Landau Theory of Second Order Phase Transitions

Transcription

1 Landau Theory o Second Order Phase Transitions Order Parameter Second order phase transitions occur when a new state o reduced symmetry develops continuously rom the disordered (high temperature) phase. The ordered phase has a lower symmetry than the Hamiltonian the phenomenon o spontaneously broken symmetry. There will thereore be a number (sometimes ininite) o equivalent (e.g. equal ree energy) symmetry related states. These are macroscopically dierent, and so thermal luctuations will not connect one to another in the thermodynamic limit. To describe the ordered state we introduce a macroscopic order parameter that describes the character and strength o the broken symmetry. Examples 1. Ising erromagnet: the Hamiltonian is invariant under all s i s i, whereas the low temperature phase has a spontaneous magnetization, and so is not. A convenient order parameter is the total average spin S = i s i or the magnetization M = µs. This relects the nature o the ordering (under the transormation s i s i we have S S) and goes to zero continuously at T c. 2. Ising antierromagnet on a simple cubic lattice: the ordering is the staggered magnetization N = n ( 1)n s n where alternate sites are labelled even or odd. 3. Heisenberg erromagnet: now the Hamiltonian is invariant under any rotation o all the spins together. The ordering is characterized by a vector order parameter. A convenient choice is again the total spin S = i s i or magnetization. 4. Heisenberg antierromagnet on a simple cubic lattice: the staggered magnetization is now a vector N = n ( 1)n s n. 5. Superluid: the broken symmetry is the invariance o the (quantum) Hamiltonian under a phase change o the wave unction. Since or a charge system a gauge transormation also changes the quantum phase, this is also known as broken gauge symmetry. The order parameter can be thought o as the one particle wave unction into which the particle Bose condense, multiplied by the number o condensed atoms. This picture is good or a non- or weakly interacting system. For a strongly interacting system a better deinition is the expectation value o the particle annihilation operator ψ( r). 6. Solid: has broken translational and rotational symmetry, and a convenient order parameter is the amplitude o the density wave. In three dimensions the liquid-solid transition is always irst order, but in two dimensions may be second order. 7. Nematic liquid crystal: consists o long molecules which align parallel to one another at low temperatures, although the position o the molecules remains disorder as in a liquid. The liquid becomes anisotropic, and a second rank tensor characterizes the strength o the anisotropy and can be used as the order parameter. Free energy expansion Since the order parameter grows continuously rom zero at the transition temperature, Landau suggested that an expansion o the ree energy in a Taylor expansion in the order parameter would tell us about the behavior 1

2 near the transition. The ree energy must be invariant under all symmetry operations o the Hamiltonian, and the terms in the expansion are restricted by symmetry considerations. The order parameter may conceivably take on dierent values ( point in dierent directions ) in dierent parts o the system (or example due to thermal luctuations), and so we irst write F = d 3 x(ψ, T ) (1) introducing the ree energy density which is a unction o the local order parameter ψ( r) and temperature, and is then expanded in small ψ. To be explicit, lets consider the Ising erromagnet, and ψ = m( r), with m( r) the magnetization per unit volume or the magnetization per spin averaged over some reasonably macroscopic volume. Since the ree energy is invariant under spin inversion, the Taylor expansion must contain only even powers o m. Hence (m, T ) = o (T ) + α(t )m β(t )m4 + γ(t ) m m. (2) The last term in the expansion gives a ree energy cost or a nonuniorm m( r), again using the idea o a Taylor expansion, now in spatial derivatives as well. A positive γ ensures that the spatially uniorm state gives the lowest value o the ree energy F. Sixth and higher order terms in m could be retained, but are not usually necessary or the important behavior near T c. Note that we keep the ourth order term, because at T c the coeicient o the second order term α(t ) becomes zero. Only quadratic derivative terms are usually needed, since γ(t c ) remains nonzero. Minimum ree energy T>T c T<T c m Figure 1: Free energy density as a unction o (uniorm) m. BelowT c new minima at nonzero ± m develop (the dashed curve denotes m(t ). I the magnetization o the system is varied (rather than the ield) the dotted line is the tie line giving the variation o the ree energy as a unction o the total magnetization density m = M/V. We expect the state that minimizes the ree energy to be the physically realized state. This is true so long as the luctuations around this most probably value are small compared to this value. We will ind that this is not always the case, and Landau s theory then corresponds to a mean ield theory that ignores these luctuations. 2

3 For the Ising erromagnet the minimum o A is given by a uniorm m( r) = m satisying α m + β m 3 = 0. (3) The solutions corresponding to a minimum are m =± α/β or α<0, (4a) = 0 or α>0. (4b) We identiy α = 0 as where the temperature passes through T c, and expand near here α(t ) a(t T c ) + β(t ) b + γ(t ) γ + (5a) (5b) (5c) so that and o (T ) + a(t T c )m bm4 + γ( m) 2 (6) ( a ) 1/2 m (Tc T ) 1/2 or T < T c. (7) b Evaluating at m gives = 0 a2 (T T c ) 2 2b (8) showing the lowering o the ree energy by the ordering. Note that (T ) deviates rom 0 quadratically, as we ound or the mean ield theory o the Ising erromagnet. This will yield a jump discontinuity in the speciic heat { c c = 0 T T c c 0 + a b T T < T c, (9) with c 0 the smooth contribution coming rom 0 (T ). We can gain useul insight into the transition by plotting (m) or a uniorm m or various temperatures, as in Fig. 1. ForT > T c the ree energy has a single minimum at m = 0. Below T c two new minima at ± m develop. Right at T c the curve is very lat at the minimum (varying as m): we might expect luctuations to be particularly important here. It is easy to add the coupling to a magnetic ield (m, T, B) = 0 (T ) + a(t T c )m bm4 + γ( m) 2 mb. (10) The magnetic ield couples directly to the order parameter and is a symmetry breaking ield: with the magnetic ield the ull Hamiltonian is not invariant under spin inversion. Now the ree energy is minimized by a non zero m. Minimizing again with respect to m, we ind above T c the diverging susceptibility χ = m B = 1 B=0 2a (T T c) 1, (11) and at T = T c or small B ( ) 1 1/3 m = B 1/3. (12) 2b Note that the exponents (power laws) are the same as we ound in the direct mean ield theory calculation. 3

4 Other Symmetries The Landau approach is readily generalized to other symmetries. For example or the Heisenberg erromagnet the order parameter is a vector m. The ree energy is expanded as a Taylor expansion in the components m x, m y, m z and each term must be invariant under rotations o m, and also the derivative term must be invariant under rotations o the space axes separately rom m. The component terms can be combined to give the simple expression in vector notation ( m, T, B) = 0 (T ) + a(t T c ) m m b( m m)2 + γ( iα i m α i m α ) 2 m B. Notice the dot product orm o the gradient term separately in the gradient and m components: this guarantees the invariance under separate rotations o spin and space coordinates. We might schematically write this term as ( m) 2, but the double summation is what is meant! Similarly or a complex order parameter ψ, we do a Taylor expansion in Re ψ and Im ψ, and demand the invariance o each term under a change o the phase. This leads to the orm (ψ, T ) = 0 (T ) + a(t T c ) ψ b ψ 4 + γ ψ ψ. Note that it is nicest to write the expression in terms o ψ, but the correct approach and the best way to avoid conusion is to establish the expansion in Re ψ and Im ψ and then simpliy. For example a term ψ 3 is not allowed, since even though it is invariant under changing the phase, it does not correspond to a term in the Taylor expansion in Re ψ, Imψ. First order transitions (a) solid T>>T m T>T m (b) T=T m T c <T<T m T=T T<T c m ρ Figure 2: (a) Free energy plots or a ree energy with cubic invariants; (b) Free energy plots in the case where the coeicient o the quatic term is negative. 4

5 We can also encounter irst order broken-symmetry transitions. For example the liquid solid transition is described by the strength o density waves ρe i q r. In three dimensions we need three density waves with wave vectors that prescribe the reciprocal lattice; we will suppose all three components have the same magnitude ρ. Since the density perturbation is added to the uniorm density o the liquid, there is no symmetry under changing sign o the density wave, and so the ree energy expansion now may have a cubic term (we suppose uniorm ρ or simplicity). 0 = a(t T c )ρ 2 + cρ bρ4 (13) Now the behavior o (ρ, T ) is rather dierent, as in Fig. 2a. At high temperatures there is a single minimum at ρ = 0 corresponding to the liquid phase. As the temperature is lowered a second minimum develops, but at a ree energy that remains higher than the liquid. Only at T m does the new ree energy become lower, and this is the melting temperature where the liquid and solid have the same ree energy. Below T m the solid is the lower ree energy. Note the jump in ρ at T m, rather than a continuous variation as at a second order transition. The temperature T c at which the minimum in the ree energy at ρ = 0 disappears (i.e. the liquid state no longer exists) is not o great physical signiicance. The Landau expansion technique may not be accurate at a irst order transition: since there is a jump in ρ at the transition, there is no guarantee that the truncated expansion in ρ is a reasonable approximation here. Indeed the liquid-solid transition is usually strongly irst order, so that the power series expansion in the order parameter is not useul, and can give misleading conclusions. Another way a irst order transition can occur, even in the absence o cubic terms in the ree energy, is i the coeicient o the quartic term turns out to be negative. Then we must extend the expansion to sixth order to get inite results (otherwise the ree energy is unbounded below). Again assuming uniorm m or simplicity: 0 = a(t T c )m b m cm6. (14) Now, see Fig. 2b two symmetric secondary minima develop or T > T c, and become the lowest minima at temperature T = T > T c which we identiy as the irst order transition temperature. Again, at T c the minimum at m = 0 corresponding to the disordered phase disappears, but this is not oten relevant to the physics. 5