dy N T (y 1, y 2 ) T (y i 1, y i )y i T (y i, y i+1 ) T (y N, y N+1 ) 1 dy 1 dy N+1 Φ L ν (y 1 ) T (y 1, y 2 ) T (y i 1, y i ) Z P } {{ } ν i 1 T (y i, y i+1 ) T (y N, y N+1 ) Φ R ν (y N+1 ), (14.14) } {{ } N i+1 y i 136

144 14 Unbinding the double helix after insertion of a complete set of states (cf. above); the braces denote the number of times I can perform an integration and obtain, respectively, a right eigenfunction with an argument smaller by one, or a left eigenfunction with an argument larger by one, as well as a factor Λ ν. The remaining integral must be performed explicitly: < y i > = 1 Λ N ν M νν Z P ν M 00 (14.15) where the second line is exact in the thermodynamic limit, and I have used the abbreviation M νµ = dyφ L ν (y)yφ R µ (y). (14.16) The spectrum of T: Gradient-expansion approximation; analogy with quantum mechanics Suppose that the displacement field does not change appreciably over a lattice constant. This is certainly reasonable at low temperatures. Note that this does not exclude large displacements per se. Nonlinearity is explicitly allowed, but the displacement field must be smooth. The assumption is certainly reasonable at low temperatures. I set y = x + z, Φ R φ and rewrite (14.6) as e β[ɛ ν V (x)] φ ν (x) = = + [ 2π βµω 2 0 { dz e 1 2 βµω2 0 z2 φ ν (x) + zφ ν(x) + 1 } 2 z2 φ ν(x) ] 1/2 { φ ν (x) + 1 } 2βµω0 2 φ ν(x) (14.17) where higher terms in the gradient expansion have been neglected and the Gaussian integrals have been performed; this is meaningful as long as the width of the Gaussians is smaller than the range of the Morse potential, i.e. βµω 2 0/a 2 > 1. (14.18) The factor in front of the r.h.s. of (14.17) can be absorbed in the eigenvalue by defining ɛ ν = ɛ ν + 1/(2β)] ln[2π/(βµω 2 0)]. Now, for many practical purposes, when it comes to calculating matrix elements, the relevant magnitude of ɛ V (x) is D, the depth of the Morse well (or some other characteristic energy in the case of another potential). The key to this statement is that one does not need to consider large negative values of x, where V (x) is huge, because at such x, both the exact eigenfunction Φ and its approximation φ can be expected to be negligible. If then βd 1 1 it is reasonable to expand the exponential and keep only the first term. Dividing both sides by β, I obtain a Schrödinger-like equation, 1 2µ(βω 0 ) 2 φ ν(x) + [V (x) ɛ ν ] φ ν (x) = 0. (14.19) Before continuing the discussion of (14.19) and its properties, I pick up the bits and pieces (cf (14.2), (14.3), (14.13) ) of the thermodynamic free energy (per site) f = 1 βn ln(z KZ P ) 1 ( ) 2π β ln + βω f, (14.20) 0 1 Note that, in connection with (14.18), this defines a temperature window D < k B T < µω 2 0 /a2 for the validity of the overall approximation scheme. 137

145 14 Unbinding the double helix where f = ɛ 0. The first term in (14.20) is the free energy of the small oscillations (transverse phonons in this context). It is a term smooth in temperature (constant specific heat!) and therefore irrelevant to any phase transition. Any nontrivial physics is hidden in the second term, which is identical with the the smallest eigenvalue of (14.19). A couple of comments are in order. First, (14.19) would be a literal (i.e. quantummechanical) Schrödinger equation, if I substituted 1/(βω 0 ) by h. I will come back to that point. Second, I can get a dimensionless potential (and eigenvalue) by dividing both sides of (14.19) by D. In other words, the relevant dimensionless parameter is δ 2 = 2µ a 2 h 2 D (quantum mechanics) 2µβ 2 ω 2 0 a 2 D (statistical mechanics). In terms of δ, the bound state spectrum of (14.19) is given [38] by ] 2 (14.21) ɛ n [1 D = 1 n + 1/2 δ n = 0, 1,..., int(δ 1/2). (14.22) There is at least one bound state if δ > 1/2. For 1 δ > 1/2 there is exactly one bound state. And if δ becomes equal to, or smaller than 1/2, there is no bound state at all. The value δ c = 1/2 is critical. In quantum mechanical language, if a particle has a mass which is lighter than a critical mass µ c = h 2 a 2 /(8D), it cannot be confined in the Morse well. Quantum fluctuations will drive it out 2. Thermodynamic free energy In the context of statistical mechanics, δ c corresponds, via (14.21), to a critical temperature T c = 2(ω 0 /a) 2µD. The free energy is given by 1 T > T c f D = ( ) T T c T < T c, (14.23) where in the upper line I have made use of the fact that the bottom of the continuum part of the spectrum is at ɛ = D. The free energy f is non-analytic at T = T c, where its second derivative is discontinuous (i.e. there is a jump in the specific heat). This corresponds to a second order transition, according to the Ehrenfest classification scheme 3. The order parameter. DNA melting as a thermodynamic instability In order to gain some further insight into the physics involved 4 it is useful to examine the average displacement (14.15), determined by the ground-state (GS) eigenfunction φ 0 (x) = e ζ/2 ζ δ 1/2 (14.24) 2 This is a general property of asymmetric one-dimensional wells; symmetric wells will support a particle in a bound state, no matter how low its mass. 3 Note that the term second order is meant literally in this case, not just as a metaphor for the absence of a latent heat (for which the term continuous transition would be appropriate). 4 The mathematical analogy between the behavior of the spectral gap which occurs in a point (d = 0) system and the singularity in the free energy of a classical chain (d = 1) is an example of a deeper analogy which relates quantum to thermal fluctuations; the formal correspondence h 1/(βω 0 ) manifests a farreaching analogy between d-dimensional quantum mechanics and (d + 1)-dimensional classical statistical mechanics. The analogy is most fruitful at d = 1, because of the interplay and the richness of exact available results which based either in the transfer-matrix approach of 2-dimensional classical statistics or on the Bethe-Ansatz developed for 1-d quantum spin systems. 138

146 14 Unbinding the double helix where ζ = 2δe ax. It is straightforward to see that, as T approaches T c from below, the eigenfunction extends towards larger and larger positive values of x: where φ 0 (x) e λx (14.25) λ = 1 δ δ c (14.26) is a (transverse) characteristic length which measures the spatial extent of the GS eigenfunction. As a consequence, we can estimate that < y >, which is dominated by the large values of the argument, will also behave as < y > (δ δ c ) 1 ( 1 T T c ) 1. (14.27) As the critical temperature is approached from below, particles cease to be confined to the minimum of the Morse well. They perform larger and larger excursions to the flatter part of the potential. At T c the transition is complete; the average transverse displacement is infinite. Particles move, on the average, on the flat top of the Morse potential. Unwinding ( melting ) of the DNA has occurred. In the language of critical phenomena < y > is the order parameter. In ordinary phase transitions, where one goes from an ordered to a disordered phase, the order parameter m vanishes at the transition point, i.e m (T c T ) β with a positive critical exponent β 5. DNA melting is really an instability - rather than an order-disorder transition. It is therefore not surprising that the corresponding critical exponent β extracted from (14.27) is negative (-1). Experimental data on DNA denaturation do not deliver < y > directly. The experimental order parameter is the helical fraction, i.e. the probability that a given base pair is still bound; technically one uses an (instrumentation-dependent) cutoff y 0 and measures P (y > y 0, T ). For the model presented here, this function approaches zero smoothly (linearly) as T T c, independently of the choice of y Nonlinear structures (domain walls) and DNA melting In discussing how adsorbed atoms arrange themselves on a substrate, we examined a number of possibilities: a uniform structure, commensurate with the substrate, and a soliton lattice. We found that the commensurate-incommensurate phase transition occurred when the mismatch between the competing lattice periodicities made the soliton lattice energetically favored. The DNA denaturation - as described within the model Hamiltonian (14.1) - is a thermal, not a parametric instability. Nonetheless, it will prove useful to examine the existence and properties [39] of competing nonlinear structures of (14.1). In this section I will use dimensionless variables ay n y n ; the energy will be measured in units of D. The total potential energy will then be Φ = [ ] N 1 2R (y n+1 y n ) 2 + V (y n ) n=0 where R = D/(Ka 2 ) is a dimensionless coupling constant. 5 not to be confused with the inverse temperature; this is the standard notation of critical phenomena (14.28) 139

147 14 Unbinding the double helix Local equilibria Definition Local equilibria are defined by static solutions of the equations of motion, i.e. by extrema Φ y n = 0 n. (14.29) of the total potential energy. Their spatial patterns, for a given boundary condition y 0 = 0, y N+1 = L are described by a second-order difference equation y n+1 2y n + y n 1 + RV (y n ) = 0 n = 1, N. (14.30) Fixed point There is only one fixed point y n = 0 of the map. Note however that it is compatible only with the boundary condition L = 0. Note further that the energy associated with the fixed point configuration is zero. n Stability criteria The stability of equilibria is governed by the spectra of the corresponding N N Hessian matrix h ij = 2 Φ (14.31) y i y j where the derivative is evaluated at the extrema defined by (14.30). Let Λ ν, ν = 1,, N denote the eigenvalues of the matrix h. If, for a given extremum, the eigenvalues are all positive, then the extrema is a local minimum. If they are all negative it is a local maximum. If some are negative and some are positive it is a local saddle point. I will not discuss the interesting marginal case where an eigenvalue vanishes, since it does not arise in the context of this particular problem. Picturing and classifying equilibria What do these equilibria look like? A picture can be given by looking at the full set of solutions of (14.30), without fixing the value of L, and then choose the ones that correspond to the boundary condition y N+1 = L. This can be done by noting that (14.30) for unspecified L is equivalent to all realizations of the two-dimensional map p n+1 = p n + RV (y n ) y n+1 = y n + p n+1, (14.32) where n = 1,, N, y 1 = p 1 + y 0, y 0 = 0 and p 1 is unspecified. The set of all orbits of the map thus derived is shown in Fig (14.2). Note that there are two kinds of orbits. Stable ones (drawn with full points) and saddles (drawn with open points), where all but one eigenvalues of the Hessian are positive. It is then possible to isolate those orbits which correspond to a given L. They are shown in Fig. (14.2.1). We note that they start off at a value very close to zero, remain there for a few sites, and then suddenly take off with a constant linear slope. The equilibria pictured here represent in some sense interpolations - or domain walls (DWs - between the bound (y 0) and the unbound (y 1) phase. 140

148 14 Unbinding the double helix y p Figure 14.2: The unstable manifold of the FP of the map (14.32) for R = 10.1 and N = 28. Black squares belong to stable equilibria, red open circles belong to unstable equilibria. The horizontal line at y = 44 demonstrates the multivaluedness of the manifold as a function of y (4 stable and 3 unstable equilibria with that value of y; details in the inset). The vertical lines are drawn at p min and p max, the minimal and maximal asymptotic slopes of DWs (from [39] y n E p n Figure 14.3: The 8 stable equilibria corresponding to N = 28, y 0 = 0, y N+1 = L = 80. Not shown are 7 unstable equilibria enmeshed between the stable ones. Inset: total energies for both stable (black squares) and unstable (red open circles) equilibria. The continuous curve corresponds to a theoretical estimate which does not distinguish between stable and unstable equilibria (cf. text); (from [39]). Configuration of lowest energy for a given L 0. For a given slope p of the unbound segment, there are L/p unbound sites. The excess energy from them is ( ) p 2 L E(p) = 2R + 1 p and has a minimum at p = p = (2R) 1/2. Thus, the minimum energy required to create a DW at a given L is ( ) 1/2 2 E (L) = L. (14.33) R As long as this energy is not available, the system will, if left to itself, prefer the equilibrium available at the fixed point. In order to maintain the transverse displacement L, one must apply an external force f = de (L) dl = ( ) 1/2 2. (14.34) R This is exactly what happens in single-molecule experiments which achieve mechanical DNA denaturation or, as commonly called, DNA unzipping. 141

149 14 Unbinding the double helix Λ ν N=32, L=100 0 * ν Figure 14.4: Eigenvalue spectra of the Hessians for (i) the fixed point (open squares) and (ii) the DW with minimal energy (open circles) for L = 100. In both cases N = 32. The DW s spectrum consists of bands of optical and acoustic phonons, localized respectively in the bound and unbound portions of the chain, and a single local mode in the gap; both bands are well described (to order O(1/L)) by the corresponding free phonon dispersion curves (dotted); (from [39]) Thermodynamics of domain walls At any finite temperature, we will have to consider the competition of the two possible structures: the one corresponding to the fixed point, and the corresponding to the DW with the least energy. Strictly speaking, in the latter case we are considering the totality of all possible values of L. Free energy associated with a given minimum For small displacements around any given local minimum {ȳ n }, i.e. y n = ȳ n + u n the total potential energy will be given, to quadratic order in the displacements, by Φ(u) E(ȳ) + 1 h ij u i u j, 2 where E(ȳ) is the energy of the local minimum. The associated configurational part of the partition function will be ( N ) Z(ȳ) = e βe(ȳ) du m e β 2 i,j m=1 i,j h iju i u j N ( ) 1/2 2π = e βe(ȳ) (14.35) βλ ν ν=1 where the product runs over all eigenvalues of the Hessian. Note that the eigenvalues must be strictly positive, not just nonnegative. The free energy associated with any given minimum will then be F (ȳ) = T ln Z(ȳ) = E(ȳ) T N ( ) 2π ln (14.36) 2 βλ ν Comparison of free energies The spectra of (i) the fixed point and (ii) the stable DW with the minimal energy at some finite L are shown in Fig ν=1 142

150 14 Unbinding the double helix Now consider the difference in free energies between the DW with the minimal energy at some finite L and the fixed point F (L) = E (L) T 2 N ( ) Λ 0 ln ν where the star in the superscript denotes the DW and the 0 the fixed point. ν=1 Λ ν (14.37) The second term represents the difference in entropies. Formation of the DW generates a gain in entropy. The quantity 1 2 N ( ) Λ 0 ln ν L p σ(r) ν=1 Λ ν is generally proportional to the number L/p of unbound sites. The extra entropy comes exclusively from the unbound part. It is due to the fact that the acoustic phonons which live in the unbound region have lower frequencies than the optical phonons which live in the bound state defined by fixed point. Combining terms, the difference in free energy can be written as which becomes zero at F (L) = [2 T σ(r)] L p (14.38) T c (R) = 2 σ(r) (14.39) and negative at higher temperatures. Thus, if the temperature is raised beyond T c (R) a DW of any length can be formed spontaneously - since it generates a net gain in free energy. Denaturation can occur spontaneously. Alternatively, we may look at the derivative p = df (L) dl = [2 T σ(r)] 1 p (14.40) which represents the unzipping force at a finite temperature T. Spontaneous thermal denaturation is, in this sense, equivalent to the vanishing of the unzipping force. For not too large values of R, the proportionality constant is ( ) σ(r) = ln R/ R/2. (14.41) In the continuum limit, R 1, this gives a T c = 2(2/R) 1/2, which is exactly the critical temperature found in Section In summary, what I have presented in this lecture is an alternative picture of the DNA instability, based on the underlying, competing nonlinear equilibrium structures (domain walls vs. fixed point). The results suggest that the domain wall, via the entropic gain it generates, can overcome the energetic cost of its production. In other words, spontaneous formation of a DW at the instability temperature is what really drives DNA denaturation. 143

151 15 Pulse propagation in nerve cells: the Hodgkin-Huxley model 15.1 Background The physics of electric pulse propagation in nerve cells [40] has a long and interesting history. Helmholtz measured the signal velocity on a frog s sciatic nerve in Bernstein succeeded shortly thereafter (1868) in detecting the complete shape of the pulse, the action potential V (t) as a function of time. The concept of a nerve cell, or neuron as an independent functional unit was established through extensive anatomical studies by Ramón y Cajal in the beginning of the twentieth century. Typically, a neuron (Fig. 15.1) consists of an input collecting part with a dendritic structure (dendrites), a cell nucleus, and an output fiber (axon) which transports and relays the signals. The membrane of the nerve cell, whose Figure 15.1: A nerve cell. existence was experimentally confirmed by Fricke in the 1920 s, is permeable to K + and Na + ions. If the nerve is at rest, the inner and outer surfaces of the membrane carry, respectively, net negative and positive electric charges. The corresponding resting potential (Ruhepotential) V in V out is of the order of 50mV The Hodgkin-Huxley model A significant breakthrough in our understanding of pulse propagation in nerve cells is due to experimental and theoretical work done by Hodgkin and Huxley (HH) in the 1950 s on the giant 1 axon of the Atlantic squid (Loligo pealei). 1 The diameter of the squid s axon is of the order of 1.5mm, which is about 50 times as thick as that of most animals, including humans. 144

152 15 Pulse propagation in nerve cells: the Hodgkin-Huxley model The axon membrane as an array of electrical circuit elements HH s schematic view of a cylindrical axon membrane as an electrical circuit element is summarized in Fig The constitutive equations for the ion transport are: Figure 15.2: Upper part: Schematic view of the axon interior and membrane. Lower part: a membrane element of length x, viewed as a piece of electrical circuit with a capacitance, independent K + and Na + ion channels for the flow of a transverse current (across the membrane) with a nonlinear conductance, and a leak - channel with linear conductance. In the HH experiments, the inner part of the membrane was held at a spatially uniform potential V (voltage clamp). This allowed a detailed analysis of the ion gate properties. Ohm s law for the longitudinal current flow (along the axon): I V πd 2 = σ /4 x (15.1) where d = mm is the diameter of the axon and σ = 2.9 S/m the axoplasm conductivity. Kirchoff s law, applied to a membrane element of length x and diameter d with a capacitance C = cπd x, can be written as I(x) = (CV ) + J + I(x + x) t where J = jπd x is the total transverse current (across the membrane). In differential form, this can be rewritten in terms of the transverse current density j as 1 I πd x = c V t j. (15.2) where c = 1µF/cm 2 is the membrane capacitance per unit surface. Eliminating the current from (15.1) and (15.2) one obtains σd 2 V 4 x 2 = c V t + j (15.3) which, under general conditions, is a driven, nonlinear diffusion equation. 145

153 15 Pulse propagation in nerve cells: the Hodgkin-Huxley model Ion transport via distinct ionic channels According to HH, the transverse current per unit length consists of distinct ionic components, and a leakage current j = j Na + j K + j L (15.4) with j Na = G Na g Na (V V Na ) j K = G K g K (V V K ) j L = G L (V V L ), (15.5) where V Na = 115 mv, V K = 12 mv are the electrochemical potentials for Na and K ions respectively, and V L = 10.6 mv is adjusted so that the total current j vanishes if V = 0. Note that this condition really corresponds to the rest state, i.e. V = V in V out + 65mV. The electrochemical potentials of the squid axon are fairly typical: the corresponding values for the frog s sciatic nerve are V Na = 122mV, V K 0mV. The G i s are linear conductances, G Na = 120 ms cm 2, G K = 36 ms cm 2, G L = 0.3 ms cm 2. Finally, the g i s are nonlinear dimensionless functions of the potentials, to be discussed below Voltage clamping In order to study the details of nonlinear transport, HH developed an experimental technique, called space clamping. The technique consisted of piercing the axon with a thin metallic electrode, so that the inside of the membrane could be held at a spatially uniform potential. In this case, using (15.3),(15.4) and (15.5), one obtains c V t = G Na g Na (V V Na ) + G K g K (V V K ) + G L (V V L ). (15.6) Voltage clamping was a further technical development which made it possible to control the uniform voltage at any desired level. Clamping was a major conceptual breakthrough in the electrophysics of nerve cells because it facilitated a detailed analysis of experimental data and made possible the extraction of crucial information on the unknown functions g i Ionic channels controlled by gates The experimental findings of HH led them to conclude that the two ionic channels are controlled by gates. Moreover, a phenomenological description of the data could be only be achieved by postulating the existence of different types of gates. Thus where m, n, h [0, 1] are gating functions. g K = n 4 (15.7) g Na = m 3 h (15.8) Gating functions The value of any gating function p corresponds to the probability that the corresponding gate is in its open state. The time evolution of any gating function is governed by a first-order ordinary differential equation dp dt = α(1 p) βp = p p 0 τ (15.9) 146

154 15 Pulse propagation in nerve cells: the Hodgkin-Huxley model n 0 m 0 h msec τ n -1 τ m -1 τ h V Figure 15.3: The HH gating functions at equilibrium. n 0 and m 0 are of the activation type, h 0 is of the deactivation type. Inset: the inverse relaxation times which correspond to the gating functions. Note that the m-gate is considerably faster than either the n or the h gates V (mv) where αdt is the probability that the closed gate will open within the time dt, and βdt the probability that the open gate will close during the same time interval. Both α and β are, in general, nonlinear functions of the membrane potential V. The second form of the equation, where the new parameters are defined as 1 τ p 0 = = α + β β α, (15.10) shows clearly that the gating function approaches an equilibrium value p 0 (V ) for any given membrane potential within a characteristic time τ(v ). Gates are classified as either of the activation type, if or of the deactivation type, if HH gate parameters lim p(v ) = 1, V lim p(v ) = 0. V The gating function n which controls the flow of K + ions is of the activation type, with α n = 0.01(10 V ) e 10 V 10 1 Here α n and β n are measured in msec 1, V in mv., β n = 0.125e V 80. (15.11) The flow of Na + is controlled by both an activation-type gate m, with α m = 0.1(25 V ) e 25 V 10 1 and a deactivation-type gate h, with α h = 0.07e V 20, β h =, β m = 4e V 18, (15.12) 1 e 30 V (15.13) The corresponding values of the gating functions at equilibrium are shown in Fig

155 15 Pulse propagation in nerve cells: the Hodgkin-Huxley model Membrane activation is a threshold phenomenon The solutions of the HH equations under space-clamping conditions for a variety of initial membrane potentials are shown in Fig A spike always forms, provided that the initial stimulus is above a certain threshold 6.5mV. The spike amplitude and width are roughly independent of the initial stimulus. 120 V (mv) V(0) (mv) Figure 15.4: The membrane potential as a function of time for a variety of initial stimuli. Note that a spike will always form if the initial stimulus is above a certain threshold; amplitude and width of the spike are roughly independent of the strength of the stimulus. The threshold lies between 6 and 7 mv t (msec) A qualitative picture of ion transport during nerve activation On the basis of the HH model, the following qualitative picture emerges for the temporal evolution of the action potential during nerve activation (cf. Fig : At zero membrane potential, m 0 (0) = 0.053, h 0 (0) = 0.596; the product m 3 h is very small, hence the Na + current will be very small. Since n 0 (0) = 0.318, the K + current will also be small. As the membrane voltage is turned on, the m-gate of the Na + channel is rapidly activated (cf. Fig ), within a characteristic time msec. Sodium ions flow into the axon. As V approaches the electrochemical potential of sodium, V Na = 115 mv, the influx of sodium ions becomes small. At high values of V the h-gate becomes deactivated. The Na + channel closes within a characteristic relaxation time of 1 8 msec. While this happens, the slower potassium gate n becomes activated, with a characteristic time of 2 5 msec. K + ions begin to flow out of the axon. The membrane depolarizes Pulse propagation It is now possible to look for propagating solutions of the nonlinear diffusion equation (15.3), i.e. solutions of the form V (x vt); since electrodes are usually placed at a fixed point in 148

156 15 Pulse propagation in nerve cells: the Hodgkin-Huxley model V (mv) relaxation (space clamping) G (ms/cm 2 ) K Na t (msec) j (ma/cm 2 ) -0.5 Figure 15.5: The thick curve shows the membrane potential (left y-scale) as a function of time. The pair of thinner curves (right y-scale) represents the potassium and sodium currents. Inset: the potassium and sodium conductances. Note that the sodium conductance peak occurs much earlier in time t (msec) space and follow the temporal evolution of a pulse, it is more convenient to look - equivalently - for solutions of the form V (t x/v). Moreover, in order to keep everything first-order in time, it is convenient to revert to (15.1) and (15.2) instead of (15.3). This leads to a system of 5 coupled ODEs ( (15.1) and (15.2) plus the three equations of the type (15.9) which determine the time evolution of the gating functions): dv dt 1 di πdv dt dn dt dm dt dh dt 4 = πσd 2 vi 4c = πσd 2 vi + j Na(V ) + j K (V ) + j L (V ) = n n 0(V ) τ n (V ) = m m 0(V ) τ m (V ) = h h 0(V ) τ h (V ) (15.14) Figure 15.6: A propagating pulse solution of the HH equations corresponds to a homoclinic trajectory (from [40]). The above system of coupled ODEs has an equilibrium point S = (0, 0, n 0 (0), m 0 (0), h 0 (0)) 149

157 15 Pulse propagation in nerve cells: the Hodgkin-Huxley model in the five-dimensional space (V, I, n, m, h). A pulse-like solution vanishes as t ±. It corresponds to a homoclinic trajectory in the 5-dimensional space. This is shown schematically in Fig. 15.6, where the three dimensions m, n, h are collapsed onto one. The trajectory starts off at S at t = ; for a generic value of v it will eventually wander off to unbounded values of voltage and/or current. If v has the right value, it will return to S in the limit t. With the HH parameters, this occurs for two values of v. The first, v = 18.8 Figure 15.7: Propagating pulse solutions of the HH equations (from [40]). m/sec, corresponds to a stable pulse with an amplitude of approximately 90 mv. The second, v = 5.7 m/sec, corresponds to an unstable pulse with a smaller amplitude (Fig. 15.7). The velocity of the stable pulse is comparable to the measured velocity, 21.2 m/sec, of the squid s action potential. 150

158 16 Localization and transport of energy in proteins: The Davydov soliton 16.1 Background. Model Hamiltonian Energy storage in C=O stretching modes. Excitonic Hamiltonian Davydov s proposal [41] was an attempt to deal with localization and transport of energy in alpha-helical proteins. He viewed the three strands of the alpha helix as roughly independent one-dimensional chains, and assumed that energy from ATP hydrolysis (0.42 ev) could be conveniently stored in the C=O (Amide-I) stretching vibration. He then argued that if the energy had to hop from one site to the next, following a linear (excitonic) model Hamiltonian H exc = { )} E 0 B nb n J (B n+1 B n + B nb n+1, (16.1) n where the B n s are boson operators representing the C=O stretching mode at the nth site, and J represents the hopping parameter, it would very soon (in the order of a few picoseconds) be dissipated due to linear dispersion - and thus cease to be available where really needed, e.g. for muscular contraction Coupling to lattice vibrations. Analogy to polaron Davydov then speculated whether the self-trapping effect, which had been proposed by Landau in 1933 in the context of electrons coupled to the lattice, and known in solid-state physics as the polaron, could be applicable in the bosonic system (16.1). We have already dealt with a similar situation in conjugated polymers, where the coupling to the lattice vibrations produces stable, propagating excitations (solitons and polarons). Lattice vibrations are acoustic phonons represented by the Hamiltonian H ph = n p 2 n 2M k n (u n+1 u n ) 2 (16.2) where u n is the displacement of the nth site from its equilibrium position, M is the mass associated with each unit of the alpha-helix, and k is a spring constant associated with the longitudinal motion along the chain. Coupling of the exciton modes to the lattice vibrations may occur because the energy stored at a given site changes with the distance between adjacent sites, i.e. with the length of the hydrogen bond connecting the nth to the n + 1st site of the strand E 0 E 0 + χ(u n+1 u n ). The above coupling generates an interaction Hamiltonian of the form H int = χ n The total Hamiltonian is the sum of the three terms (u n+1 u n )B nb n. (16.3) H = H exc + H ph + H int. (16.4) 151

159 16 Localization and transport of energy in proteins: The Davydov soliton 16.2 Born-Oppenheimer dynamics The dynamics of the coupled exciton-phonon system described by the Hamiltonian (16.4) can be considerably simplified if we make use of the Born-Oppenheimer approximation. This is not an unreasonable assumption, since acoustic vibrations are slow compared to the excitonic modes. It allows us to treat the lattice displacements as classical variables and simplifies the mathematical computations Quantum (excitonic) dynamics For a given set of lattice displacements {u n } we denote the excitonic wave function by Ψ >= n α n (t)b n 0 > (16.5) where 0 > is the bosonic vacuum state and the amplitudes α n (t) depend parametrically on the lattice configuration {u n }. Normalization of the quantum state demands that α n 2 = 1. (16.6) n The time evolution proceeds according to the time-dependent Schrödinger equation i h Ψ >= Ĥ Ψ > (16.7) t where Ĥ = H exc + H int is the quantum part of the Hamiltonian (remember, at this stage H ph is just a c-number!); with the Ansatz (16.5) the Schrödinger equation (16.7) can be written as a set of coupled equations for the complex amplitudes i h α n t = E 0 α n + χ(u n+1 u n )α n J(α n+1 + α n 1 ). (16.8) The total energy The expectation value of the excitonic part of the energy can be expressed in terms of the amplitudes as < Ψ Ĥ Ψ >= n [E 0 + χ(u n+1 u n )] α nα n J n α n(α n+1 + α n 1 ). (16.9) The total energy of a given exciton-phonon configuration {u n, α n } is ɛ =< Ψ Ĥ Ψ > +H ph({u n }). (16.10) The limiting case χ = 0 In the limiting case χ = 0, (16.8) reduces to i h α n t which has plane wave solutions of the form = E 0 α n J(α n+1 + α n 1 ), (16.11) α (q) n (t) = 1 N e i(qx ɛq h t) (16.12) 152

160 16 Localization and transport of energy in proteins: The Davydov soliton with total energy ɛ q = E 0 2J cos qa hω q (16.13) where the a is the lattice constant (the distance between successive sites along a single strand of the helix). These plane waves are the excitons which, according to Davydov, would exhibit dispersion over a time scale much too short to be relevant for biological energy transport. The group velocity of excitons is given by v g = ω q q = 2Ja sin qa. (16.14) h In the long-wavelength limit the exciton energy takes the form We identify as the exciton s effective mass. ɛ q = E 0 2J + m = h2 4Ja 2 v2 g. (16.15) h2 2Ja Lattice motion The dynamics of the classical lattice coordinates is described by Mü n = ( ) H ph + < u Ψ Ĥ Ψ > n = k(u n+1 + u n 1 2u n ) + χ( α n 2 α n 1 2 ). (16.16) Coupled exciton-phonon dynamics It will prove convenient to define a new complex amplitude via α n = φ n e ī h (E 0 2J)t. This allows us to rewrite (16.8) and (16.16) as and respectively. i h φ n = J(φ n+1 + φ n 1 2φ n ) + χ(u n+1 u n )φ n (16.17) Mü n = k(u n+1 + u n 1 2u n ) + χ( φ n 2 φ n 1 2 ) (16.18) The general problem of coupled phonon-exciton dynamics involves the solution of the above set of coupled ODEs The Davydov soliton The heavy ion limit. Static Solitons In the limit M we may assume that ions do not move. left-hand side of (16.18) equal to zero, whereupon This allows us to set the u n+1 u n = χ k φ n 2, (16.19) 153

161 16 Localization and transport of energy in proteins: The Davydov soliton which transforms (16.17) to and the total energy (16.10) to i h φ n = J(φ n+1 + φ n 1 2φ n ) χ2 k φ n 2 φ n, (16.20) E 0 φ n 2 J n n The continuum approximation φ n(φ n+1 + φ n 1 ) χ2 2k φ n 4. (16.21) If the amplitudes vary smoothly from site to site, we can approximate the set of discrete variables {φ n } by a continuum field φ(x). The dynamics (16.20) takes the form of the continuum field equation iφ τ + φ xx + 1 σ 0 φ 2 φ = 0, (16.22) where σ 0 = kj/χ 2 and I have introduced the dimensionless time τ = Jt/ h. We recognize (16.22) as the nonlinear Schrödinger (NLS) equation. In section we derived a family of one soliton solutions of the form φ(x) = κ(x vτ) σ 0 κ sech 2 with arbitrary κ and v. In the present context, since we assumed that ions do not move, v must vanish. Furthermore, the normalization condition fixes the value of κ = 1/(2 2σ 0 ). dx φ(x) 2 = 1 n Form of the soliton The exact form of the static soliton is given by φ(x) = 1 sech x e iτ/(16σ 2 8σ0 4σ 0 0 ). Note that the spatial extent of the soliton (in units of the lattice constant) is of the order of 4σ 0. With the standard parameter values [42] σ 0 is estimated to be between 1.6 and 7.4; this appears to justify the use of the continuum approximation. Energy considerations The total energy of the soliton can be calculated from (16.21). The first term in (16.21) contributes E 0. In the second term, we can use a Taylor expansion which produces a contribution 2J + J dx φ 2 = 2J + χ4 48Jk 2. Finally, the third term produces a contribution χ2 2k χ4 dx φ 4 = 24Jk

162 16 Localization and transport of energy in proteins: The Davydov soliton Collecting terms, I obtain the total energy of a static soliton ɛ(v = 0) = E 0 2J χ4 48Jk 2 (16.23) which lies below the excitonic band (16.13). Thus the soliton is expected to be a more stable excitation (cf. the similar argument made with the SSH kink, or the TLM polaron in conjugated polymers) Moving solitons It is straightforward to generalize the analysis of the previous subsection to the case of moving solitons. The system of coupled ODEs (16.17) and (16.18) can be written in the continuum limit as i hφ t + Jφ xx χu x φ = 0 Mu tt ku xx + χ ( φ 2) x = 0. (16.24) If we look for propagating solutions of the type u(x vt), the second equation has a first integral, M(v 2 c 2 )u x = χ φ 2 where Mc 2 = k and I have assumed boundary conditions decaying at infinity to set the integration constant equal to zero. The last equation, when introduced into the first of (16.24) yields i h J φ t + φ xx + 1 σ φ 2 φ = 0 (16.25) which contains only the φ field. The parameter σ = σ 0 (1 v 2 /c 2 ) now depends on the soliton velocity. Again, we recognize (16.25) as the NLS equation, with a family of normalized (cf. above) solutions φ(x, t) = ψ(x, t) e iθ where ψ(x, t) = 1 ( ) x vt x0 sech 8σ 4σ ( J θ = 16 hσ 2 + h ) 4J v2 t + hv 2J x + θ 0. The moving Davydov soliton is a coherent localized excitation which couples the quantum (excitonic) to the vibrations of the underlying lattice. Note that the above analysis is only valid for positive σ. This restricts soliton velocities to v < c. Energy of the moving soliton Again, it is possible to calculate the total energy involved in the coupled exciton-phonon system. The contributions can be read off (16.10) in the continuum limit: ɛ(v) = E 0 2J + J dx φ 2 + χ dx u x φ 2 + k dx u 2 x + M dx u 2 t 2 2 ( = E 0 2J + J dx ψ 2 + θ 2 x ψ 2) dx J σ φ J σ 0 dx σ σ φ J σ 0 v 2 dx σ c 2 σ φ 4 155

163 16 Localization and transport of energy in proteins: The Davydov soliton E 0 2J + J dx +O(v 4 /c 4 ) E 0 2J + J ( ψ 2 + θ 2 x ψ 2) J dx 2 σ ψ4 + J v2 c 2 ( 1 48σ 2 + h2 v 2 ) 4J 2 J 24σ 2 + J v 2 12σ 2 c 2 dx σ ψ4 E 0 2J χ4 48Jk m sv 2 + O(v 4 /c 4 ) (16.26) where the sum of the first three terms is the energy (16.23) of the static soliton and ( ) m s = m 1 + Mχ4 6 h 2 (16.27) k 3 is the soliton s effective mass (where I have reintroduced the lattice constant a in order to restore the proper units to m ). 156

164 17 Nonlinear localization in translationally invariant systems: discrete breathers The existence of localized states in condensed matter systems has traditionally been associated with the presence of impurities and disorder, i.e. with material behavior which breaks the translational invariance of the perfect crystal. One of the major discoveries of the last two decades in nonlinear science is that localization may also occur as a consequence of nonlinearity in pure, translationally invariant systems of any dimensionality. A key contribution to this field was made by Sievers and Takeno [43] who proposed the existence of intrinsic localized modes in anharmonic crystals. I will first present their argument, which is approximate and makes use of the rotating-wave approximation (RWA). I will then present further evidence for the existence of nonlinear localized excitations - called discrete breathers by some authors - based on numerical calculations by Flach and coworkers [44] and give a plausibility argument which underlies the exact mathematical proof given by MacKay and Aubry [45]. For more details consult the review articles by Flach [46] and Aubry [47] The Sievers-Takeno conjecture The starting point is the one-dimensional Hamiltonian which describes nearest-neighbor atoms coupled via nonlinear springs; the anharmonicity is quartic: H = [ p 2 n 2M K 2(u n+1 u n ) ] 2 K 4(u n+1 u n ) 4 (17.1) n I try to solve the equations of motion Mü n = K 2 (u n+1 + u n 1 2u n ) + K 4 [(u n+1 u n ) 3 + (u n 1 u n ) 3 ] (17.2) using the rotating-wave approximation, known from the theory of nuclear magnetic resonance. The idea is to make the Ansatz u n = α(ξ n e iωt + ξ ne iωt ) (17.3) and neglect fast oscillations which occur from terms of order e 3iωt. In the spirit of the RWA, such oscillations presumably average to zero over the time scales of interest, defined by the inverse of the frequency ω. I will further assume that the ξ n s are real, so that I keep track of the e iωt terms only. Note that there are no terms of order e ±2iωt. The Ansatz results in a second order recurrence equation for the amplitudes: 2ξ n ξ n+1 ξ n 1 + λ [ (ξ n ξ n+1 ) 3 + (ξ n ξ n 1 ) 3] = ω2 J ξ n (17.4) where J = K 2 /M and λ = 3K 4 α 2 /K 2. The value of J can be set equal to unity without loss of generality. 157

165 17 Nonlinear localization in translationally invariant systems: discrete breathers I now rewrite (17.4) as where [ω 2 δ mn D mn ]ξ n = V n (ξ n 1, ξ n, ξ n+1 ) (17.5) V n (ξ n 1, ξ n, ξ n+1 ) = λ [ (ξ n ξ n+1 ) 3 + (ξ n ξ n 1 ) 3] the matrix elements D mn = 2 if m = n, D mn = 1 if m = n ± 1 and D mn = 0 otherwise describe the dynamical matrix of the harmonic chain with nearest neighbor springs. Now we have used the inverse of the matrix ω 2 I D in discussing the problem of a single impurity. The matrix ( ω 2 D ) 1 mn = G( m n, ω2 ) = 1 e iq(m n) N ω 2 ω 2, (17.6) q q where the sum runs over all eigenvectors of the dynamical matrix, is known as the lattice Green function. In the particular case we are discussing here, ω 2 q = 2(1 cos q) and G has been shown, for frequencies above the band, i.e, ω 2 > 4, to be of the form G(n, ω 2 ) = ( 1) n 1 g(n), where ω2 { g(n) = (1 y) 1/2 2 [1 12 ]} n y y (1 y)1/2 ( y 4 ) n, (17.7) where y = 4/ω 2 and the last line gives the leading order asymptotic expansion for y 1. Use of the lattice Green function allowed Sievers and Takeno to rewrite (17.5) as ξ m = n G(m n, ω 2 )V n (ξ n 1, ξ n, ξ n+1 ) (17.8) and exploit the rapid convergence properties of the sum in the r.h.s. of (17.8) which results from the exponential decay of the Green functions. Let us see how this works in detail: We look for symmetric solutions of the type ξ n = ( 1) n η n with η 0 = 1. 1 Because of (17.4), η 0 and η 1 are related by the equation 1 + η 1 + λ(1 + η 1 ) 3 = 1 2 ω2 = 2 y. (17.9) In terms of the η s, (17.8) can be written as η m = λ ω 2 n= {g(m n) + g(m n + 1)} (η n + η n+1 ) 3, which can be further symmetrized by making use of the property g( n) = g(n), to where ( ) 2 η m = y 1 η 1 g(m) λy A mn (η n + η n+1 ) 3 m 1, (17.10) n=1 A mn = g(m n) + g(m n + 1) + g(m + n) + g(m + n + 1). 1 Note that this is permissible since the scale of the amplitude has already been fixed by α in the original Ansatz (17.3), 158

166 17 Nonlinear localization in translationally invariant systems: discrete breathers Note that the m = 0 equation (which I did not write down) is not really independent, since it must be equivalent to (17.9). The system of coupled nonlinear equations (17.9) and (17.10) can in principle be solved numerically to yield the amplitudes and the eigenfrequency. The numerical procedure would presumably converge fast, due to the exponential decay of the A mn s. In practice, only a few terms are expected to contribute to the sum. However, one can already make some statements using the asymptotic properties of the Green functions in the limit y 1. To leading order in y, the m = 1 equation (17.10) gives η which can be used in (17.9) to compute the eigenvalue. The result y = 4 ω 2 = λ gives a consistent value of y 1 if λ 4/27. For sufficiently strong nonlinearities, one therefore has ω λ. (17.11) Numerical evidence of localization An illuminating picture of nonlinear localization as local integrability was obtained through numerical simulations by Flach and coworkers [44]. I summarize some of their findings. The starting point is the one-dimensional Hamiltonian H = n [ p 2 n ] 2 C(u n+1 u n ) 2 + V (u n ) (17.12) with a moderately weak harmonic interparticle coupling strength C = 0.1 and a nonlinear on-site potential The equations of motion V (u) = u 2 u u4. ü n = C(u n+1 + u n 1 2u n ) V (u n ) i = 0, ±1, ± N/2 (17.13) were numerically integrated for a lattice of N = 3000 sites, subject to periodic boundary conditions. The initial condition was spatially localized, i.e. all particles started at rest, u n = 0 n and all but one of them (at the n = 0 site) were at the equilibrium positions corresponding to the absolute minimum of (17.12). A measure of the energy density at the lth site is given by e l = 1 2 u2 l + V (u l ) + C 4 [ (ul+1 u l ) 2 + (u l u l 1 ) 2]. (17.14) Note that the sum over all e l s is by definition a conserved quantity, the total energy. 159

167 17 Nonlinear localization in translationally invariant systems: discrete breathers Figure 17.1: The temporal evolution of e (5). The solid line shows the total energy. In the inset, the energy distribution around the central site, measured for 1000 < t < 1150 (from [44] ) Diagnostics of energy localization An empirical diagnosis of localization can be made in terms of the quantities m e (2m+1) = e l, (17.15) which provide a measure of the energy residing in the first 2m + 1 central sites. If these can be shown to remain constant over long periods of time, one may reasonably conjecture the presence of a localized oscillation which keeps the energy around the central sites. Fig describes the temporal evolution of e (5). There is some radiation, amounting to less than 1% of the total energy, which occurs during the first few hundred time units. After these transients decay however, the energy stays remarkably constant. m Internal dynamics It is possible to obtain additional information about the internal dynamics of the localized oscillation by looking at the Fourier spectra of the few central sites. The numerical results are shown in Fig The spectra of the central site, l = 0, shows a dominant peak at ω 1 = The spectra of the sites l = ±1 show a dominant peak at ω 2 = All other peaks of both spectra can be obtained as sums or differences of these two fundamental frequencies. This remarkable result suggests that we are, in effect, dealing with a system of two degrees of freedom. This suggestion can be tested in some more detail by looking at the reduced dynamical system with two degrees of freedom ü 0 = V (u 0 ) + 2C(u 1 u 0 ) (17.16) ü 1 = V (u 1 ) + C(u 0 2u 1 ) (17.17) which is obtained from the full dynamics (17.13) by assuming all particles with l > 1 to remain at rest, and exploiting the symmetry u 1 = u

168 17 Nonlinear localization in translationally invariant systems: discrete breathers Figure 17.2: Fourier spectra of u 0 (t > 1000). Inset: spectra of u 1. All peak frequencies are either sums or differences of the two fundamental frequencies. (From [44] ). Fig shows a Poincaré cut of this reduced dynamical system. Note the presence of regular motion (tori) embedded in a sea of chaos. Flach and coworkers [44] made, and tested, the following remarkable conjecture: If the frequencies of a torus lie outside the phonon band of the linearized version of (17.13), the torus should correspond to a localized oscillation of the full system. Indeed, a choice of initial condition from the islands 1 or 2 generates a localized oscillation in the full system (17.13). A choice from island 3, or from a chaotic trajectory, generates a nonlocalized pattern. This is shown in detail in Fig Towards exact discrete breathers Consider a Hamiltonian of the type (17.12) which describes a system of N weakly coupled nonlinear oscillators. The spatial dimensionality is not important for the arguments which follow. Let the state of the system be described at any given time by the 2N-dimensional vector X = {x 1, x N ; p 1 p N }. At zero coupling strength it is possible to excite a single oscillator at the lth site and leave all other particles at rest. The motion of the system will be periodic, with a given period 161

169 17 Nonlinear localization in translationally invariant systems: discrete breathers Figure 17.3: Poincaré cut of the reduced dynamical system at E = (From [44] ). Figure 17.4: Temporal evolution of e (5) for a variety of initial conditions chosen according to their properties in the reduced system. Localization occurs for 4 initial conditions chosen from fixed points in islands 1 and 2 in Fig. 17.3, the larger torus in island 1, and the torus in island 2 (solid lines). Initial conditions from the torus in island 3 (longdashed line), or from a chaotic trajectory (dashed-dotted line) lead to decaying e (5). The upper short-dashed line shows the total energy of all simulations. (From [44] ). T. I denote this by X 0 (t) = X 0 (t + T ). Now consider what happens when a weak coupling is turned on. The time evolution of the system over a time T, generated by the full Hamiltonian, transforms any initial state vector X(0) (denoted from now on as X for simplicity) to X(T ). Let F [ X] = X(T ) X formally denote the operator which performs this time evolution. Periodic orbits of the interacting system correspond to roots of the 2N coupled equations F [ X] = 0. (17.18) Since a weak coupling was assumed, it is not unreasonable to expect the initial condition X 0 of the decoupled system - which is known to lead to a periodic orbit there - to lie near the root of (17.18). We could then use this a starting point for a Newton-like iteration 162

170 17 Nonlinear localization in translationally invariant systems: discrete breathers procedure 2, i.e. proceed to construct a next iterate X 1 near the original guess X 0 by demanding that F [ X 1 ] = F [ X 0 ] + M 0 ( X 1 X 0) = 0 where the elements of the 2N 2N matrix M 0 are given by Mmn 0 = F m Xn 0 = X m(t ) X n δ mn X= X 0. (17.19) This is equivalent to demanding X 1 = X 0 [ M 0] 1 F [ X 0 ] which can be continued as X j+1 = X j [ M j] 1 F [ X j ] (17.20) until convergence to the true discrete breather is achieved. This can also be a practical method to construct exact breather solutions to machine numerical accuracy. It is successful, provided that (a) the matrix M is invertible, and (b) that the breather frequency ω b = 2π/T has no resonances of the type nω b = ω q (17.21) with the phonon band. Note that this generally allows breathers with frequencies above the phonon bands without any restrictions, but may impose severe limits to breathers whose frequencies lie below a phonon band. For example, in the case of the Hamiltonian (17.12), breather frequencies must lie outside the frequency ranges (cf. Fig. 17.5). 1 < nω b < (1 + 4C) 1/2 n = 1, 2, 3, 1.5 Figure 17.5: Forbidden frequency bands (in color) for DBs in the case of the Hamiltonian (17.12). The dotted line represents the linear phonon dispersion curve. Note that the bands with n > 5 merge, leaving no frequency region allowed for DBs. Frequency k Details of the existence proof for discrete breathers can be found in Ref. [45]. 2 The Newton procedure for locating roots of f(x) = 0 starts from a guess x 0 with f(x 0 ) not too far from zero, and iterates successively, ( ) df 1 x j+1 = x j f(x j ). dx x=x j Provided that the guess does not lead away from the true root, the procedure is rapidly (quadratically) convergent. 163

171 A Impurities, disorder and localization In the following, I will try to illustrate, with the help of characteristic examples, how disorder can lead to localization of eigenstates. The point is not to present exact criteria for localization; this would be beyond the scope of these lectures. Nonetheless, the simple examples treated should make clear that (i) an isolated eigenstate, i.e. one which is outside the phonon (or electron, depending on the problem) bands, will tend to be localized, and (ii) the introduction of disorder will transform most eigenstates from extended to localized ones. In other words, these examples serve to illustrate the obvious, i.e. that breaking translational invariance will introduce some degree of localization; at the same time, they remind us the basic condition for the existence of isolated localized states, i.e. that the frequency of oscillation should lie outside any bands. A.1 Definitions A.1.1 Electrons Consider a one-dimensional tight-binding electron Hamiltonian H = { } ɛ n c nc n t n,n+1 (c n+1 c n + h.c) n (A.1) where the on-site energies ɛ n and the hopping amplitudes t n,n+1 may depend on the lattice site. We look for eigenstates of (A.1) in the subspace of one-electron states: H Ψ >= E Ψ > Ψ >= n ψ n c n 0 > The amplitudes ψ n must then satisfy the difference eigenvalue equation Limiting cases are (A.2) t n,n 1 ψ n 1 t n,n+1 ψ n+1 + ɛ n ψ n = Eψ n. (A.3) the translationally invariant case t n = t 0, ɛ n = ɛ 0 n. The eigenstates are plane waves with the corresponding eigenvalues. diagonal disorder t n,n+1 = t 0 n. off-diagonal disorder ɛ n = ɛ 0 n. ψ (q) n = 1 N e iqn E q = ɛ 0 2t 0 + 2t 0 (1 cos q) a single impurity of either type, e.g ɛ n = ɛ 0 if n α, ɛ α = ɛ. 164

172 A Impurities, disorder and localization A.1.2 Phonons Lattice vibrations in disordered harmonic lattices are described by the same mathematics. Consider the Hamiltonian H = n p 2 n 2µ n k n,n+1 (x n+1 x n ) v n x 2 n 2 n n (A.4) where, in general, masses and spring constants depend on the lattice site; note that I have added a harmonic on-site term. The coefficients v n will of course vanish in the usual harmonic chain with nearest-neighbor springs only. The classical equations of motion corresponding to (A.4) are µ n ẍ n = k n,n+1 x n+1 + k n,n 1 x n 1 (k n,n+1 + k n,n 1 + v n )x n ; (A.5) if we look for normal modes of (A.5) with the Ansatz the amplitudes must satisfy the difference eigenvalue equation x n (t) = 1 µn ψ n e iωt, (A.6) t n,n 1 ψ n 1 t n,n+1 ψ n+1 + ɛ n ψ n = ω 2 ψ n, (A.7) where t n,n+1 = (µ n µ n+1 ) 1/2 k n,n+1 and ɛ n = (k n,n+1 + k n,n 1 + v n )/µ n. Limiting cases of interest are the translationally invariant case: k n,n+1 = k, µ n = µ, v n = v are plane waves ψ n (q) = 1 e iqn N n. The eigenstates with the corresponding eigenvalues mass disorder: k n,n+1 = k spring disorder: µ n = µ ωq 2 = 1 [v + 2k(1 cos q)]. µ n, {µ n } random. n, {k n } random. a single impurity of either type, e.g µ n = µ if n α, µ α = µ (isotopic mass impurity). on-site single impurity (corresponds to diagonal disorder at a single site): k n,n+1 = k, µ n = µ, n, v n = v if n α, v α = v. Note that in the most general case considered here, one has to diagonalize a tridiagonal N N real symmetric matrix. This can be done with very efficient numerical techniques[48]. A.2 A single impurity A.2.1 An exact result The case of a lattice impurity which modifies only a single diagonal element of the dynamical matrix has been treated exactly by M. Lax [49] and provides substantial insight. I present the original derivation and elaborate on the special case of one dimension. 165

173 A Impurities, disorder and localization Consider the more general problem where one knows the spectrum of a given dynamical matrix D, D ij e j (p) = λ(p) e i (p) i = 1, N (A.8) and is interested in the spectrum of a modified matrix D + B, where the modification B is of reduced dimensionality, i.e. B rs 0 only if r, s = 1, k; k N. Let ψ be an eigenvector of the modified matrix, corresponding to an eigenvalue Λ: It follows that (D ij + B ij ) ψ j = Λψ i (Λδ ij D ij ) ψ i = B ij ψ j ψ i = (ΛI D) 1 ij B jkψ k = (ΛI D) 1 ir B rsψ s = e i (p) e r(p) Λ λ(p) B rsψ s p (A.9) where in the last line I have used the standard representation of the inverse of D in terms of its spectrum. Let me now consider the special case where k = 1, i.e. B ij = bδ iα δ jα (on-site lattice impurity at the site α, cf. above). The condition (A.9) becomes ψ i = b p e i (p) e α(p) Λ λ(p) ψ α. (A.10) If the unperturbed matrix describes a translationally invariant system, the eigenvectors will be plane waves, i.e. e j (p) = 1 e i p R j (A.11) N where p is the wavevector corresponding to the eigenvalue index p. (Note that up to now there are no restrictions to dimensionality.) Using the plane-wave form of the eigenvectors, I rewrite (A.10) as which, for i = α, becomes ψ i = b N p e i p( R i R α ) ψ α. (A.12) Λ λ(p) 1 1 N Λ λ(p) = 1 b p. (A.13) Locating states outside the band Eq. (A.13) is an N th order algebraic equation. It is straightforward to see that a root must exist between any pair of successive eigenvalues λ(p) < Λ < λ(p + 1). This provides N 1 roots. The Nth root will therefore lie outside the band. In this case however, one 166

174 A Impurities, disorder and localization can readily substitute the sum in (A.13) by an integral. Let us see how this works in the one-dimensional case with an eigenvalue spectrum where (A.13) can be written as N 2π p π 0 λ(p) = v + 2 (1 cos p), (A.14) dp π π π dp N π π 1 Λ v cos p = 1 b 0 dp. (A.15) If b > 0, we look for a state above the band, Λ > v + 4. The integral has the value [2(Λ v 4)] 1/2, therefore Λ = v b 2 /2. As b 0, the eigenvalue merges with the band. Although this is not strictly the case (because changing a single mass modifies two off-diagonal as well as a diagonal element of the dynamical matrix), it is very similar to what happens if one introduces an impurity with a mass lighter than the rest. Fig. A.1b shows a numerical calculation in that case. If b < 0, the situation is analogous to what happens with a heavy impurity. The eigenvalue lies below the phonon band, and merges with it as b 0. Fig. A.1b shows a numerical calculation with a heavy impurity. States outside the band are localized In the one- I now return to (A.12) to extract information regarding the eigenvectors. dimensional case discussed above, (A.12) can be written as ψ π n = b ψ 0 = b π π 0 dp 2π dp π e ipn Λ λ(p) cos(pn) Λ λ(p) (A.16) where I have assumed that the eigenvalue Λ lies outside the band, and that the impurity is at the site 0. The imaginary part of the integral vanishes because λ(p) is an even function of p. Now take the case b > 0. I use the result Λ = v b 2 /2 (cf. above) and obtain ψ n = b π dp cos(pn) ψ π 1 + b2 4 + cos p. The integral can be evaluated exactly, yielding where ξ = ψ n ψ 0 = ( 1) n e n/ξ, (A.17) 1 arccosh(1 + b 2 /4) The value of the eigenvector decreases exponentially with the distance from the impurity. 1 In other words, the state is localized. This is a generic feature of isolated eigenstates which lie outside bands of extended states. 1 Note however that the localization length ξ diverges as b 0, i.e. as the eigenvalue approaches the band.. 167

175 A.2.2 Numerical results A Impurities, disorder and localization A number of other simple cases can be worked out analytically. Here I show some numerical results for the isotopic mass impurity. Note that, in the absence of analytical results such as (A.17), one needs a handy criterion to determine the degree of localization of a given eigenvector. One such criterion is the participation ratio, defined as a) b) frequency harmonic chain N=64, plus on-site u^2 3.5 isotopic impurity m ' /m = amplitude site eigenvalue index participation ratio frequency harmonic chain N=64, plus on-site u^2 3.5isotopic impurity m ' /m = amplitude site eigenvalue index participation ratio Figure A.1: Isotopic mass impurity in a harmonic crystal with on-site interaction: (a) heavy impurity, (b) light impurity. Inset: the amplitude of the localized state. P = N j ψ j 4 1. (A.18) The idea behind this particular criterion is that the squared amplitude of a normalized extended state is typically 1/N, therefore its P should be of order unity; a state which is localized over a couple of lattice constants contributes only amplitudes of order unity, and therefore its P should be of order 1/N. A state which is localized over a significant length scale, say ξ lattice constants, will have a P of the order of ξ/n. Therefore, the participation ratio provides a measure of the localization length. This is important when we interpret numerical results: a statement like eigenstates of disordered one-dimensional systems are always localized is not very useful. Perhaps some states have localization lengths which are comparable to the system size; this is bound to influence macroscopic transport properties. Therefore, getting detailed information about participation ratios, localization lengths, etc. is essential in understanding the effects of disorder. Fig. A.1 shows the vibrational spectrum of a one-dimensional lattice with an on-site potential (v 0) and a single isotopic mass impurity (heavy and light). Fig. A.2 shows the spectrum of a heavy or light mass impurity in the case where v = 0, i.e. there is no on-site potential. The difference is that the heavy mass has nowhere to go; there are no states below the band. In this case, all states remain extended. 168

176 A Impurities, disorder and localization frequency amplitude a) b) site harmonic chain N=64 isotopic impurity m ' /m = 5 inset: "most" localized EV eigenvalue index 0.6 participation ratio frequency harmonic chain N=64 isotopic impurity m ' /m = 0.5 amplitude participation ratio site eigenvalue index Figure A.2: as in previous figure, no on-site term. The light mass impurity does not generate a localized state, because there are no states below the phonon band. Inset: right: the localized state, left: the most localized state (the one with the lowest participation ratio) is still an extended state. A.3 Disorder Here I show numerical results obtained for a selection of random distribution of potential parameters. A.3.1 Electrons in disordered one-dimensional media Figs. A.3 and A.4 show one-electron spectra of disordered one-dimensional system (A.3). The disorder is of the diagonal type, i.e. the t n,n+1 = 1 and ɛ n = 2 + W r n, where r n is a random number between 1/2 and 1/2, and and the strength of the disorder increases from W = 1 to W = 4. Fig. A.5a is a histogram of the density of states for W = 2; Fig. A.5b is a histogram of participation ratios for W = 1, 2, 4. Note the drastic increase of localized states which occurs at W = 4. A.3.2 Vibrational spectra of one-dimensional disordered lattices Figs. A.6 and A.7 show the effect of mass disorder in the one-dimensional harmonic lattice. The masses are generated according to µ i = e W ri, where r i is a random number between 0.5 and 0.5 and the strength of the disorder is W = 4. Note the proliferation of localized states. Only the very lowest frequencies correspond to extended states. 169

177 A Impurities, disorder and localization a) b) 0.15 frequency amplitude site Schr discr, N=128, plus on-site u^2 no disorder participation ratio frequency amplitude site participation ratio eigenvalue index eigenvalue index Figure A.3: Spectrum of one-electron states : (a) reference (no disorder), (b) diagonal disorder W = 1. a) b) frequency amplitude participation ratio site eigenvalue index frequency amplitude site participation ratio eigenvalue index Figure A.4: Spectrum of one-electron states : (a) as in previous, W = 2, (b) W =

178 A Impurities, disorder and localization a) b) 18 W= W number of states number of states energy inv. partic. Figure A.5: Spectrum of one-electron states : participation ratios, W = 1, 2, 4. (a) Density of states W = 2, (b) Histogram of a) b) frequency amplitude harm lattice mass disorder site participation ratio Number of states 50 B eigenvalue index frequency Figure A.6: Mass disorder in the harmonic lattice W=4 : (a) spectrum, and degree of localization, (b) Frequency Histogram. 171

179 A Impurities, disorder and localization a) b) 800 C Number of states Inv participation ratio E Inverse participation ratio frequency Figure A.7: Mass disorder in the harmonic lattice W=4 : (a) The vast majority of states are localized. (b) A detailed view of localization vs. frequency: Localization lengths can become significantly large at low frequencies. There is a low frequency regime where theory predicts that the localization length grows as the inverse square of the frequency (dotted line). At the very lowest frequencies this theoretically predicted localization length is limited by finite size effects. 172

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