Quenches in statistical physics models

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1 Quenches in statistical physics models Leticia F. Cugliandolo Sorbonne Universités, Université Pierre et Marie Curie Laboratoire de Physique Théorique et Hautes Energies Institut Universitaire de France leticia/seminars In collaboration with Jeferson Arenzon, Thibault Blanchard, Alan Bray, Federico Corberi, Ingo Dierking, Michikazu Kobayashi, Ferdinando Insalata, Marcos-Paulo Loureiro, Marco Picco, Yoann Sarrazin, Alberto Sicilia and Alessandro Tartaglia. Heidelberg, September 2016

2 Thermal quenches in the stochastic Gross-Pitaevskii equation : morphology of the vortex network Leticia F. Cugliandolo Sorbonne Universités, Université Pierre et Marie Curie Laboratoire de Physique Théorique et Hautes Energies Institut Universitaire de France leticia@lpthe.jussieu.fr leticia/seminars In collaboration with Michikazu Kobayashi Heidelberg, September 2016

3 Quenches in statistical physics models passing by critical percolation! What is it about? Open system Stochastic dissipative dynamics Statistical physics framework Out of equilibrium percolation fractal properties coarsening - phase ordering kinetics

4 Phenomenon The talk is on Dynamics following a fast quench of a control parameter across an equilibrium phase transition. The equilibrium phases are known on both sides of the transition, i.e. the possible asymptotic states are known. The dynamic mechanism is understood, the system tries to order locally in one of the competing states.

5 Interests and goals Practical & fundamental interest, e.g. Mesoscopic structure effects on the opto-mechanical properties of phase separating glasses Cooling rate effects on the density of topological defects in cosmology and condensed matter Some issues The role played by the initial conditions & short-time dynamics Full geometric characterisation of the structure When does the usual dynamic scaling regime set in? The role played by the cooling rate that are related to each other.

6 Second-order phase transition Z 2 symmetry breaking, g the control and ϕ the order parameter In the picture: annealing with finite rate, and scalar order parameter. Infinitely fast quench: g 0 > g c for t < 0 and g < g c for t > 0 e.g. g = T/J in a classical spin model coupled to a bath at temperature T

7 Phenomenon This talk is on the stochastic dynamics of a U(1) complex field theory (similar to the 3d XY lattice model) after an instantaneous quench from high to low temperature There is a 2nd order phase transition, and in magnetic language the equilibrium phases are the paramagnet at high T ( ψ = 0) and a (degenerate) ferromagnet at low T ( ψ = 0). The dynamic mechanism is understood.

8 Relativistic bosons; 4 He, type II superconductors, cosmology, etc. c 2 ψ + 2 ψ + 2iµ ψ = g(ψ 2 ρ)ψ c is the velocity of light, ρ and g parameters in (Mexican hat) potential. Limits µ 0 : c 2 ψ + 2 ψ = g( ψ 2 ρ)ψ Goldstone c : 2iµ ψ + 2 ψ = g( ψ 2 ρ)ψ Gross-Pitaevskii models

9 Relativistic bosons; 4 He, type II superconductors, cosmology, etc. c 2 ψ + 2 ψ + 2iµ ψ = g(ψ 2 ρ)ψ The energy functional E = d 3 x (c 2 ψ 2 + ψ 2 gρψ 2 + gψ 4 ) is conserved under the dynamics. The energy is minimised by the static configuration ψ = ρ e iχ with χ = ct There are static vortex solutions, e.g. ψ( x) = f(r) e inθ with f(0) = 0 and f(r ) = ρ, and n Z (thin tubes at the centre of which the field vanishes and the phase turns around). Tsubota, Kasamatsu & Kobayashi 13, Kobayashi & NItta 15, etc.

10 Stochastic noise and dissipation added c 2 ψ + 2 ψ + 2iµ ψ γ ψ = g(ψ 2 ρ)ψ γt ξ Langevin-like dynamics γ viscosity, ξ complex Gaussian white noise in normal form ξ i ( x, t) = 0 and ξ i ( x, t 1 )ξ j ( y, t 2 ) = δ ij δ (3) ( x y)δ(t 1 t 2 ) Passage to Fokker-Planck formalism allows to show that the dynamics takes the system to lim t P (ψ, t) = P GB (ψ) e βe Kobayashi & LFC 16

11 Relativistic bosons; also 4 He, type II superconductors, cosmology, etc. c 2 ψ + 2 ψ + 2iµ ψ γ ψ = g(ψ 2 ρ)ψ γt ξ Langevin-like dynamics γ viscosity, ξ Gaussian white noise in normal form In the limit c, the stochastic Gross-Pitaevskii equation (2iµ γ) ψ = 2 ψ + g(ψ 2 ρ)ψ + γt ξ Gardiner et al 00s

12 3d XY lattice model Archetypical classical magnetic example H = J ij s i s j J > 0 ferromagnetic coupling constant. ij sum over nearest-neighbours on a 3d lattice s i planar spins: two components with constant modulus angle θ i. Second order phase transition with spontaneous symm breaking at T c > 0. Order parameter: spin-alignment, m N 1 i s i. No intrinsic spin dynamics, Monte Carlo rules mimic coupling to thermal bath. Non-conserved order parameter dynamics [ towards ] etc. allowed.

13 Statics Phase transition and order parameter in the field equation m L = (T c T) β T L 3 m = ijk ψ ijk critical temperature T c = 2.26 critical exponent β = Kobayashi & LFC 16 T c and critical exponents from kurtosis (Binder parameter), susceptibility, specific heat, etc. Values compatible w/results from simulations Ballesteros et al. 96, Hasenbusch & Török 99 and ϵ expansion Guida & Zinn-Justin 98, Täuber & Diehl 14 for models in the same universality class.

Vorticity (zeros of the field) & reconnection conventions

Typical choices: maximal & stochastic reconnection rules

14 Vorticity (zeros of the field) & reconnection conventions 2πv p = plaq [ θ] 2π = 0, ±1,... ( 0 when the field turns around on a plaquette) One field configuration with two possible line structures Typical choices: maximal & stochastic reconnection rules while just one choice in Kajantie et al. 00, Bittner, Krinner & Janke 05, Kobayashi & LFC 16

15 Vortex configurations In equilibrium 0.6 T c 0.8 T c T c 1.2 T c Periodic boundary conditions (torus) implies that the vortex lines are closed, i.e. loops. Stochastic reconnection rule. All vortex loops in blue, the longest one in red.

16 Number density of vortex loops in equilibrium, T -dependence N (S) T = 0.8 T c T c 1.2 T c 2.0 T c l 5/2 l Stochastic rule l High & low T At T T c, red data points, very short loops exponential decay Kobayashi & LFC 16

17 Number density of vortex loops in equilibrium, T -dependence N (S) T = 0.8 T c T c 1.2 T c 2.0 T c l 5/2 l Stochastic rule l High & low T At T T c, yellow data points, very long loops two power law decays l 5/2 and l 1 Kobayashi & LFC 16

18 Number density of vortex loops in equilibrium, L-dependence N (S) /L L = l 5/2 l Stochastic rule l High T At T T c - for l L 2 Gaussian statistics l 5/2 cfr. Flory 41, de Gennes 79, Vachaspati & Vilenkin 84 - for l L 2 fully-packed loops large-scale statistics l 1 cfr. Nahum & Chalker 12

19 Number density of vortex loops in equilibrium T T c N(l) l α L [ (c1 L 3 ) n + c n 2 l (α L 1)n ] 1/n with α L = 5/2, c 1 and c 2 finite constants, n = 6 sharp crossover. N (S) l L = l 3/2 1st term l L 2 N l α LL 3 Nl (l/l 2 ) 3/2 2nd term l L 2 N l 1 Nl ct l/l 2 Kobayashi & LFC 16

20 Number density of vortex loops in equilibrium, T -dependence N (S) T = 0.8 T c T c 1.2 T c 2.0 T c l 5/2 l Stochastic rule l High & low T At T = T c, black data points another power-law with a cut-off? Kobayashi & LFC 16

21 Percolation Purely geometric problem Percolating red cluster highlighted in black

22 Percolation Purely geometric problem Take a lattice Λ in d spatial dimensions. Define a site occupation variable n i = 1, 0 with probability p, 1 p Question : What is the probability of there being a cluster of occupied nearestneighbour sites that crosses a sample with linear size L from one end to another in at least one direction? In the limit L there is a continuous phase transition at p c such that = 0 if p p c lim P (p, L) L > 0 if p > p c p c depends on Λ and d. At p c the spanning cluster has fractal properties that are well characterised

23 Percolation Purely geometric problem Take a lattice Λ in d spatial dimensions. Define a site occupation variable n i = 1, 0 with probability p, 1 p Question : What is the probability of there being a cluster of occupied nearestneighbour sites that crosses a sample with linear size L from one end to another in at least one direction? In the limit L there is a continuous phase transition at p c such that = 0 if p p c lim P (p, L) L > 0 if p > p c p c depends on Λ and d. The distribution of finite size clusters is algebraic at p c, n(a) A α A

24 Percolation Purely geometric problem Take a lattice Λ in d spatial dimensions. Define a site occupation variable n i = 1, 0 with probability p, 1 p Question : What is the probability of there being a cluster of occupied nearestneighbour sites that crosses a sample with linear size L from one end to another in at least one direction? In the limit L there is a continuous phase transition at p c such that = 0 if p p c lim P (p, L) L > 0 if p > p c p c depends on Λ and d. The distribution of finite length interfaces is algebraic at p c, n(l) l α L

Localised-extended transition The thermodynamic phase transition (from non-analyticity of free-energy density) does not necessarily coincide with the geometric transition between a phase

25 Localised-extended transition The thermodynamic phase transition (from non-analyticity of free-energy density) does not necessarily coincide with the geometric transition between a phase with localised loops and one with extended loops. Well-known fact in the context of statistical physics models, such as the finite dimensional Ising models, where spin clusters are studied

26 - in d = 3, T L < T c Müller-Krumbhaar 74 Localised-extended transition The thermodynamic phase transition (from non-analyticity of free-energy density) does not necessarily coincide with the geometric transition between a phase with localised loops and one with extended loops. Well-known fact in the context of statistical physics models, such as the finite dimensional Ising models where spin clusters are studied, - in d = 2, T L = T c but criticality is not the same. Kasteleyn & Fortuin 69, Coniglio & Klein 81, Swendsen & Wang 87

27 Number density of vortex loops in equilibrium, determine T L N (S) T = 0.95 T c 0.96 T c 0.97 T c 0.98 T c 0.99 T c l l Stochastic rule close to T (S) L At T (S) L = 0.98 T c, percolation of vortex loops, algebraic decay l 2.17 Kajantie et al. 00, Bittner et al. 05, Kobayashi & LFC 16

28 Number density of vortex loops in equilibrium: finite size scaling 10 2 L = l N/L ml TL T l Stochastic rule (S) At TL (S) close to TL = 0.98 Tc, percolation of vortex loops, algebraic decay ℓ 2.17 Kajantie et al. 00, Bittner et al. 05, Kobayashi & LFC 16

29 Number of non-contractible loops (on the torus) 3 T c 2 Nnon cont T L 1 L = T At T (S) L number density = 0.98 T c, percolation of vortex loops, algebraic decay l 2.17 of their

30 T L < T c Confirmed with other observables. Independent of the reconnection rule. Independent of the lattice spacing. Static feature, therefore independent of the dynamic equation used to sample the loops.

31 Dynamics after a quench with g the control parameter In the picture: annealing with finite rate. Infinitely fast quench: T T c for t < 0 and T = 0 for t > 0

32 Progressive elimination of vortex loops after a quench T T c T = 0 t = 0 t = 3 t = 5 As ρ vortex the reconnection rule loses importance Kobayashi & LFC 16

33 Interests and goals Practical & fundamental interest, e.g. Mesoscopic structure effects on the opto-mechanical properties of phase separating glasses Cooling rate effects on the density of topological defects in cosmology and condensed matter Some issues The role played by the initial conditions & short-time dynamics Full geometric characterisation of the structure When does the usual dynamic scaling regime set in? The role played by the cooling rate that are related to each other.

34 Number density of vortex loops after a quench to T = 0 N l max l t = l 2.17 l 1 l 5/2 After t p 6 there are three algebraic regimes l - l < l max short loops - l max < l < l percolation-like l l l fully-packed loops large-scale statistics l 1

35 Number density of vortex loops after a quench to T = 0 N t = l 2.17 l l l Time t longer than t p 6 still three (time-dependent) algebraic regimes l before maximum N(l, t) l 2.17 percolation-like after maximum l 1 after last crossover

36 Number density of vortex loops after a quench to T = 0 N t = l 2.17 l l l Note also that the first two length-scale regimes are translated down in time, while the last one is not.

37 Time-dependence of the total loop number Nloop/L L = t Various system sizes, L = 40, 100, 500 Dotted line N loop /L 3 = t ζ with ζ = 1

38 Dynamic scaling Assume that at t p N(l, t p ) l α L[(c 1 (t p )L 3 ) n + c n 2 l(α L 1)n ] 1/n with α L 2.17, the one of critical line percolation at T L. Describing the two algebraic decays l α L controlled by c 1 (t p ), c 2 and L. and l 1 with a cross-over The parameter n = 6 is enough to render the crossover sharp. From N loop t ζ with ζ 1, and l p = l(t p ), assume l(t, l p ) l 2 p + γ v (t p t)

39 Analytic arguments We followed counting arguments in Arenzon, Bray, LFC & Sicilia 07 For very long lines, the number of which does not change in time, N(l, t) ln( l 2 + γ 2 v(t t p ), t p ) l2 + γ 2 v(t t p ) Implies a crossover from l 1 to l α L at a time-dependent length-scale l (t) (c 1 (t p )/c 2 ) 1/(α L 1) γ 2 v(t t p ) Kobayashi & LFC 16

40 Analytic arguments We followed counting arguments in Arenzon, Bray, LFC & Sicilia 07 For t t p, the weight of l < l (t) diminishes, (N loop (t) t ζ ), N(l, t) (γ 2 vt) ζ ln( l 2 + γ 2 v(t t p ), t p ) l2 + γ 2 v(t t p ) that, using N(y, t p ) c 1 (t p )L 3 y α L, can be further simplified to (γ 2 vt) ζ+α L/2 N(l, t) c 1(t p )L 3 l/(γ 2 vt) 1/2 [1 + l 2 /(γ 2 vt)] (1+α L)/2

41 Analytic arguments (γ 2 vt) ζ+α L/2 N(l, t) c 1(t p )L 3 l/(γ 2 vt) 1/2 [1 + l 2 /(γ 2 vt)] (1+α L)/2 two length-scale limits (γ 2 vt) ζ+α L/2 N(l, t) L 3 l/(γ 2 vt) 1/2 (l/(γ 2 vt) 1/2 ] α L l ξ d (t) l ξ d (t) with ξ d (t) (γ 2 vt) 1/2

42 Number of vortex loops after a quench to T = 0 - power laws N t = l 2.17 l l l The argument above justifies the l 1 (dotted) and l 2.17 (dashed) regimes.

43 Number of vortex loops after a quench to T = 0 - scaling t = t 2 N l/t 1/2 The dotted line is the analytic functional form with ζ = 1

44 Summary & conclusions Statics of the 3d U(1) field theory We solved numerically the stochastic dynamics of a 3d complex field. We found that it approaches equilibrium (as it should). We studied the phase transition and critical exponents and found values in agreement with previous analytic predictions and numerical simulations for this universality class. We showed that the thermodynamic phase transition does not coincide with the geometric one for line percolation in equilibrium configurations. We characterised the number density of vortex lengths in equilibrium at all temperatures.

45 Summary & conclusions Dynamics of the 3d U(1) field theory We performed instantaneous quenches from T T c to T = 0 and we studied the vortex dynamics. We focused on the time evolution of the number density of vortex lengths. We found that the initial high temperature double algebraic form rapidly evolves to a triple algebraic form with an intermediate regime with the statistics of the critical percolating configurations. This is similar to, but also different from, what we found in 2d statistical spin models Blanchard, LFC & Picco 14, Tartaglia, LFC & Picco 15. We characterised the functional form of the time-evolving number density with semi-analytic arguments.

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