Quantum quenches in the thermodynamic limit

Quantum quenches in the thermodynamic limit Marcos Rigol Department of Physics The Pennsylvania State University Correlations, criticality, and coherence in quantum systems Evora, Portugal October 9, 204 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 / 34

Outline Introduction Quantum quenches Many-body quantum systems in thermal equilibrium 2 Numerical Linked Cluster Expansions Direct sums Resummations 3 Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-v -t -V chain (thermalization) Quenches in XXZ chain from a Neel state (QA vs GGE) 4 Conclusions Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 2 / 34

Quenches in one-dimensional superlattices Quantum dynamics in a D superlattice Trotzky et al. (Bloch s group), Nature Phys. 8, 325 (202). Initial state 000... 00 Unitary dynamics under the Bose-Hubbard Hamiltonian Experimental results ( ) vs exact t-dmrg calculations (lines) without free parameters local observables (top) vs nonlocal observables (bottom) Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 3 / 34

Relaxation dynamics after turning off a superlattice Density profile Momentum profile 0.24 2.5 τ=0 τ=0 n 0.8 0.2 0.06 n k 2.5 0.5 0 20 0 0 0 20 x/a 0 π π/2 0 π/2 π ka MR, A. Muramatsu, and M. Olshanii, PRA 74, 05366 (2006). MR, V. Dunjko, V. Yurovsky, and M. Olshanii, PRL 98, 050405 (2007). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 4 / 34

Unitary dynamics after a sudden quench If the initial state is not an eigenstate of Ĥ ψ 0 α where Ĥ α = E α α and E 0 = ψ 0 Ĥ ψ 0, then a few-body observable O will evolve following O(τ) ψ(τ) Ô ψ(τ) where ψ(τ) = e iĥτ/ ψ 0. What is it that we call thermalization? O(τ) = O(E 0 ) = O(T ) = O(T, µ). One can rewrite O(τ) = α,α C α C αe i(e α Eα )τ/ O α α where ψ 0 = α C α α. Taking the infinite time average (diagonal ensemble ˆρ DE α C α 2 α α ) τ O(τ) = lim dτ Ψ(τ τ τ ) Ô Ψ(τ ) = C α 2 O αα Ô DE, 0 α which depends on the initial conditions through C α = α ψ 0. Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 5 / 34

Description after relaxation (lattice models) Hard-core boson (spinless fermion) Hamiltonian L ) Ĥ = t (ˆb iˆb i+ + H.c. + V ˆn iˆn i+ t (ˆb ) iˆb i+2 + H.c. 0.6 0.5 i= Nonintegrable: t = V 0 Dynamics vs statistical ensembles n(k) 0.5 + V ˆn iˆn i+2 Integrable: t = V = 0 time average thermal GGE n(k) 0.4 0.25 0.3 initial state time average thermal 0.2 -π -π/2 0 π/2 π ka MR, PRL 03, 00403 (2009), PRA 80, 053607 (2009),... 0 -π -π/2 0 π/2 π ka MR, Dunjko, Yurovsky, and Olshanii, PRL 98, 050405 (2007),... Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 6 / 34

Eigenstate thermalization Eigenstate thermalization hypothesis [Deutsch, PRA 43 2046 (99); Srednicki, PRE 50, 888 (994).] b b in an The expectation value hα O αi of a few-body observable O eigenstate of the Hamiltonian αi, with energy Eα, of a many-body b at the mean energy Eα : system is equal to the thermal average of O b b ME (Eα ). hα O αi = hoi Nonintegrable Integrable (ρ GGE = P m λm Iˆm ).5 0 2 0-8 -6-4 -2 0.5 ρ(e) exact ρ(e) microcan. ρ(e) canonical - ρ(e) exact ρ(e) microcan. ρ(e) canonical 0.5 0 - ρ(e)[j ] n(kx=0) 2-0 e ZGGE ρ(e)[j ] n(kx=0) 3 0.5 0-8 -6 E[J] -4-2 0 E[J] MR, Dunjko, and Olshanii, Nature 452, 854 (2008). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 7 / 34

Time fluctuations and their scaling with system size δn k0. t =V =0.24 L=2 L=24 δn k0. t =V =0.2 0 δn k0. 0 δn k0. t =V =0.03 t =V =0 0 0 20 40 60 80 00 τ Bounds Relative differences (struct. factor) (G) P. Reimann, PRL 0, 90403 (2008). k δn(τ) = N(k, τ) N diag(k) (G) Linden et al., PRE 79, 0603 (2009). k N (N) Cramer et al., PRL 00, 030602 (2008). diag(k) (N) Venuti&Zanardi, PRE 87, 0206 (203). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 8 / 34

Time fluctuations Are they small because of dephasing? Ô(t) Ô(t) = α,α α α C α C αe i(e α Eα )t O α α α,α α α e i(e α Eα )t O α N α states N 2 states N states O typical α α Otypical α α Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 9 / 34

Time fluctuations Are they small because of dephasing? Ô(t) Ô(t) = α,α α α C α C αe i(e α Eα )t O α α α,α α α e i(e α Eα )t O α N α states N 2 states N states O typical α α Otypical α α Time average of Ô Ô = α C α 2 O αα α O αα Oαα typical N states One needs: O typical α α Otypical αα Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 9 / 34

Time fluctuations Are they small because of dephasing? Ô(t) Ô(t) = α,α α α C α C αe i(e α Eα )t O α α α,α α α e i(e α Eα )t O α N α states N 2 states N states O typical α α Otypical α α Time average of Ô Ô = α C α 2 O αα α O αα Oαα typical N states One needs: O typical α α Otypical αα MR, PRA 80, 053607 (2009) Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 9 / 34

Finite temperature properties of lattice models Computational techniques for arbitrary dimensions Quantum Monte Carlo simulations Polynomial time Large systems Finite size scaling Sign problem Limited classes of models Exact diagonalization Exponential problem Small systems Finite size effects No systematic extrapolation to larger system sizes Can be used for any model! High temperature expansions Exponential problem High temperatures Thermodynamic limit Extrapolations to low T Can be used for any model! Can fail (at low T ) even when correlations are short ranged! Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 / 34

Finite temperature properties of lattice models Computational techniques for arbitrary dimensions Quantum Monte Carlo simulations Polynomial time Large systems Finite size scaling Sign problem Limited classes of models Exact diagonalization Exponential problem Small systems Finite size effects No systematic extrapolation to larger system sizes Can be used for any model! High temperature expansions Exponential problem High temperatures Thermodynamic limit Extrapolations to low T Can be used for any model! Can fail (at low T ) even when correlations are short ranged! DMFT, DCA, DMRG,... Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 / 34

Linked-Cluster Expansions Extensive observables Ô per lattice site (O) in the thermodynamic limit O = c L(c) W O (c) where L(c) is the number of embeddings of cluster c Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 2 / 34

Linked-Cluster Expansions Extensive observables Ô per lattice site (O) in the thermodynamic limit O = c L(c) W O (c) where L(c) is the number of embeddings of cluster c and W O (c) is the weight of observable O in cluster c O(c) is the result for O in cluster c W O (c) = O(c) s c W O (s). O(c) = Tr ˆρ GC c = Zc GC Z GC c = Tr } {Ô ˆρ GC c, and the s sum runs over all subclusters of c. exp (Ĥc µ ˆN c)/k B T {exp } (Ĥc µ ˆN c)/k B T Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 2 / 34

Linked-Cluster Expansions In HTEs O(c) is expanded in powers of β and only a finite number of terms is retained. Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 3 / 34

Linked-Cluster Expansions In HTEs O(c) is expanded in powers of β and only a finite number of terms is retained. In numerical linked cluster expansions (NLCEs) an exact diagonalization of the cluster is used to calculate O(c) at any temperature. (Spins models in the square, triangular, and kagome lattices) MR, T. Bryant, and R. R. P. Singh, PRL 97, 87202 (2006). MR, T. Bryant, and R. R. P. Singh, PRE 75, 068 (2007). (t-j model in the square lattice) MR, T. Bryant, and R. R. P. Singh, PRE 75, 069 (2007). PRL 98, 207204 (2007), PRB 76, 84403 (2007), PRB 83, 3443 (20), PRA 84, 0536 (20), PRB 84, 2244 (20), PRB 85, 06440 (202), PRA 86, 023633 (202), PRL 09, 20530 (202), CPC 84, 557 (203), PRB 88, 2527 (203),... Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 3 / 34

Numerical Linked Cluster Expansions i) Find all clusters that can be embedded on the lattice Bond clusters c L(c) 2 2 3 2 4 4 5 4 6 2 7 4 8 4 9 8 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 5 / 34

Numerical Linked Cluster Expansions i) Find all clusters that can be embedded on the lattice ii) Group the ones with the same Hamiltonian (Topological cluster) No. of bonds topological clusters 0 2 3 2 4 4 5 6 6 4 7 28 8 68 9 56 0 399 02 2 2732 3 7385 4 20665 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 5 / 34

Numerical Linked Cluster Expansions i) Find all clusters that can be embedded on the lattice ii) Group the ones with the same Hamiltonian (Topological cluster) iii) Find all subclusters of a given topological cluster No. of bonds topological clusters 0 2 3 2 4 4 5 6 6 4 7 28 8 68 9 56 0 399 02 2 2732 3 7385 4 20665 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 5 / 34

Numerical Linked Cluster Expansions i) Find all clusters that can be embedded on the lattice ii) Group the ones with the same Hamiltonian (Topological cluster) iii) Find all subclusters of a given topological cluster iv) Diagonalize the topological clusters and compute the observables No. of bonds topological clusters 0 2 3 2 4 4 5 6 6 4 7 28 8 68 9 56 0 399 02 2 2732 3 7385 4 20665 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 5 / 34

Numerical Linked Cluster Expansions i) Find all clusters that can be embedded on the lattice ii) Group the ones with the same Hamiltonian (Topological cluster) iii) Find all subclusters of a given topological cluster iv) Diagonalize the topological clusters and compute the observables v) Compute the weight of each cluster and compute the direct sum of the weights E 0-0.2-0.4-0.6 Heisenberg Model QMC 00 00 2 bonds 3 bonds -0.8 0. 0 MR et al., PRE 75, 068 (2007). B. Tang et al., CPC 84, 557 (203). T Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 5 / 34

Numerical Linked-Cluster Expansions Square clusters c L(c) 2 3 4 /2 No. of squares topological clusters 0 2 3 2 4 5 5 5 2 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 6 / 34

Numerical Linked-Cluster Expansions Square clusters c L(c) 2 /2 0-0.2 Heisenberg Model QMC 00 00 2 bonds 3 bonds 3 4 5 2 E -0.4-0.6-0.8 0. 0 T MR et al., PRE 75, 068 (2007). 4 sites 5 sites 4 squares 5 squares B. Tang et al., CPC 84, 557 (203). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 6 / 34

Resummation algorithms We can define partial sums n O n = S i, with S i = L(c i ) W O (c i ) c i i= where all clusters c i share a given characteristic (no. of bonds, sites, etc). Goal: Estimate O = lim n O n from a sequence {O n }, with n =,..., N. Wynn s algorithm: where ε (k ) n ε ( ) n = 0, ε (0) n = O n, ε (k) n = ε (k 2) n+ + = ε (k ) n+ ε (k ) n. Brezinski s algorithm [θ ( ) n = 0, θ (0) n = O n ]: θ (2k+) n where 2 θ (k) n = θ (2k ) n + θ n (2k) = θ (k) n+2 2θ(k) n+ + θ(k) n., θ (2k+2) n = θ (2k) ε (k ) n n+ + θ(2k) n+ θ(2k+) n+ 2 θ (2k+) n Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 8 / 34

Resummation results (Heisenberg model) E 0-0.2-0.4 Energy (square lattice) QMC 00 00 Wynn 6 Euler ED 6 C v 0.6 0.4 C v (square lattice) QMC Wynn 6 Euler ED 6 ED 9-0.6 0.2-0.8 0. 0 T 0 0. 0 T MR, T. Bryant, and R. R. P. Singh, PRE 75, 068 (2007). B. Tang, E. Khatami, and MR, Comput. Phys. Commun. 84, 557 (203). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 9 / 34

Diagonal ensemble and NLCEs The initial state is in thermal equilibrium in contact with a reservoir ˆρ I a c = e (Ec a µ I N c a )/T I a c a c Zc I, where Zc I = e (Ec a µi N c a )/T I, a a c (Ea) c are the eigenstates (eigenvalues) of the initial Hamiltonian ĤI c in c. MR, PRL 2, 7060 (204). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 2 / 34

Diagonal ensemble and NLCEs The initial state is in thermal equilibrium in contact with a reservoir ˆρ I a c = e (Ec a µ I N c a )/T I a c a c Zc I, where Zc I = e (Ec a µi N c a )/T I, a a c (Ea) c are the eigenstates (eigenvalues) of the initial Hamiltonian ĤI c in c. At the time of the quench ĤI c Ĥc, the system is detached from the reservoir. Writing the eigenstates of ĤI c in terms of the eigenstates of Ĥc where ˆρ DE c lim τ τ τ 0 dτ ˆρ(τ) = α W c α α c α c a Wα c = e (Ec a µ I N c a )/T I α c a c 2, α c (ε c α) are the eigenstates (eigenvalues) of the final Hamiltonian Ĥc in c. Z I c MR, PRL 2, 7060 (204). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 2 / 34

Diagonal ensemble and NLCEs The initial state is in thermal equilibrium in contact with a reservoir ˆρ I a c = e (Ec a µ I N c a )/T I a c a c Zc I, where Zc I = e (Ec a µi N c a )/T I, a a c (Ea) c are the eigenstates (eigenvalues) of the initial Hamiltonian ĤI c in c. At the time of the quench ĤI c Ĥc, the system is detached from the reservoir. Writing the eigenstates of ĤI c in terms of the eigenstates of Ĥc where ˆρ DE c lim τ τ τ 0 dτ ˆρ(τ) = α W c α α c α c a Wα c = e (Ec a µ I N c a )/T I α c a c 2, α c (ε c α) are the eigenstates (eigenvalues) of the final Hamiltonian Ĥc in c. Using ˆρ DE c in the calculation of O(c), NLCEs allow one to compute observables in the DE in the thermodynamic limit. MR, PRL 2, 7060 (204). Z I c Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 2 / 34

Models and quenches Hard-core bosons in D lattices at half filling (µ I = 0) L ) Ĥ = t (ˆb iˆb i+ + H.c. + V ˆn iˆn i+ t (ˆb ) iˆb i+2 + H.c. i= + V ˆn iˆn i+2 Quench: T I, t I = 0.5, V I =.5, t I = V I = 0 t = V =.0, t = V NLCE with maximally connected clusters (l = 8 sites) Energy: E DE = Tr[Ĥ ˆρDE ] Convergence: (O ens ) l = Oens l O ens 8 O8 ens (E DE ) 7 0-2 0-4 0-6 0-8 0-0 0-2 Convergence of E DE with l (E DE ) l 0-2 0-4 0-6 0-8 0-0 0-2 0-4 T I = T I =5 2 7 2 7 l 2 7 2 7 l t =V =0 t =V =0. t =V =0.25 t =V =0.5 0-4 0. 0 00 T I Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 23 / 34

Few-body experimental observables in the DE Momentum distribution ˆm k = e ik(j j ) ˆρ jj L jj 0.7 0.6 (m k ) 8 0.5 0.4 Initial t =V =0, DE t =V =0, GE t =V =0.5, DE t =V =0.5, GE 0 π/4 π/2 3π/4 π k Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 24 / 34

Few-body experimental observables in the DE Momentum distribution ˆm k = e ik(j j ) ˆρ jj L jj Differences between DE and GE k δ(m) l = (m k) DE l (m k ) GE 8 k (m k) GE 8 (m k ) 8 0.7 0.6 0.5 0.4 (m) 7 0-0 -3 0-5 0-0 -3 DE GE 0. 0 00 T I Initial t =V =0, DE t =V =0, GE t =V =0.5, DE t =V =0.5, GE 0 π/4 π/2 3π/4 π k δ(m) l 0-2 0-3 0-3 0-4 t =V =0 t =V =0.025 t =V =0. t =V =0.5 T I =2 T I =0 0 2 3 4 5 6 7 8 9 l Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 24 / 34

Failure of the GGE based on local quantities XXZ (integrable) Hamiltonian ( ) Ĥ = J σi x σi+ x + σ y i σy i+ + σz i σi+ z i Quench starting from the Neel state to different values of Quench action solution differs from GGE based on local quantities: B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, MR, and J.-S. Caux, PRL 3, 7202 (204). B. Pozsgay, M. Mestyán, M. A. Werner, M. Kormos, G. Zaránd, and G. Takács, PRL 3, 7203 (204). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 26 / 34

Failure of the GGE based on local quantities XXZ (integrable) Hamiltonian ( ) Ĥ = J σi x σi+ x + σ y i σy i+ + σz i σi+ z i Quench starting from the Neel state to different values of Quench action solution differs from GGE based on local quantities: B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, MR, and J.-S. Caux, PRL 3, 7202 (204). B. Pozsgay, M. Mestyán, M. A. Werner, M. Kormos, G. Zaránd, and G. Takács, PRL 3, 7203 (204). Can we use NLCEs for ground states (pure states)? Using the parity symmetry of the clusters: a e c = 2 (...... +...... ) a o c = 2 (............ ) W c,e/o α = αc e/o a e/o c 2 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 26 / 34

Results for spin-spin correlations <σ z σ2 z > <σ z σ2 z > -0.25-0.35-0.45-0.86-0.88-0.9-0.958-0.96-0.962 <σ z σ2 z >-0.956-0.98 DE GE QA Resum -0.98 0 2 3 4 5 6 7 8 l <σ z σ2 z >-0.979 = =4 =7 =0 MR, PRE 90, 0330(R) (204) (a) (b) (c) (d) <σ z σ3 z > <σ z σ3 > <σ z σ3 z > <σ z σ3 z > 0.2 0.5 0. 0.05 0.8 0.77 0.73 0.94 0.92 0.9 0.97 0.96 (e) (f) (g) (h) 0.95 0 2 3 4 5 6 7 8 l Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 27 / 34

Quench action, GGE, and NLCE (a) 0.8 0.6 0.4 0.2 0.02 0 σ z σz 3-0.09 2 0 0. 3 5 7 9.0.02.03 δ σ z σz 3 B. Wouters et al., PRL 3, 7202 (204). 0.3 0.2 σ z σz 3 (b) σ z σz 3 sp σ z σz 3 GGE σ z σz 3 NLCE δ σ z σ z 3 = σ z σ z 3 GGE / σ z σ z 3 QA Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 28 / 34

Conclusions NLCEs provide a general framework to study the diagonal ensemble in lattice systems after a quantum quench in the thermodynamic limit. NLCE results suggest that few-body observables thermalize in nonintegrable systems while they do not thermalize in integrable systems. As one approaches the integrable point DE-NLCEs behave as NLCEs for equilibrium systems approaching a phase transition. This suggests that a transition to thermalization may occur as soon as one breaks integrability. The GGE based on known local conserved quantities does not describe observables after relaxation, while the QA does, as suggested by the NLCE results. New things to be learned about integrable systems! Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 29 / 34

Collaborators Michael Brockmann (ITP Amsterdam) Jean-Sébastien Caux (ITP Amsterdam) Jacopo De Nardis (ITP Amsterdam) Davide Fioretto (ITP Amsterdam) Bram Wouters (ITP Amsterdam) PRL 3, 7202 (204). Supported by: Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 30 / 34

Finite temperature properties of lattice models Computational techniques for arbitrary dimensions Quantum Monte Carlo simulations Polynomial time Large systems Finite size scaling Sign problem Limited classes of models DQMC of a 2D system with: U = 6t, V = 0.04t, T = 0.3t and 560 fermions Density -5-0 -5 0 5 0 5 Density fluctuations.2 0.9 0.6 0.3 0.0-5 -0-5 0 5 0 5 0-2.7.8 0.9 0.0 Spin correlations -5-0 -5 0 5 0 5 Pairing correlations 3.2 2.4.6 0.8 0.0-5 -0-5 0 5 0 5 0-2.4.6 0.8 0.0 S. Chiesa, C. N. Varney, MR, and R. T. Scalettar, PRL 06, 03530 (20). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 3 / 34

Dispersion of the energy in the DE Dispersion of the energy E 2 = L ( Ĥ2 Ĥ 2 ) 8 E 2 0.9 0.8 0.7 0.6 DE GE δ( E 2 ) l Deviations from the GE δ(o) l = ODE l 0. 0.05 T I = O GE 8 O8 GE 0 t =V =0 2 t =V =0.5 6 0 l 4 8 t =V =0.5, diff. init. state 0.5 0 00 T The dispersion of the energy (and of the particle number) in the DE depends on the initial state independently of whether the system is integrable or not. Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 32 / 34

Few-body experimental observables in the DE 0.03 nn kinetic energy K = t i ˆb iˆb i+ Differences between DE and GE δ(k) l = KDE l K GE 8 K GE 8 δ(k) 8 0.025 0.02 0.05 0.0 0.005 t =V =0 t =V =0.025 t =V =0. t =V =0.5 0 0 00 T δ(k) l 0-2 0-3 0-4 0-0 -3 t =V =0 t =V =0.025 t =V =0. t =V =0.5 T I =2 T I =0 0 2 3 4 5 6 7 8 9 l Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 33 / 34

NLCEs vs exact diagonalization (equilibrium, t = 0) (E GE ) l (K GE ) l 0 2 0-2 0-4 0-6 0-8 0-0 0-2 0-4 0 2 0-2 0-4 0-6 0-8 0-0 0-2 V =0 V = T= T=5 δ(e ED ) L δ(k ED ) L 0.3 0. 0.03 8 20 22 L 0.07 0.04 0.02 8 20 22 L ( E 2 GE ) l 0 2 0-2 0-4 0-6 0-8 0-0 0-2 0-4 0 2 0-4 5 9 3 7 0-4 5 9 3 7 l l ED from L. Santos and MR, PRE 82, 0330 (200). (m GE ) l 0-2 0-4 0-6 0-8 0-0 0-2 δ(m ED ) L 0.02 0.0 0.005 8 20 22 L Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 34 / 34