Symmetries in large deviations of additive observables: what one gains from forgetting probabilities and turning to the quantum world

Transcription

1 Symmetries in large deviations of additive observables: what one gains from forgetting probabilities and turning to the quantum world Marc Cheneau 1, Juan P. Garrahan 2, Frédéric van Wijland 3 Cécile Appert-Rolland 4, Bernard Derrida 5, Alberto Imparato 6 1 Institut d Optique, Palaiseau 2 Nottingham University 3 MSC, Paris 4 LPT, Orsay 5 LPS, ENS, Paris 6 Aarhus University Nice June 9th 2015 Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

2 Introduction Motivations Classical and quantum dynamics Correspondence generator of stochastic classical system [particles hopping] Hamiltonian of quantum XXZ chain (Well known at least in the stat. mech. community.) Use: dictionary between regimes of large deviations of dynamical observables phases across a Quantum Phase Transition Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

3 Introduction Motivations Classical and quantum dynamics Correspondence generator of stochastic classical system [particles hopping] Hamiltonian of quantum XXZ chain (Well known at least in the stat. mech. community.) Use: dictionary between regimes of large deviations of dynamical observables phases across a Quantum Phase Transition Perspectives opened ; questions raised finite-size effects large/small scale spectrum import/export techniques from/to stat. mech. (I will ask questions to you.) Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

4 Exclusion Process System Exclusion Processes generic settings maximal occupation N α β ρ 0 ρ 1 γ 1 L δ Configurations: occupation numbers {n i } Exclusion rule: 0 n i N Markov evolution for the probability P({n i }, t) t P({n i }, t) = [ W(n i n i )P({n i}, t) W(n i n i)p({n i }, t) ] n i Large deviation function of additive observables A e sa e t ψ(s) ( determining P(A, t)) A = total current Q on time window [0, t] = #j umps j umps A = total activity K on time window [0, t] = #j umps + j umps Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

5 Exclusion Process System Exclusion Processes generic settings maximal occupation N α β ρ 0 ρ 1 γ 1 L δ Configurations: occupation numbers {n i } Exclusion rule: 0 n i N Markov evolution for the probability P({n i }, t) t P({n i }, t) = [ W(n i n i )P({n i}, t) W(n i n i)p({n i }, t) ] n i Large deviation function of additive observables A e sa e t ψ(s) ( determining P(A, t)) A = total current Q on time window [0, t] = #j umps j umps A = total activity K on time window [0, t] = #j umps + j umps Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

6 Exclusion Process System Exclusion Processes generic settings maximal occupation N α β ρ 0 ρ 1 γ 1 L δ Configurations: occupation numbers {n i } Exclusion rule: 0 n i N Markov evolution for the probability P({n i }, t) t P({n i }, t) = [ W(n i n i )P({n i}, t) W(n i n i)p({n i }, t) ] n i Large deviation function of additive observables A e sa e t ψ(s) ( determining P(A, t)) A = total current Q on time window [0, t] = #j umps j umps A = total activity K on time window [0, t] = #j umps + j umps Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

7 Exclusion Process System Operator representation [Schütz & Sandow PRE ] maximal occupation N α β ρ 0 ρ 1 γ 1 L δ Evolution of probability vector P: t P = W P W = 1 k L 1 [ σ + k σ k+1 + σ k σ+ k+1 ˆn kň k+1 ˆn k+1 ň k ] + α [ σ 1 + ň ] [ ] 1 + γ σ 1 ˆn 1 + δ [ σ + L ň ] [ L + β σ L ˆn ] L σ ± = σ x ± iσ y and σ z = ˆn N 2 are spin operators (of spin j = N 2 ) Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

8 Exclusion Process System Operator representation [Schütz & Sandow PRE ] maximal occupation N α β ρ 0 ρ 1 γ 1 L δ Evolution of probability vector P: t P = W P W = 1 k L 1 [ σ + k σ k+1 + σ k σ+ k+1 ˆn kň k+1 ˆn k+1 ň k ] + α [ σ 1 + ň ] [ ] 1 + γ σ 1 ˆn 1 + δ [ σ + L ň ] [ L + β σ L ˆn ] L σ ± = σ x ± iσ y and σ z = ˆn N 2 are spin operators (of spin j = N 2 ) densities ρ 0 = α α+γ ; ρ 1 = δ δ+β ; contact rates a 0 = α γ ; a 1 = δ β Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

9 Exclusion Process System Operator representation [Schütz & Sandow PRE ] maximal occupation N α β ρ 0 ρ 1 γ 1 L δ Evolution of probability vector P: t P = W P W = 1 k L 1 [ σ + k σ k+1 + σ k σ+ k+1 ˆn kň k+1 ˆn k+1 ň k ] + α [ σ 1 + ň ] [ ] 1 + γ σ 1 ˆn 1 + δ [ σ + L ň ] [ L + β σ L ˆn ] L σ ± = σ x ± iσ y and σ z = ˆn N 2 are spin operators (of spin j = N 2 ) XXX spin chain Hamiltonian (up to boundary terms and constants). Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

10 Exclusion Process Large deviations Operator representation for large deviations maximal occupation N α β ρ 0 ρ 1 γ 1 L δ e sk e tψ(s) with ψ(s) = max Sp W s W s = [ e s σ + k σ k+1 + e s σ k σ+ k+1 ˆn ] kň k+1 ˆn k+1 ň k 1 k L 1 + α [ e s σ 1 + ň [ 1] + γ e s σ1 ˆn ] 1 + δ [ e s σ + L ň [ L] + β e s σ L ˆn ] L for the activity K: XXZ spin chain Hamiltonian Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

11 Exclusion Process Large deviations Operator representation for large deviations maximal occupation N α β ρ 0 ρ 1 γ 1 L δ e sq e tψ(s) with ψ(s) = max Sp W s W s = [ e +s σ + k σ k+1 + e s σ k σ+ k+1 ˆn ] kň k+1 ˆn k+1 ň k 1 k L 1 + α [ e s σ 1 + ň [ 1] + γ e +s σ1 ˆn ] 1 + δ [ e +s σ + L ň [ L] + β e s σ L ˆn ] L for the current Q: asymmetric XXZ spin chain Hamiltonian Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

12 Exclusion Process Non-equilibrium drive from boundaries Example 1: use of rotational symmetry map non-equilibrium current fluctuations to equilibrium current fluctuations Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

13 Exclusion Process Local rotation Mapping non-eq to eq [Imparato, VL, van Wijland, PTPS ] Large deviations of the current W(s) = invariant by rotation { }}{ Sk S k+1 1 k L 1 + constant + α [ S + 1 ň ] [ ] 1 + γ S 1 ˆn 1 + δ [ S + ] [ ] L es ň L + β S L e s ˆn L ψ(s) = max Sp W(s) Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

14 Exclusion Process Local rotation Mapping non-eq to eq [Imparato, VL, van Wijland, PTPS ] Large deviations of the current W(s) = invariant by rotation { }}{ Sk S k+1 1 k L 1 Local transformation Q 1 W(s)Q = + α [ S + 1 ň ] [ ] 1 + γ S 1 ˆn 1 + δ [ S + ] [ ] L es ň L + β S L e s ˆn L 1 k L 1 Sk S k+1 ψ(s) = max Sp W(s) [non-hermitian due to boundaries] + α [ S + 1 ň 1] + γ [ S 1 ˆn ] [ 1 ] [ ] +δ S + L es ň L + β S L e s ˆn L }{{} describes contact with reservoirs of same densities Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

15 Exclusion Process Local rotation SO(3) symmetry [Imparato, VL, van Wijland, PTPS ] Detailed transformation: Q = 1 + xs x iys y + zs z (on one site) (invertible) performs a rotation of the vector S = (S x, S y, S z ) of spin operators Q 1 S x Q = (RS) 1 Q 1 S y Q = (RS) 2 Q 1 S z Q = (RS) 3 for some SO(3) rotation matrix R. Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

16 Exclusion Process Local rotation SO(3) symmetry [Imparato, VL, van Wijland, PTPS ] Detailed transformation: Q = 1 + xs x iys y + zs z (on one site) (invertible) performs a rotation of the vector S = (S x, S y, S z ) of spin operators Q 1 S x Q = (RS) 1 Q 1 S y Q = (RS) 2 Q 1 S z Q = (RS) 3 for some SO(3) rotation matrix R. Form of the matrix: (Cayley form) R = (I + A)(I A) 1 A = 0 iz y iz 0 ix y ix 0 Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

17 Exclusion Process Local rotation Large deviations [Imparato, VL, van Wijland, PTPS ] Result: (transforming all sites) Q 1 W res (s; ρ 0, ρ 1 ; a 0, a 1 )Q = W res (s ; ρ 0, ρ 1; a 0, a 1 ) with primed variables ρ 0 = (1 + x)ρ 0 x z 1 x ρ 1 = (x + e s z(1 e s )) e s = x + e s + z(e s 1) 1 + xe s + z(e s 1) [ x + e s + z(1 e s ) ] ρ 1 x z 1 x 2 Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

18 Exclusion Process Local rotation Summary [Imparato, VL, van Wijland, PRE ] Symmetric exclusion process α β ρ 0 ρ 1 γ 1 L δ Local transformation α β ρ ρ 1 L γ System in equilibrium Prob stationn. non-eq (current) Prob stationn. eq (current ) δ Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

19 Exclusion Process Local rotation Probabilistic interpretation Measure ˆP(n, s, t) biased by e sq Mapping: ˆP(n,s, t; ρ 0, ρ 1 ; a 0, a 1 ) = n e tw(s;ρ 0,ρ 1 ;a 0,a 1 ) P init = n Q e tw(s ;ρ 0,ρ 1 ;a 0,a 1 ) Q 1 P }{{} init }{{} new projection new initial state condition Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

20 Exclusion Process Local rotation Probabilistic interpretation Measure ˆP(n, s, t) biased by e sq Mapping: Question: ˆP(n,s, t; ρ 0, ρ 1 ; a 0, a 1 ) = n e tw(s;ρ 0,ρ 1 ;a 0,a 1 ) P init = n Q e tw(s ;ρ 0,ρ 1 ;a 0,a 1 ) Q 1 P }{{} init }{{} new projection new initial state condition What is the mathematical embedding (in terms of process&prob.)? (Duality, Radon-Nykodym? caveat: prob. not preserved) Generalization: higher dimensions generic network and current more than two reservoirs see also: Derrida & Gerschenfeld (ω variable) Akkermans, Bodineau, Derrida & Shpielberg (1d LDF for d > 1) Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

21 Exclusion Process Periodic Boundary Conditions Example 2: exclusion process on a ring Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

22 Exclusion Process Periodic Boundary Conditions. Focus on a simple situation Simple exclusion process (SSEP): maximal occupation N = 1 Periodic boundary conditions Fixed total particle number N 0 density: ρ 0 = N 0 /L Ring geometry 2 1 L 0. Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

23 Exclusion Process Periodic Boundary Conditions Focus on a simple situation Simple exclusion process (SSEP): maximal occupation N = 1 Periodic boundary conditions Fixed total particle number N 0 density: ρ 0 = N 0 /L L 1 [ W s = e s( σ + k σ k+1 + ) ] σ k σ+ k+1 ˆnk (1 ˆn k+1 ) (1 ˆn k )ˆn k+1 k=1 = L 1 e s 2 2 H L 1 [ H = σ x k σk+1 x + σy k σy k+1 + σz k k+1] σz k=1 with = e s Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

24 Exclusion Process Periodic Boundary Conditions Classical/Quantum dictionary SSEP Quantum Spin Chain local occupation number n k (1 k L) local spin σk z (1 k L) n k = 0, 1, σk z = 1, 1, (fixed) total occupation N 0 ρ 0L (fixed) total magnetization M m 0L (mesoscopic) density ρ(x) (0 x 1) (mesoscopic) magnet. m(x) (0 x 1) Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

25 Exclusion Process Periodic Boundary Conditions Classical/Quantum dictionary SSEP Quantum Spin Chain local occupation number n k (1 k L) local spin σ z k (1 k L) n k = 0, 1, σ z k = 1, 1, (fixed) total occupation N 0 ρ 0L (fixed) total magnetization M m 0L (mesoscopic) density ρ(x) (0 x 1) (mesoscopic) magnet. m(x) (0 x 1) evolution operator ferromagnetic XXZ Hamiltonian (J xy = 1) W s = L 1 e s L H H = [ ( ] J xy σ x k σk+1 x + σ y k k+1) σy + Jzσkσ z k+1 z k=1 L 1 [ ] = σkσ x k+1 x + σ y k σy k+1 + σz kσk+1 z counting factor = e s of the activity K cumulant generating function k=1 anisotropy = J z along direction Z ground state energy ψ(s) = max Sp W s = L 1 2 e s 2 E L(s) E L (s) = min Sp H Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

26 Exclusion Process Microscopic solution Bethe Ansatz [Appert, Derrida, VL, van Wijland, PRE ] Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

27 Exclusion Process Microscopic solution Bethe Ansatz [Appert, Derrida, VL, van Wijland, PRE ] Coordinate Bethe Ansatz: Integrability known from long ; difficulty: L eigenvector of components eigenvalue N 0 [ ] xi A(P) ζp(i) P i=1 ψ(s) = 2N 0 + e s[ ζ ζ N0 ] e s Bethe equations N 0 ζi L = j=1 j i [ 1 ] 2es ζ i + ζ i ζ j 1 2e s ζ j + ζ i ζ j [ ] ζ 1 ζ N0 Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

28 Exclusion Process Microscopic solution Bethe Ansatz [Appert, Derrida, VL, van Wijland, PRE ] , : finite-size solution : infinite-size limit Repartition of Bethe roots in the complex plane Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

29 Exclusion Process Microscopic solution Finite-size effects [Appert, Derrida, VL, van Wijland, PRE ] large deviation function ψ(s) = 2Lρ 0 (1 ρ 0 )s + L 2 F(u) +... with u = L 2 ρ }{{}}{{} 0 (1 ρ 0 )s order 0 finite-size universal function (singular in u = π2 2 ) F(u) = k 2 ( 2u) k B 2k 2 Γ(k)Γ(k + 1) Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

30 Exclusion Process Microscopic solution Finite-size effects [Appert, Derrida, VL, van Wijland, PRE ] large deviation function ψ(s) = 2Lρ 0 (1 ρ 0 )s + L 2 F(u) +... with u = L 2 ρ }{{}}{{} 0 (1 ρ 0 )s order 0 finite-size universal function (singular in u = π2 2 ) u Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

31 Exclusion Process Microscopic solution Finite-size effects [Appert, Derrida, VL, van Wijland, PRE ] large deviation function ψ(s) = 2Lρ 0 (1 ρ 0 )s + L 2 F(u) +... with u = L 2 ρ }{{}}{{} 0 (1 ρ 0 )s order 0 finite-size universal function (singular in u = π2 2 ) non-analyticity dynamical phase transition at s c = π 2 2L 2 ρ 0 (1 ρ 0 ) Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33 u

32 Macroscopic approach Analogy with quantum mechanics Macroscopic limit a way to derive MFT A reminder: propagator in quantum mechanics final e ith initial Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

33 Macroscopic approach Analogy with quantum mechanics Macroscopic limit a way to derive MFT A reminder: propagator in quantum mechanics final e ith initial = = dz 1... dz n final e i th z n z n 1 e i th z n 2... DpDq exp{i 1 ħ S[p, q] } }{{} action... z 1 e i th initial p = p(x, t) and q = q(x, t) are generically space- & time-dependent fields. semi-classical limit recovered in the large 1 ħ limit [saddle-point] Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

34 Macroscopic approach Hydrodynamic limit Macroscopic limit [Tailleur, Kurchan, VL, JPA ] For exclusion processes Using SU(2) coherent states: ρ f e tw ρ i = ρ(t)=ρf DρDˆρ exp{l S[ˆρ, ρ] } ρ(0)=ρ i }{{} action Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

35 Macroscopic approach Hydrodynamic limit Macroscopic limit [Tailleur, Kurchan, VL, JPA ] For exclusion processes Using SU(2) coherent states: ρ f e tw ρ i = e sk ρ f e tws ρ i = ρ(t)=ρf DρDˆρ exp{l S[ˆρ, ρ] } ρ(0)=ρ i }{{} ρ(t)=ρf action DρDˆρ exp{l S s [ˆρ, ρ] } ρ(0)=ρ i }{{} Again: use saddle-point to handle the large L limit. action Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

36 Macroscopic approach Hydrodynamic limit Macroscopic limit [Tailleur, Kurchan, VL, JPA ] For exclusion processes Same S s [ˆρ, ρ] as the MSR action of the Langevin evolution: t ρ(x, t) = x [ x ρ(x, t) + ξ(x, t) ] ξ(x, t)ξ(x, t ) = 1 L ρ(x, t)( 1 ρ(x, t) ) δ(x x)δ(t t) One recovers the action of fluctuating hydrodynamics [Spohn; Bertini, De Sole, Gabrielli, Jona-Lasinio, Landim] Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

37 Macroscopic approach Large deviation function ψ(s): again [Appert, Derrida, VL, van Wijland, PRE ] Periodic boundary conditions More general fluctuating hydrodynamics 1 Q D(ρ) Lt 1 Lt Q2 c = σ(ρ) (Fourier s law) (For the SSEP, σ(ρ) = ρ(1 ρ)) Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

38 Macroscopic approach Large deviation function ψ(s): again [Appert, Derrida, VL, van Wijland, PRE ] Periodic boundary conditions More general fluctuating hydrodynamics 1 Q D(ρ) Lt 1 Lt Q2 c = σ(ρ) Saddle point evaluation e sk DρDˆρ exp{l S s [ˆρ, ρ]} (Fourier s law) (For the SSEP, σ(ρ) = ρ(1 ρ)) Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

39 Macroscopic approach Large deviation function ψ(s): again [Appert, Derrida, VL, van Wijland, PRE ] Periodic boundary conditions More general fluctuating hydrodynamics 1 Q D(ρ) Lt 1 Lt Q2 c = σ(ρ) Saddle point evaluation e sk DρDˆρ exp{l S s [ˆρ, ρ]} (Fourier s law) (For the SSEP, σ(ρ) = ρ(1 ρ)) Large deviation function [assuming uniform profile ρ(x) = ρ] ψ(s) = s K c t }{{} at saddle-point + L 2 DF(u) }{{} of quadratic fluctuations with u = L 2 s σ(ρ 0)σ (ρ 0 ) 8D 2 Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

40 Macroscopic approach Large deviation function Correspondence between the (Gaussian) integration of small fluctuations AND discreteness of Bethe root repartition. More general? Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

41 Macroscopic approach Large deviation function Correspondence between the (Gaussian) integration of small fluctuations AND discreteness of Bethe root repartition. More general? Repartition of Bethe roots for s > s c? Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

42 Macroscopic approach Large deviation function Correspondence between the (Gaussian) integration of small fluctuations AND discreteness of Bethe root repartition. More general? Repartition of Bethe roots for s > s c? Fluctuating hydrodynamics for quantum chains? Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

43 Beyond the critical point Scaling Dynamical phase transition [VL, Garrahan, van Wijland, JPA ] Rescaling of the large deviation function [singularity at λ c > 0 as L ] φ(λ) = lim L Lψ(λ/L2 ) Using the correct non-uniform saddle-point profile for λ > λ c λ c = π2 σ(ρ 0 ) (can be large!) uniform profile non-uniform profile Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

44 Beyond the critical point Scaling Dynamical phase transition [VL, Garrahan, van Wijland, JPA ] Rescaling of the large deviation function [singularity at λ c > 0 as L ] φ(λ) = lim L Lψ(λ/L2 ) Using the correct non-uniform saddle-point profile for λ > λ c λ c = π2 σ(ρ 0 ) (can be large!) uniform profile non-uniform profile see also: for LDF of Q [Bodineau, Derrida, PRE ] phase transition in WASEP for large dev. (non-stationary profile) [Jona-Lasinio et al.] generic criterion for instability Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

45 Beyond the critical point Scaling Dynamical phase transition [VL, Garrahan, van Wijland, JPA ] Optimal increasing Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

46 Beyond the critical point Scaling SSEP Quantum Spin Chain local occupation number n k (1 k L) local spin σk z (1 k L) n k = 0, 1, σk z = 1, 1, (fixed) total occupation N 0 ρ 0L (fixed) total magnetization M m 0L (mesoscopic) density ρ(x) (0 x 1) (mesoscopic) magnet. m(x) (0 x 1) evolution operator W s cumulant generating function ψ(s) ferromagnetic XXZ Hamiltonian H ground state energy E L (s) Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

47 Beyond the critical point Scaling SSEP Quantum Spin Chain local occupation number n k (1 k L) local spin σ z k (1 k L) n k = 0, 1, σ z k = 1, 1, (fixed) total occupation N 0 ρ 0L (fixed) total magnetization M m 0L (mesoscopic) density ρ(x) (0 x 1) (mesoscopic) magnet. m(x) (0 x 1) evolution operator W s cumulant generating function ψ(s) minimal activity phase (s + ) ferromagnetic XXZ Hamiltonian H ground state energy E L (s) Ising ferromagnetic order ( + ) maximal activity phase (s ) XY degenerate groundstate ( = 0) & (in fact, superp. of e iθ...) time t (steady state: t + ) dynamical partition function inverse temp. β (zero-temp. limit: β + ) partition function e sk Tr e tws Z XXZ β ( ) = Tr e β H Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

48 Beyond the critical point Scaling Sketch of derivation [VL, Garrahan, van Wijland, JPA ] Saddle-point equations for the profile ρ(x) take the form ( x ρ(x) ) 2 + EP ( ρ(x) ) = 0 Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

49 Beyond the critical point Scaling Sketch of derivation [VL, Garrahan, van Wijland, JPA ] Saddle-point equations for the profile ρ(x) take the form ( x ρ(x) ) 2 + EP ( ρ(x) ) = 0 Motion in time x of a particle of position ρ in a Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

50 Beyond the critical point Scaling Sketch of derivation [VL, Garrahan, van Wijland, JPA ] Saddle-point equations for the profile ρ(x) take the form ( x ρ(x) ) 2 + EP ( ρ(x) ) = 0 Motion in time x of a particle of position ρ in a oscillations depict the non-uniform profile ρ(x) Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

51 Beyond the critical point Scaling Excitations [Cheneau, VL, work in progress] What about solutions with more than one kink+anti-kink? φ(λ) 1.0 corresponding profiles ρ(x) 0 Λ c 4 c λ x Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

52 Beyond the critical point Smaller sizes Small sizes: the ground state Aim: experimental realizations with cold atoms non-periodic (but isolated, 1D) system smaller sizes & finite-temperature & excited state increasing L Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

53 Beyond the critical point Smaller sizes Small sizes: the full spectrum [preliminary!] L = 9 sites N 0 = 3 particles Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

54 Beyond the critical point Smaller sizes Small sizes: the full spectrum [preliminary!] L = 9 sites N 0 = 3 particles Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

55 Beyond the critical point Smaller sizes Small sizes: the full spectrum L = 9 sites N 0 = 3 particles [preliminary!] infinite-size ground state infinite-size excited states Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

56 Beyond the critical point Smaller sizes Small sizes: the full spectrum L = 9 sites N 0 = 3 particles [preliminary!] infinite-size ground state infinite-size excited states gathering(?) of microscopic eigenvalues macroscopic (L = ) states Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

57 Conclusion Summary Microscopic approach: operator formalism XXZ spin chain Bethe Ansatz Macroscopic approach: MFT, saddle-point method, dynamical phase transition Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

58 Conclusion Summary Microscopic approach: operator formalism XXZ spin chain Bethe Ansatz Macroscopic approach: MFT, saddle-point method, dynamical phase transition Questions: Finite-size crossover around a quantum phase transition? Between: Luttinger Liquid (s ) Phase-separated ferromagnet (s + ) Across the transition: continuum spectrum gaped spectrum? XXZ transition not at = 1 but at = 1 + O(L 2 ) Are scaling exponents/functions known? Are they interesting? Hydrodynamics approaches for quantum questions? Non-Hermitian operators dissipation in Lindblad? Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33

59 Conclusion Thank you for your attention! References: Marc Cheneau, Vivien Lecomte et al. work in progress (2014) Vivien Lecomte, Juan P. Garrahan, Frédéric van Wijland J. Phys. A (2012) Vivien Lecomte, Alberto Imparato, Frédéric van Wijland PTPS (2010) Cécile Appert-Rolland, Bernard Derrida, Vivien Lecomte, Frédéric van Wijland Phys. Rev. E (2008) Vivien Lecomte (LPMA Paris VI-VII) LDF symmetries and quantum mechanics Nice June 9th / 33