How To Understand The Theory Of Order In Quantum Theory

Transcription

1 Ordem e formação de padrões em sistemas com interações competitivas Daniel A. Stariolo, Porto Alegre Sergio A. Cannas, Córdoba Francisco (Pancho) Tamarit, Córdoba Mateus Michelon, Campinas Daniel G. Barci, Rio de Janeiro Ana Carolina Ribeiro-Teixeira, Porto Alegre Lucas Nicolao, Porto Alegre Orlando V. Billoni, Córdoba Santiago Pighín, Córdoba Marianela Carubelli, Córdoba Roberto Mulet, Habana Danilo Pescia s group at ETH, Zurich Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de / 12

2 Ultrathin ferromagnetic films Fe/Cu(001) ultrathin films, imaged with SEMPA. In a) the effective temperature increases from left to right. Image width=155 µm, thickness varies from 2.06ML to 1.98ML, T=294K. In b)-d) the temperature is increased from left to right for the same region. width=36µm, thickness 2.19ML, T b 298K, T c 304K and T d 305K. e)the labyrinthine pattern. In the inset a triangular V disclination and the structure factor. Thickness 1.97ML, T 293K f) The labyrinthine pattern on a stepped substrate. The disclination has a square symmetry, as does the structure factor. Parameters as in e), thickness 2.15ML. From O. Portmann et al., Nature 422, 701 (2003)

3 Diblock copolymers R. Ruiz et el. Phys. Rev. B 77, (2008)

4 Microemulsions From P. M. Chaikin and T. C. Lubensky Principles of Condensed Matter Physics

5 Physics ruled by competition between short range and long range interactions. Long range interactions frustrate the tendency for long range order coming from s.r.i. and then complex structures appear as stripes, bubbles, labyrinth, tetragonal, nematic, columnar phases. There is also competition between positional and orientational order giving rise to phases similar to liquid crystals. Examples go from classical to quantum systems like ultrathin ferromagnetic films, diblock copolymers, low dimensional electronic systems, superconductors, microemulsions, Raleigh-Bénard convection, etc. Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de / 12

6 Modulated phases in condensed matter M. Seul and D. Andelman Science 267, 476 (1995)

7 The Brazovskii model in two dimensions: stripe solutions H = 1 2 d 2 x [ κ( φ( x)) 2 r 0 φ 2 ( x) + u ] 0 2 φ4 ( x) + g 2 d 2 x d 2 x φ( x)g( x x )φ( x ) In reciprocal space H = H 0 + H i, with: H 0 = d 2 k (2π) 2 φ( k) ( r 0 + A 2 (k k 0 ) ) φ( k) H i = u 0 d 2 k 1 (2π) 2 d 2 k 2 (2π) 2 d 2 k 3 (2π) 2 d 2 k 4 (2π) 2 φ( k 1 )φ( k 2 )φ( k 3 )φ( k 4 ) δ( k 1 + k 2 + k 3 + k 4 ) First order isotropic-stripe transition induced by flucutations φ( x) = A cos( k 0 x) Anglular fluctuations of k 0 do not cost energy. K1 K2 Ky K3 t K4 Kx Brazovskii, JETP 41, 85 (1975) Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de / 12

8 Beyond Brazovskii: nematic-like order Consider a generic quartic interaction term of the form: H i = d 2 k 1 (2π) 2 d 2 k 2 (2π) 2 d 2 k 3 (2π) 2 d 2 k 4 (2π) 2 u( k 1, k 2, k 3, k 4, ) φ( k 1 )φ( k 2 )φ( k 3 )φ( k 4 ) δ( k 1 + k 2 + k 3 + k 4 ) Ky K1 K2 K3 t K4 Kx Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de / 12

9 Nematic symmetry Isotropy force the wave vectors to lay on the circle and hence u( k 1, k 2, k 3, k 4 ) = u(θ 1, θ 2, θ 3, θ 4 ). Conservation of total momentum allows to eliminate one of the four k s. Fixing two of them, the other two are automatically fixed by the requirement that the four vectors lay on the circle. Because of rotational invariance, the interaction can only depend on the difference between the two independent angles: u(θ 1, θ 2, θ 1 + π, θ 2 + π) = u(θ 1 θ 2 ) = u(θ) Finally, index permutation invariance in u( k 1, k 2, k 3, k 4 ) implies u(θ) = u(θ + π), which is a signature of nematic symmetry. Then u can be expanded in Fourier series: u(θ) = u 0 + u 2 cos(2θ) + u 4 cos(4θ) +... Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de / 12

10 Nematic energy Up to the first two terms in the expansion, the interaction energy with nematic symmetry can be written as: H 4 = d 2 x {u 0 φ 4 ( x) + u 2 tr ˆQ 2} with ˆQ ij ( x) = φ( x) ( i j 12 ) 2 φ( x) a traceless symmetric tensor which can also be represented in two dimensions as a complex nematic order parameter Q = α e i2θ The order parameter is a quadrupolar moment. Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de / 12

11 Mean field results If u 2 = 0 = Brazovskii smectic solution. Nevertheless, this solution is unstable to fluctuations in d = 2 and T Br 0. u 2 > 0 corresponds to repulsive quadrupole interactions. In this case the only solution is α(t) = 0. u 2 < 0 corresponds to attractive quadrupole interactions. In this case mean field gives a second order isotropic-nematic transition, with α(t) T c T 1/2 and T c = 2/(mk0) 2 u 0 / u 2. Smectic Nematic T Isotropic D.A.Barci and D.A.S., PRL 98, (2007) Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de / 12

12 Quasi-long-range order and experimental scales The continuous symmetry implies angular fluctuations in the director cost little energy and can destroy long range nematic order. Evaluating local fluctuations: δf F( ˆQ(x)) F(< ˆQ >) = ρ s (T) d 2 x ϕ(x) 2 KT energy Now, the mean field nematic order parameter is given by < ˆQ > = α exp( W) ( ) T L W = ln 4πρ s a Debye Waller factor with T KT = (π/8)ρ s (T KT ). Thermodynamic limit: L/a < ˆQ > 0, no nematic long range order. For the Fe/Cu(001) films in O. Portmann et al., Nature 422, 701 (2003) L 10 3 m a m W 7/32 < ˆQ > 0.8 α Long range nematic ordershould be observable! Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de / 12