Dynamics of isolated disordered models: equilibration & evolution

Transcription

1 Dynamics of isolated disordered models: equilibration & evolution Leticia F. Cugliandolo Sorbonne Universités, Université Pierre et Marie Curie Laboratoire de Physique Théorique et Hautes Energies Institut Universitaire de France Krakow, Poland, 2017

2 Question Does an isolated system reach equilibrium? Boosted by recent interest in the dynamics after quantum quenches of cold atomic systems rôle of interactions (integrable vs. non-integrable) many-body localisation novel effects of quenched disorder Same issues in classical systems? The (old) ergodicity question revisited LFC, Lozano & Nessi 17, LFC, Lozano, Nessi, Picco & Tartaglia (in prep)

3 Quantum quenches Definition & questions Take an isolated quantum system with Hamiltonian Ĥ0 Initialize it in, say, ψ 0 the ground-state of Ĥ 0 (or any ˆρ(t 0 )) Unitary time-evolution Û = e i Ĥt with a Hamiltonian Ĥ. Does the system reach a steady state? Is it described by a thermal equilibrium density matrix e βĥ? Do at least some observables behave as thermal ones? Does the evolution occur as in equilibrium? Other kinds of density matrices?

4 Classical quenches Definition & questions Take an isolated classical system with Hamiltonian H 0, evolve with H Initialize it in, say, ψ 0 a configuration, e.g. { q i, p i } for a particle system ψ 0 could be drawn from a probability distribution, e.g. Z 1 e β 0H 0 (ψ 0 ) Does the system reach a steady state? Is it described by a thermal equilibrium density matrix e βh? Do at least some observables behave as thermal ones? Does the evolution occur as in equilibrium? Other kinds of probability distributions

5 Quenches Various cases E 1 = E 1 kin + E1 pot E 2 = E 2 kin + E2 pot E 2 kin At t = 0 change in V V V Continuity in variables x(0 ) = x(0 + ) = x 0 p(0 ) = p(0 + ) = p 0 x x 0, p 0 x Jump in energy V V dashed to solid: energy extraction solid to dashed: energy injection x x

6 Classical quenches Models We chose to study classical disordered models Interesting & very well characterised equilibrium phases & relaxational dissipative dynamics rich free-energy landscapes with metastability, flat regions, large and small barriers, etc.

7 Quenched disorder Spin Disordered Potential V = P ij Jij si sj P ijk Jijk si sj sk +... A particle position ~ s = (s1,..., sn ) the exchanges Jij, Jijk, etc. taken from in a N dimensional space a probability distribution (details later) under a random potential V (~ s) Sketch for N Real variables si R Spherical constraint PN 2 i=1 si =N Connection with the following problem =2

8 Classical dynamics Coordinate-momenta pairs { s, p} and Hamiltonian H = K( p) + V ( s) Newton-Hamilton equations with the kinetic energy K( p) = 1 2m N i=1 p 2 i ṡ i = p i /m ṗ i = dv ( s)/ds i The potential energy landscape makes the models behave differently N saddles (including min/max) for two body-interactions V ( s) = i jj ij s i s j exp(nσ) saddles for more than two body interactions J ijk s i s j s k i j k With dissipation used to model domain-growth & fragile glasses, respectively

9 The quench Spin Disordered Potential V = P ij Jij si sj P ijk Jijk si sj sk +... with exchanges Jij, Jijk, etc. taken At time t =0 Same configuration s i (0), si (0) Ji01...ip 7 Ji1...ip Final parameter J from a Gaussian pdf zero mean [Jij ] = 0 and [Ji21...ip ] = p!j02 /(2N p 1 ) Initial parameter J0 The rugged landscape is stretched/contracted and pulled up/down

10 The initial conditions We chose initial states drawn from equilibrium with Hamiltonian H 0 at inverse temperature β The models have a phase transition at a finite β c The high temperature phase is a disordered one, a paramagnet (PM) The low temperature phase is different in the two-body and more than two-body interaction models : two equilibrium states like in a ferromagnet (FM) for two-body. a number diverging with N of equilibrium states with O(e NΣ ) metastable states, like in a glass in the more than two-body case. Initial conditions: disordered (PM) like or confined (FM/metastable).

11 Dynamic equations Conservative dynamics In the N limit exact causal Schwinger-Dyson equations (m 2 t z t )C(t, t w ) = dt [ Σ(t, t )C(t, t w ) + D(t, t )R(t w, t ) ] ( 2 t z t )R(t, t w ) = +β J 0 J D(t, 0)C(t w, 0), dt Σ(t, t )R(t, t w ) + δ(t t w ), with the post-quench self-energy and vertex D(t, t w ) = J 2 p 2 Cp 1 (t, t w ) Σ(t, t w ) = J 2 p(p 1) C p 2 (t, t w ) R(t, t w ) 2 and the Lagrange multiplier z t fixed by C(t, t) = 1.

12 Three body model Phase diagram 0.64 (a) T /J T 0 d 0.6 PM TAP Ageing Initial conditions T 0 s Jm 0 /(J 0 m) Energy injection Energy extraction In PM, quenches go to GB equilibrium at β f (e f ) with e f the final energy Following metastable states,gb-like equilibration at β f determined by e f Out of equilibrium relaxation with aging effects when e f = e th

13 Three body model Initial configuration in a metastable (TAP) state C(t1, t2) (a) q q 0 t 2 = χ (b) T f t 2 = C (c) 0.6 e(t) t 1 t 2 t Following metastable states, equilibration at β f fixed by e f = e kin f + e pot f

14 Three body model Energy extraction from PM to threshold C(t1, t2) (a) q t 2 = χ e(t) (b) T f T eff t 2 = C (c) t 1 t 2 t Out of equilibrium relaxation when quench parameters tuned so that e f = e th

15 Two body model Non-linear coupling through the Lagrange multiplier Diagonal in the basis of eigenvectors v µ of the interaction matrix J ij Projection of the coordinate (spin) vector on the eigenvectors s µ = s v µ with µ = 1,..., N Newton equations are m s µ (t) = [z(t) λ µ ]s µ (t) with z(t) the Lagrange multiplier that enforces the spherical constraint and λ µ the eigenvalues (distribution known from random matrix theory) Three methods to solve : for N, closed Schwinger-Dyson equations on C(t, t w ) and R(t, t w ), the global self-correlation and linear response for finite N, solve Newton equations imposing the constraint

16 Two-body model An Ansatz The projection of the spin configuration on the eigenvector v µ reads s µ (t) = s µ (0) Ω µ (0) t Ω µ (t) cos ṡ µ (0) + Ω µ (0)Ω µ (t) sin 0 t 0 dt Ω µ (t ) dt Ω µ (t ) The time-dependent frequency Ω µ (t) and Lagrange multiplier z(t) are fixed by 1 2 Ω µ (t) Ω µ (t) 3 4 ( Ωµ (t) ω µ (t) ) 2 + Ω 2 µ(t) = z(t) λ µ with initial conditions Ω µ (0) = 0 and Ω 2 µ(0) = λ max λ µ and z(t) = e f + 2 N µ λ µ s 2 µ(t) Similar to Sotiriadis & Cardy 10 for quantum O(N) model

17 Two-body model Results Recall (a) PM Ageing Phase diagram for the p = 3 model T /J0 T 0 d 0.6 TAP Phase diagram for p = 2 in preparation Ts Jm 0/(J 0m) Following confined states, GB equilibrium at β f determined by e f Quenches to PM parameters: the system does not reach GB equilibrium Instead, it approaches the Generalized Gibbs Ensemble with N quantities I µ that are conserved asymptotically and effective temperatures β µ The fluctuation-dissipation relation between the linear response and the correlation function is not satisfied

18 Fluctuation-dissipation relations Classical setting Measure Im R AB (ω) and ω C AB (ω) take the ratio and extract β AB eff (ω) In equilibrium all βeff AB (ω) should be equal to the same constant This is the fluctuation-dissipation theorem (FDT). If there is a frequency or observable dependence, the system is not in equilibrium Do these β eff (ω) play a role in closed systems too?

19 Two body model Non-linear coupling through the Lagrange multiplier The system is able to act as a bath on itself and equilibrate to a Gibbs- Boltzmann (GB) distribution functions ρ GB = Z 1 e β f H The system is not able to act as a bath on itself, it behaves as an integrable system asymptotically, and it approaches a Generalized Gbbs Ensemble (GGE) ρ GGE = Z 1 e N µ=1 β µi µ with the constants of motion 2I µ = mṡ 2 µ µ λ µs 2 µ (the kinetic + potential energy of the µth mode) and β µ fixed by I µ GGE = I µ (t = 0)

20 GGE and FDT temperatures A generic method FDR 2Im R(ω) ω C(ω) = β(ω) βµ The asympt mode freq ω 2 µ = [z as λ µ ]/m Im[ ˆR(ω µ )]/(ω µ Ĉ(ω µ )) 1/T µ N =1024 µ/n 2Im R(ω µ ) ω µ C(ωµ ) = β µ LFC. Lozano, Nessi, Picco & Tartaglia (in prep) cfr. Foini, Gambassi, Konik & LFC 17, see Laura Foini s talk on Wedn

21 Conclusions Study of the quenched dynamics of classical isolated disordered models We showed that they can equilibrate to GB measures undergo non-stationary (aging) dynamics (not shown here) approach a GGE depending on the type of model (highly interacting or quasi quadratic) and the kind of quench performed Work on the extension of these studies to the quantum models and the better understanding of the approach to a GGE is under way

22 Three-body interactions Potential energy landscape T d PM Spinodal epot/j e c e 0 Threshold Equilibrium e th (T d ) e eq (T s ) e eq T s T max th T/J T max eq

23 Glassy dynamics Non stationary relaxation & separation of time-scales Density-density correlation density response C 1e+00 q ea 1e-01 t α rapid & stationary (C st ) aging & slow (C ag ) χ 1e+00 χ ea aging & slow (χ ag ) rapid & stationary (χ st ) 1e-02 1e-01 1e+01 1e+03 t-tw 1e+05 1e-01 1e-01 1e+01 1e+03 t-tw 1e+05 C(t, t w ) χ(t, t w ) = t t w dt R(t, t ) Correlation Time-integrated linear response Analytic solution to a mean-field model LFC & J. Kurchan 93

24 Glassy dynamics Fluctuation-dissipation relation: parametric plot C 1e+00 q ea 1e-01 t α rapid & stationary (C st ) aging & slow (C ag ) e-02 1e-01 1e+00 1e+01 1e+03 1e+05 t-tw aging & slow (χ ag ) 0.6 ea χ χ ea rapid & stationary (χ st ) 0.2 1e-01 1e-01 1e+01 1e+03 t-tw 1e C q ea Analytic solution to a mean-field model LFC & J. Kurchan 93

25 FDR & effective temperatures Can one interpret the slope as a temperature? x Thermometer (coordinate x) Observable A Thermal bath (temperature T) Coupling constant k A A A A... α=1 α=2 α=3 α=μ M copies of the system 0.6 ea C q ea (1) Measurement with a thermometer with Short internal time scale τ 0, fast dynamics is tested and T is recorded. Long internal time scale τ 0, slow dynamics is tested and T is recorded. (2) Partial equilibration (3) Direction of heat-flow LFC, Kurchan & Peliti 97

26 FDT & effective temperatures Sheared binary Lennard-Jones mixture χ k (C k ) plot for different wave-vectors k, partial equilibrations. J-L Barrat & Berthier 00

27 Effective temperatures Glasses, coarsening, driven systems Different observables can depend differently (e.g. velocity vs. positions). There is a separation of time-scales, with a crossover at, roughly, ωt w (or controlled by the drive) The FDRs take a very special form: - ωt w 1 quasi-stationary relation and FDT with bath T - ωt w 1 non-stationary relation and FDR with another T. T eff (ω, t w ) crosses over from T to T that depends upon - the initial condition before the quench (disordered vs. ordered) ; - weakly on other parameters of the systems. Notion of interacting vs. non-interacting concerning partial equilibrations. LFC 11, review

28 Fluctuation-dissipation relations Quantum setting Measure Im R AB (ω) and CAB ± (ω) take the ratio and extract tanh(β AB eff (ω) ω/2) In equilibrium all βeff AB (ω) should be equal to the same constant This is the fluctuation-dissipation theorem (FDT). If there is a frequency or observable dependence, the system is not in equilibrium Do these β eff (ω) play a role in closed systems too?

29 Plan 1. Introduction. 2. Fluctuation-dissipation relations Measurements of effective temperatures and properties. Relation to free-energy densities and entropy. Fluctuation theorems. 3. Quantum quenches. 4. Integrable systems and Generalized Gibbs Ensembles. LFC, Kurchan & Peliti 97 ; Foini, LFC & Gambassi 11 & 12 ; Foini, Gambassi, Konik & LFC 16 ; de Nardis, Panfil, Gambassi, LFC, Konik & Foini 17 Thanks to the joint autumn programs at KITP 2015

30 Plan 4 Integrable systems and Generalized Gibbs Ensembles. - As a test of non-thermal equilibrium - Integrable non-interacting systems Foini, LFC & Gambassi 11 & 12 - Integrable interacting systems Foini, Gambassi, Konik & LFC 16 de Nardis, Panfil, Gambassi, LFC, Konik & Foini 17 Thanks to the joint autumn programs at KITP 2015

31 Fluctuation-dissipation relations Quantum Ising chain The initial Hamiltonian Ĥ Γ0 = i ˆσ x i ˆσ x i+1 + Γ 0 ˆσ z i i The initial state ψ 0 ground state of Ĥ Γ0 Instantaneous quench in the transverse field Γ 0 Γ Evolution with ĤΓ. Iglói & Rieger 00 Reviews: Karevski 06 ; Polkovnikov et al. 10 ; Dziarmaga 10 Observables : correlation and linear response of local longitudinal and transverse spin, etc. Specially interesting case Γ c = 1 the critical point. Rossini et al. 09

32 Quantum quench T eff from the FDR (quench to Γ c = 1) Im R(ω) = tanh(β eff (ω)ω /2) C+ (ω) T eff T z 1 / Teff T M eff T z eff T E eff ω βeff z (ω) βm eff (ω) ct Foini, LFC & Gambassi 11 & 12 Similar ideas in Bortolin & Iucci 15 (hard core bosons) Chiocchetta, Gambassi & Carusotto 15 (photon/polariton condensates)

33 Summary Fluctuation-dissipation relations Use of fluctuation-dissipation relations to check for deviations from Gibbs-Boltzmann equilibrium in the dynamics of closed quantum systems

34 Fluctuation-dissipation relations Can they be used to infer the steady state density operator?

35 From the FDR to the GGE Lehman representation The correlation and linear response are C(t 2, t 1 ) = 1 2 [Â(t2), (t 1 )] + R(t 2, t 1 ) = i [Â(t2), (t 1 )] θ(t 2 t 1 ) The expectation value is calculated over a generic density matrix ˆρ (units such that =1 and [ ˆX,Ŷ ] ± ˆXŶ ±Ŷ ˆX ) Taking a Fourier transform wrt to t 2 t 1 C(ω) = π δ(ω + E n E m ) A nm 2 (ρ nn + ρ mm ) m,n 0 Im R(ω) = π δ(ω + E n E m ) A nm 2 (ρ nn ρ mm ) m,n 0 where the sums run over a complete basis of eigenstates { n } n 0 of the Hamiltonian Ĥ with increasing eigenvalues E n, and X mn = m ˆX n.

36 From the FDR to the GGE Lehman representation Note that C(ω) and Im R(ω) are non-zero only if ω takes values within the discrete set {E m E n } m,n 0 (due to the delta functions) and A nm 0. In Gibbs-Boltzmann equilibrium ρ nn exp( βe n ) since there is a single charge, ˆQ1 = Ĥ, and for any bosonic Â, the FDT holds ω Im R(ω) = tanh(βω/2) C(ω) In contrast, for the GGE, ρ nn exp( k λ kq kn ) with Q kn n ˆQ k n. By properly choosing  we can extract the λ k s from the corresponding FDR.

37 From the FDR to the GGE Example: a non-interacting integrable model Take a non-interacting Hamiltonian in its diagonal form Ĥ = k ɛ k ˆη k ˆη k, ˆη k s are creation operators for excitations of energy ɛ k The number operators ˆQ k = ˆη k ˆη k are the (commuting) conserved charges. The GGE density matrix is ˆρ e k λ k ˆQ k with ˆQ k = ˆη k ˆη k, while β k λ k /ɛ k defines a mode-dependent inverse effective temperature. For  = k (α k ˆη k + α k ˆη k ) Im R(ω k ) C(ω k ) = tanh(λ k /2) with ω = ω k ɛ k (in the absence of degeneracies with respect to k and α k C)

38 Quantum quench Hard-core bosons in one dimension Consider the Lieb-Liniger model with density ϱ Ĥ c = dx [ ˆφ x (x) x ˆφ(x) + c ˆφ (x) ˆφ ] (x) ˆφ(x) ˆφ(x) initialized in the ground state of Ĥ c=0 and evolved with Ĥc Mapping to hard-core bosons, that after a Jordan-Wigner transformation become free fermions, Ĥ c = k ɛ ˆf k ˆf k k with ɛ k = k 2 The conserved charges are ˆQ k = ˆf ˆf k k = 4ϱ 2 /(ɛ 2 k + 4ϱ2 ) and the Lagrange multipliers λ k = ln[ɛ 2 k /(4ϱ2 )]

39 Quantum quench Hard-core bosons in one dimension Consider again the c = 0 to c quench of the Lieb-Liniger model In the stationary limit (and for q 0, π) C(q, t) = e i(ɛ k ɛ k q ) t ( n k q + n k n k q n k ) 2 k R(q, t) = iθ(t) k e i(ɛ k ɛ k q )t (n k q n k ) The Fourier transform picks ω = ɛ k ɛ k q with two solutions k 1,2 (q, ω) Measuring at frequency ω and wave-vector q related by ω = 2ɛ (q+π)/2, a single mode is selected, k 1 = k 2 = (q + π)/2, and the FDR becomes Im R(q, ω q )/C(q, ω q ) = tanh λ q with λ q = ln[ɛ 2 q/(4ϱ 2 )]

40 Quantum quench Hard-core bosons in one dimension

41 Quantum quench The one dimensional Bose gas Consider the Lieb-Liniger model with density ϱ Ĥ c = dx [ ˆφ x (x) x ˆφ(x) + c ˆφ (x) ˆφ ] (x) ˆφ(x) ˆφ(x) initialized in the ground state of Ĥ c=0 and evolved with Ĥc<+, now an interacting problem. Bethe Ansatz solution: lim t O = ϑ GGE O ϑ GGE with the eigenstate ϑ GGE characterised by a mode occupation ϑ GGE (λ) computed, for this problem, in De Nardis, Wouters, Brockmann & Caux 14

42 Quantum quench The one dimensional Bose gas Let us parametrize ϑ GGE (λ) as and ɛ(λ F )=0. ϑ GGE (λ) = 1 1+e ɛ(λ) One particle-hole kinematics at slow momentum k k F = πϱ The FDR of the density-density correlation and linear response is Im R(k, ω)/ C(k, ω) = tanh(k λ ɛ(λ)/(2πρ t (λ)) λ(k,ω) Without entering the technical details, ɛ(λ), ρ t (λ) and λ(k, ω) depend on ϑ GGE (λ). Computing the left-hand-side one can reconstruct ϑ GGE (λ) and compare it to the exact form derived by De Nardis et al 14

43 Quantum quench Hard-core bosons in one dimension #num N =10,k =0.2k F N =20,k =0.2k F N =20,k =0.6k F #num #GGE ϑ num from FDR & ϑ GGE from direct calculation. Error due to the low k expansion present in both evaluations. de Nardis, Panfil, Gambassi, LFC, Konik, Foini 17

44 Summary Fluctuation-dissipation relations The FDRs of carefully chosen, but quite natural, observables contain the GGE effective temperatures. They can be used to measure them or, even more generally, to infer the steady state density matrix

45 References 4 Integrable systems and Generalized Gibbs Ensembles. - As a test of non-thermal equilibrium - Integrable non-interacting systems Foini, LFC & Gambassi 11 & 12 - Integrable interacting systems Foini, Gambassi, Konik & LFC 16 de Nardis, Panfil, Gambassi, LFC, Konik & Foini 17 Thanks to the joint autumn programs at KITP 2015

46 Two-time observables Correlations 0 τ t=0 t t preparation time w waiting time measuring time time The two-time correlation between two observables Â(t) and ˆB(t w ) is C AB (t, t w ) Â(t) ˆB(t w ) expectation value in a quantum system,... = Tr... ˆρ/Trˆρ or the average over realizations of the dynamics (initial conditions, random numbers in a MC simulation, thermal noise, etc.) in a classical system.

47 Two-time observables Linear response h 0 t δ t + δ w w 2 2 t The perturbation couples linearly to the observable ˆB at time t w Ĥ Ĥ h(t w) ˆB The linear instantaneous response of another observable R AB (t, t w ) δ Â(t) h δh(t w ) h=0 Â(t) is Similarly in a classical system

48 Fluctuation-dissipation theorem Gibbs-Boltzmann density operator ˆρ = Z 1 e βĥ C BA ( ω) = e βω CAB (ω) and then Im R AB (ω) = [ 1 ±1 tanh(β ω/2)] CAB ± (ω) Bosons Fermions Classical limit : Im R AB (ω) = βω C AB (ω)

49 Quantum quench T eff from the longitudinal spin FDR e-01 1e-05 1e-09 C x (τ) R x (τ) Insets 0.6 1e e τ/τ c τ 2 sin(4τ + φ) τ C x (τ) A c e τ/τ c [1 a c τ 2 sin(4τ + φ c )] R x (τ) A R e τ/τ c [1 a R τ 2 sin(4τ + φ R )] Foini, LFC & Gambassi 11

50 Quantum quench T eff from FDT? For sufficiently long-times such that one drops the power-law correction β x eff Rx (τ) d τ C x +(τ) τ ca R A c A constant consistent with a classical limit but T x eff(γ 0 ) T e (Γ 0 ) Morever, a complete study in the full time and frequency domains confirms that T x eff(γ 0, ω) T z eff(γ 0, ω) T e (Γ 0 ) (though the values are close). Fluctuation-dissipation relations as a probe to test thermal equilibration No equilibration for generic Γ 0 in the quantum Ising chain

51 FDT & effective temperatures Role of initial conditions T > T found for quenches from the disordered into the glassy phase (Inverse) quench from an ordered initial state, T < T 2d XY model or O(2) field theory Binary Lennard-Jones mixture Berthier, Holdsworth & Sellitto 01 Gnan, Maggi, Parisi & Sciortino 13

52 Quantum quenches Expectations Non-integrable systems expected to eventually thermalise. Integrable systems? role of initial states ; non critical vs. critical quenches, etc. Definition of T e from ψ 0 Ĥ ψ 0 = Ĥ T e = Z 1 β e Tr Ĥe β eĥ Just one number, it can always be done Comparison of dynamic and thermal correlation functions, e. g. C(r, t) ψ 0 ˆφ( x, t) ˆφ( y, t) ψ 0 vs. C(r) ˆφ( x) ˆφ( y) Te. Calabrese & Cardy ; Rigol et al ; Cazalilla & Iucci ; Silva et al, etc. But the functional form of correlation functions can be misleading!

53 Quantum quenches Questions Non-integrable systems expected to thermalise. Integrable systems? role of initial states ; non critical vs. critical quenches, etc. Definition of T e from ψ 0 Ĥ ψ 0 = Ĥ T e = Z 1 β e Tr Ĥe β eĥ Just one number, it can always be done Comparison of dynamic and thermal correlation functions, e. g. C(r, t) ψ 0 ˆφ( x, t) ˆφ( y, t) ψ 0 vs. C(r) ˆφ( x) ˆφ( y) Te. Calabrese & Cardy ; Rigol et al ; Cazalilla & Iucci ; Silva et al, etc. Proposal: put qfdt to the test to check whether T eff = T e exists Foini, LFC & Gambassi 11 & 12