Out-of-equilibrium Correlated Systems : Bipartite Entanglement as a Probe of Thermalization
OOut-of-equilibrium Correlated Systems : BipartiteEntanglement as a Probe of Thermalization
Didier Poilblanc
Laboratoire de Physique Th´eorique UMR5152, CNRS and Universit´e de Toulouse,F-31062 France
Abstract.
Thermalization play a central role in out-of-equilibrium physics ofultracold atoms or electronic transport phenomena. On the other hand, entanglementconcepts have proven to be extremely useful to investigate quantum phases of matter.Here, it is argued that bipartite entanglement measures provide key information onout-of-equilibrium states and might therefore offer stringent thermalization criteria.This is illustrated by considering a global quench in an (extended) XXZ spin-1/2chain across its (zero-temperature) quantum critical point. A non-local bipartition of the chain preserving translation symmetry is proposed. The time-evolution afterthe quench of the reduced density matrix of the half-system is computed and itsassociated (time-dependent) entanglement spectrum is analyzed. Generically, thecorresponding entanglement entropy quickly reaches a ”plateau” after a short transientregime. However, in the case of the integrable XXZ chain, the low-energy entanglementspectrum still reveals strong time-fluctuations. In addition, its infinite-time averageshows strong deviations from the spectrum of a Boltzmann thermal density matrix. Incontrast, when the integrability of the model is broken (by small next-nearest neighborcouplings), the entanglement spectra of the time-average and thermal density matricesbecome remarkably similar.PACS numbers: 75.10.Jm,05.30.-d,05.30.Rt a r X i v : . [ c ond - m a t . s t r- e l ] A p r ut-of-equilibrium Correlated Systems : Bipartite Entanglement as a Probe of Thermalization
1. Introduction
Rapid progress in the field of ultracold atoms [1] offer brend new perspectives torealize controlled experimental setup to investigate out-of-equilibrium physics. Real-time observation of quantum dynamics of isolated systems have become possible [2].In addition, ultracold atoms loaded on optical lattices [3, 1] or laser-cooled Coulombcrystals of charged ions [4] offer very clean experimental implementation of simple latticemany-body Hamiltonians and provide simulators for Condensed Matter. The ability todynamically change parameters [5] in these Hamiltonians on short time scales could beexploited to realize quantum information processing [1] or cooling [6] devices.In electronic condensed matter systems, relaxation towards steady states play acentral role in many transport phenomena, like e.g. in electric transport resulting fromthe application of a sudden voltage bias at the edges of a quantum dot [7] or of aHubbard chain [8]. Spin chains also offers simple generic systems to investigate out-of-equilibrium physics as e.g. heat transport [9]. However, conceptually, thermalization [10]of non-equilibrium isolated quantum many-body systems after e.g. a sudden change ofHamiltonian parameters (quantum quench) is still poorly understood, despite recentwork on correlated bosons [11, 12] in one-dimension (1D). Generally, whether somelocal observables approach steady values and whether their time average equal thecorresponding thermal average are often used as criteria of thermalization. However,the fact that thermalization occur for certain local observables (according to the abovecriteria) does not at all guarantee that other observables will also meet the criteria.Independently, quantum information concepts have been applied with great successto several domains of Condensed Matter [13], giving new type of physical insightson exotic quantum phases. Quantum entanglement of a A/B bipartition of a many-body (isolated) quantum system can be characterized by the groundstate (GS) reduceddensity matrix (RDM) ρ A obtained from tracing out the B part. The correspondingentanglement Von Neumann (VN) entropy S VN = − Tr { ρ A ln ρ A } offers an extraordinarytool [14], e.g. to identify underlying conformal field theory (CFT) structure in one-dimensional systems. Another central quantity is the entanglement spectrum (ES)defined by the (positive) eigenvalues of a (dimensionless) pseudo-Hamiltonian H definedby the implicit relation ρ A = exp ( −H ). Remarkably, ES faithfully reflect CFTstructures [15], topological symmetries [16] or properties of edge states in fractionalquantum Hall states [17] or low-dimensional quantum magnets [18].So far, time evolution of entanglement has been investigated only in very simplecases e.g. for a small segment in a 1D system after global or local quenches [20, 13, 14].However, the full potential of entanglement measures has not been fully exploited toinvestigate thermalization of many-body systems. Because for a given bipartition (i.e.into two halves) of the whole system all the information of the GS is contained inits Schmitt decomposition and the ES is a (convenient) way of arranging the Schmittcoefficients, the ES contains the whole information of the state. The main goal here istherefore to use the bipartite ES to take the place of local observables to investigate ut-of-equilibrium Correlated Systems : Bipartite Entanglement as a Probe of Thermalization non-local observables is addressed at once, (ii) finitesize scaling can be performed and (iii) reduced density matrices benefit from numerousconserved quantities (total momenta and total spin) allowing for a direct comparison ofthe ES separately in each symmetry sector and hence an ultimate comparison betweentime-averaged and Boltzman thermal density matrices. Note that non-local real-spacepartitions have also been used e.g. to define non-local order in gapless spin chains [19].Practically, a genuine correlated anisotropic 1D spin chain and the time evolutionof its reduced density matrix, entanglement entropy and ES after a global quench areconsidered using exact (Lanczos and full) diagonalization techniques. For the integrablecase we have considered, the reduced density matrix exhibits strong time-fluctuationsand its infinite-time average significantly deviates from a thermal density matrix, despitethe fact that the entanglement entropy reaches a well-defined entropy plateau. Incontrast, thermalization, as defined by the above criteria, seems to be possible whenintegrability is broken by adding (small) extra terms to the Hamiltonian.
2. Model and setup
Let us now consider the 1D anisotropic spin-1/2 Heisenberg (so-called XXZ) model(Fig. 1), H = J z (cid:88) i S zi S zi +1 + 12 J xy ( S + i S − i +1 + S − i S + i +1 ) , (1)whose (1D) parameter space can be mapped on a (half) unit circle assuming J xy = cos θ and J z = sin θ . Alternatively the system can be viewed as a (half-filled) hard-coreboson chain with hopping t = J xy / V = J z . Itsremarkable phase diagram obtained by Haldane [21] and shown in Fig. 1(b) exhibits aQuantum Critical Point (QCP) located exactly at the SU(2)-symmetric point θ = π/ | V | > t ( θ < − π/
4) will not be of interesthere. Ultimately, we shall consider adding next NN hopping t (cid:48) and repulsion V (cid:48) (seeFig. 1(a)) in order to break integrability.The key to construct extensive quantities is to realize a non-local partition of thechain into ”even” and ”odd” sites. In other words, if the chain is drawn in a zig-zagfashion as in Fig. 1(a), the A and B parts form the two edges of the system which becomeexplicit for t (cid:48) (cid:54) = 0 and V (cid:48) (cid:54) = 0. One consider here finite chains of N (= 16, 20 and 24)sites with periodic boundary conditions, as shown in Fig. 2. The groundstate RDM ofthe subsystems ρ A = ρ B can be computed using translation symmetry of each L = N/ ut-of-equilibrium Correlated Systems : Bipartite Entanglement as a Probe of Thermalization Figure 1. (Color online) (a) A N-site (extended) XXZ spin-1/2 chain (drawn here asa ”zig-zag” on a periodic ribbon) is partitioned into two identical A and B subsystemsof L = N/ J xy = cos θ and J z = sin θ . (c) The CDW groundstate for θ = π/ t = 0 ( J xy = 0) prepared beforethe quench : all bosons (up spins) are located on A and the B sites are empty (downspins). site subsystem [18] (each symmetry class is labelled by a momentum K = 2 πn/L ) andthe conservation of the Z-component of the total spin (i.e. the number of bosons), S ZA + S ZB = 0. Interestingly, liquid and insulating bosonic phases are characterized byqualitatively different entanglement properties. First, the entanglement entropy perunit length , shown in Fig. 2(a) for a N = 24 site chain, shows a (kink-like) maximum atthe SU(2)-symmetric QCP and vanishes (in the thermodynamic limit) for the classicalCDW (Ising) configuration obtained when θ → π/
2. Indeed, as shown in Fig. 1(c), for θ = π/ /L finite size entropyas shown in Fig. 2(a). Secondly, each quantum phase is uniquely characterized by itsES defined as the spectrum of H = − ln ρ A : Fig. 2(b-c) (Fig. 2(d-e) ) for parameters inthe CDW phase (LL phase) shows very distinctive features and a clear gapped (lineargapless) spectrum. In addition, at the QCP (Fig. 2(d)) one observes a SU(2) multipletstructure. ut-of-equilibrium Correlated Systems : Bipartite Entanglement as a Probe of Thermalization - p/2 - p/3 - p/6 p/6 p/3 p/2q S V N ( q ) / S m a x CDW Insulator (Ising AF)Phase Separation (Ising FM) (a)
XXZ chain of N=24 sites Luttinger Liquid (XY phase)
SU(2) (b)(c)(d)(e) - ! /2 ! /2 ! momentum K
024 024 S AZ =0|S AZ |=1|S AZ |=2|S AZ |=3|S AZ |=4|S ZA |=5|S AZ |=6 - ! /2 ! /2 ! momentum K " $ " (d) % = ! /4 (c) % = ! /3 (b) % =2 ! /5 (e) % = ! /6 " $ " " $ " " $ " Figure 2. (Color online) GS properties of the XXZ chain; (a) VN entanglemententropy vs θ computed on a N=24 site ring. The entropy is normalized by the maximumvalue S max = L ln 2 ( L = N/ excitation spectra (for the4 values of θ shown in (b)) as a function of momentum K along the ribbon. Theeigenvalues ξ α of H = − ln ρ A are labelled according to the Z-component of the totalspin (i.e. the number of bosons N A = L/ − S ZA ) of the A subsystem (legend of symbolson graph). Note the CDW (LL ), doubly-degenerate (unique) GS of H (of energy ξ )carries N A = 0 or N A = L ( N A = L/
2) hardcore bosons. ut-of-equilibrium Correlated Systems : Bipartite Entanglement as a Probe of Thermalization t F ( t ) = | < f ( ) | f ( t ) > | ; S V N ( t ) / S m a x t t (a) q: p/2 -> 2p/5 perturbative regime non-perturbative regime (b) q: p/2 -> p/3 (c) q: p/2 -> p/6 Figure 3. (Color online) (a-c) Squared-fidelity (shaded) and VN entanglement entropyof various non-equilibrium states obtained after different sudden quenches of the N = 24 sites chain Hamiltonian, θ init → θ f as shown on plots ( t (cid:48) = V (cid:48) = 0). Thecontinuous lines (dashed red line) correspond to a symmetric (non-symmetric) initialstate (see text). The dotted (green) line is the GS entropy for θ = θ f .
3. Time evolution and bipartite entanglement entropy
Let us now consider a sudden quantum quench of the system at time τ = 0 (quasi-adiabatic quenches will be treated later on). For simplicity, the initial state | φ (0) (cid:105) ischosen either (i) as one of the two (zero entropy) degenerate GS at θ = θ init = π/ τ > t (spin-flip term in spin language) is switched on, i.e. the value of θ discontinuously jumps at τ = 0 to its ”final” value θ f (see Fig. 1(b)).The time evolution of the system wavefunction, | φ ( τ ) (cid:105) = exp ( − iτ H ( θ f )) | φ (0) (cid:105) (2)is easily computed by (arbitrary) time steps of δτ (from 0 .
01 to 0 .
8) with arbitrary goodprecision [22] (typically better than 10 − ) by Taylor expanding the time evolution ut-of-equilibrium Correlated Systems : Bipartite Entanglement as a Probe of Thermalization − iδτ H ( θ f )). Hence, for finite size systems under consideration, an exact computation of the time-dependent RDM, ρ A ( τ ) = Tr B | φ ( τ ) (cid:105)(cid:104) φ ( τ ) | (3)can be done. Results for the squared-fidelity F ( τ ) = |(cid:104) φ (0) | φ ( τ ) (cid:105)| and the entanglemententropy, S VN ( τ ) = − Tr { ρ A ( τ ) ln ρ A ( τ ) } (4)are shown in Fig. 3 for increasingly ”strong” quenches corresponding to θ f = 2 π/ π/ π/
6. After a very short transient regime the squared-fidelity drops sharply andthe entanglement entropy raises to a more or less well defined plateau, whose averagevalue exceeds the value of the GS of the final Hamiltonian (f-GS). In the N → ∞ limit,one expects a non-perturbative regime [23] where F ( τ ) ∼ − N →
0. However, on finitesystem and for small quench, F ( τ ) can remain large, as seen e.g. in Fig. 3(a). In thatregime, integrability of the model can play an important role [24]. Therefore, fromnow on, one will assume a sufficiently large quench, let say θ f = π/
6, to observe timeevolutions on finite clusters generic of the thermodynamic limit . This corresponds infact to a quench ”across” the QCP at θ = π/ O of the A subsystem is given by˜ O ( τ ) = Tr( ρ A ( τ ) O ). Time-average like ¯ O = (cid:104) ˜ O (cid:105) , where (cid:104) G (cid:105) ≡ lim T →∞ T (cid:82) T + G ( τ ) dτ ,can then be rewritten as Tr( ρ ave A O ), involving the time-averaged RDM ρ ave A = (cid:104) ρ A (cid:105) . Next,time-averages will be performed in a ∆ τ = 40 time interval using a mesh of 50 points(excluding the initial short transient regime).Bipartite entanglement can provide precise characterization of the system after thequench, in the thermodynamic limit, e.g. showing whether or not it reaches a (quasi-)steady state. One finds that the time fluctuations of S VN ( τ ) shown in Fig. 4(a) vanish inthe thermodynamic limit as revealed by the finite size scaling of Fig. 4(c). Interestingly,as seen in Fig. 4(a), we notice that the VN entropy of the average density matrix ρ ave A , S aveVN = − Tr { ρ ave A ln ρ ave A } , differs from (cid:104) S VN (cid:105) , the time-average of the entanglemententropy. A proper finite size scaling of the two quantities shown in Fig. 4(b) proves thatthis fundamental property holds in the thermodynamic limit.Incidentally, it is also important to distinguish between ”quantum” fluctuations( O − ¯ O ) = Tr { ρ ave A ( O − ¯ O ) } and ”time” fluctuations (cid:104) ( ˜ O − ¯ O ) (cid:105) . An extreme exampleis the case of S ZA where ˜ S ZA ( τ ) = 0 at all times (from the time-conserved A ↔ B symmetry) while quantum fluctuations are finite as shown in Fig. 4(c). ut-of-equilibrium Correlated Systems : Bipartite Entanglement as a Probe of Thermalization t S V N ( t ) / S m a x R e l a t i v e f l u c t ua t i on s N242016242016
S[< r A >] (a) (b) (c)S AZ S VN (x 20) Figure 4. (Color online) (a) Time-dependent VN entanglement entropy after a sudden θ init = π/ → θ f = π/ t (cid:48) = V (cid:48) = 0), for different chain lengths N as shownon plots. Dashed-dotted (dashed) lines corresponds to (cid:10) S VN (cid:11) ( S aveVN ). (b) Finite-size scaling of the averages shown in (a). (c) Finite-size scaling of the relative timefluctuations of the entanglement entropy (red squares) and quantum fluctuations of S ZA (normalized by L/ τ = 40 time intervalusing a mesh of 50 points. The data of b) and c) are well fitted assuming ∼ − αL finitesize corrections.
4. Entanglement spectra: time evolution within the ”entropy plateau”
In order to understand the physical origin of the difference between (cid:104) S VN (cid:105) and S aveVN it isinteresting to inspect more closely the behavior of ρ A ( τ ) within the ”entropy plateau”regime as a function of time τ . For this purpose, it is convenient to use the (bipartite) time-dependent ES defined by the (positive) eigenvalues of the (dimensionless) pseudo-Hamiltonian H ( τ ) defined as, ρ A ( τ ) = exp ( −H ( τ )) . (5)As for the equilibrium GS, the spectrum of H is computed separately in every sector ofthe momentum K = πL/ (using translation invariance of the A and B half-systems) andof the z-component of the total spin. Fig. 5 shows ”typical” entanglement spectra taken ut-of-equilibrium Correlated Systems : Bipartite Entanglement as a Probe of Thermalization - ! /2 ! /2 ! momentum K
012 012 - ! /2 ! /2 ! momentum K - ! /2 ! /2 ! momentum K " $ " (b) % =3.2 (a) % =0.8 (c) % =16 (e) % =32 " $ " " $ " " $ " (f) % =40 (d) % =24 Figure 5. (Color online) ”Snapshots” at different times τ of the ES of ρ A . (a) lieswithin the transcient regime and (b-f) lie within the ”entropy plateau”. Symbols aresimilar to Fig. 2(b-e). at different times. At short times, some memory of the initial state is clearly visible asin Fig. 5(a). At longer times, let’s say τ > .
5, it is remarkable to find very differentspectra at different times although all such spectra lead roughly to a very comparablevalue of the entanglement entropy. In other word, ρ A ( τ ) fluctuates strongly in timeon a constant entropy ”hyper-surface” of the space of density matrices with Tr ρ = 1.This naturally explains why (cid:104) S VN (cid:105) and S aveVN differ substantially. Our finite size scalingsuggests that this is a fundamental phenomenon and not a finite size effect.
5. Entanglement spectra: comparison between infinite-time average andthermal ensembles
The RDM ρ A contains all relevant information about the subsystem, much beyondany local observable. As argued in the introduction, its associated ES is thereforean ultimate observable to investigate thermalization. Furthermore, the existence of conserved quantities such as momentum and particle number makes the comparison ut-of-equilibrium Correlated Systems : Bipartite Entanglement as a Probe of Thermalization - ! /2 ! /2 ! momentum K - ! /2 ! /2 ! momentum K - ! /2 ! /2 ! momentum K " $ " " $ " (a) N=16 (b) N=20 (c) N=24 Figure 6. (Color online) Entanglement spectra of the time-averaged
RDM obtainedafter a θ init = π/ → θ f = π/ N = 16,20 and 24 ( t (cid:48) = V (cid:48) = 0). Symbols are similar to Fig. 2(b-e) and time-averages areperformed as for Fig. 4. between time-average and thermal density matrices much finer, providing a stringentthermalization criterion (based on the close similarity between these two quantities).The ES of ρ ave A computed for XXZ chains (in the entropy plateau) with 16, 20 and24 sites, and shown in Fig. 6(a-c), reveal a smooth convergence with system size, anaccumulation of low pseudo-energy levels at K = 0 and a small pseudo-energy gap, insharp contrast with the ES of the f-GS shown in Fig. 2(e).In order to probe thermalization, we now wish to compare ρ ave A to the thermalaverage of ρ A . Thermal averages can be defined as: ρ λA = (cid:48) (cid:88) α w λα Tr B | Ψ α (cid:105)(cid:104) Ψ α | , (6)where the prime means the sum is restricted to eigenstates of H ( θ f ) with non-zero overlap with | φ (0) (cid:105) , accounting for conserved quantities like momentum and z-component of total spin, and hidden conservation laws in the integrable case. Bothcanonical ( w can α = Z exp ( − βE α ) where E α are eigenenergies associated to | Ψ α (cid:105) ) and ut-of-equilibrium Correlated Systems : Bipartite Entanglement as a Probe of Thermalization - ! /2 ! /2 ! momentum K - ! /2 ! /2 ! momentum K - ! /2 ! /2 ! momentum K " $ " (d) Time aver. (e) Canonical " $ " " $ " " $ " (f) Microcan. (a) Time aver. (b) Canonical (c) Microcan. V’=t’=0 t’=0.5 tV’=0.2 V
Figure 7. (Color online) ES of ρ ave A (a,d), ρ can A (b,e) and ρ micro A (c.f) obtained by full diagonalization of N = 20 site integrable ( t (cid:48) = V (cid:48) = 0) and non-integrable chains( θ init = π/ → θ f = π/ /β (cid:39) .
07 (b),∆ E = 0 . /β (cid:39) .
48 (e) and ∆ E = 0 . microcanonical ( w micro α = Cst in an energy window [ E − ∆ E, E + ∆ E ]) thermalensembles can be considered. Note, the effective temperature 1 /β of the canonicalensemble is (implicitly) given by constraining the conserved mean-energy to be E = − ( V − V (cid:48) ) L/
2. Note that a full diagonalization of H ( θ f ) is now required since theprevious formula (6) involves the complete set of eigenstates | Ψ α (cid:105) , hence limiting theavailable chain lengths to N = 20. The infinite-time average ρ ave A can be computedexactly also using (6) with w ave α = |(cid:104) Ψ α | φ (0) (cid:105)| which, in contrast to the thermalaverages, depends now on the initial state. Incidentally, this enables to control thevery good accuracy of the previous approximate averaging procedure, as shown by adirect comparison between Fig. 6(b) and Fig. 7(a).The results shown in Figs. 7 reveal striking differences between integrable andnon-integrable Hamiltonians. For the latter, surprisingly good agreement between thelow-energy ES of ρ ave A and ρ can A is seen, suggesting that the Eigenstate ThermalizationHypothesis [10] may apply up to the level of the (bipartite) entanglement properties of ut-of-equilibrium Correlated Systems : Bipartite Entanglement as a Probe of Thermalization
6. Quasi-adiabatic quenches
The system after the quench possesses an excess of entanglement entropy (per unitlength) compared to the f-GS. There is in fact an interesting correlation between thisexcess entropy and the ”speed” at which the quench is performed. A smooth quench canbe realized by considering a time-dependent Hamiltonian H ( θ ( τ )) where θ ( τ ) decreasescontinuously from θ init = π/ θ f = π/ , T f ]. One can choose e.g. θ ( τ ) = (1 − ˜ w τ ) θ init + ˜ w τ θ f , (7)˜ w τ = ( w τ − w ) / ( w T f − w ) , (8) w τ = 1 / (1 + exp (( T f / − τ ) /T )) , (9)and T = T f /
10. As shown in Fig. 8, under increasing the characteristic time T theaverage value of the entropy plateau decreases towards the GS value, as expectedfor increasingly adiabatic processes. Interestingly, the deviation (cid:104) S VN (cid:105) − S aveVN alsosimultaneously decreases (to reach zero for a fully adiabatic process).Sudden and continuous quenches also give very different ES of the time-averagedreduced density matrix ρ ave A . Data for continuous and sudden quenches are comparedin Fig. 9. Remarkably, the spectra in the quasi-adiabatic case resemble very closely theequilibrium f-GS spectrum of Fig. 2(e) of the main paper with a clear linear envelope,in contast to the spectra obtained for a sudden quench. However, we note that therequirement (e.g. on the time scale T ) to have a quasi-adiabatic process becomes moreand more stringent for increasing system size because of the vanishing of the finite-sizegap (in the XY phase).
7. Summary
To summarize, the concept of bipartite entanglement is introduced in a genuine out-of-equilibrium isolated many-body system. I propose that it provides a simple andcomplete probe of thermalization, beyond the investigation of simple local observables.In this framework, absence of thermalization is diagnosed whenever the reduced densitymatrix deviates (once averaged over a sufficiently large time interval) noticeably fromthe thermal (reduced) density matrix taken at an effective temperature imposed onlyby the initial conditions. Generically, we find that the system approaches quickly anentanglement entropy plateau. However, integrable and non-integrable chains behavequite differently within this plateau. The RDM of the integrable chain stronglyfluctuates in time and does not thermalize according to the above criterium. This is tobe contrasted to the case where extra terms are introduced in the chain Hamiltonianto break the integrability: in such a case, the ES of the RDM and the spectrum of ut-of-equilibrium Correlated Systems : Bipartite Entanglement as a Probe of Thermalization t S V N ( t ) / S m a x T =0T =2T =4 f-GS entropy q : p/2 -> p/6 "quasi-adiabatic""sudden" N=24
Figure 8. (Color online) VN entanglement entropy of various non-equilibrium statesobtained after a θ init = π/ → θ f = π/ t (cid:48) = V (cid:48) = 0) over different time scales characterized by T . The arrows indicate thedifference between the time-averaged entropy (cid:10) S VN (cid:11) (dot-dashed lines) and the VNentropy of ρ ave A , S aveVN (dashed lines). Time-averages have been performed using 50 binsin a ∆ τ = 40 time-interval within the plateaux. the thermal density matrix become very similar, a strong hint that thermalization canoccur up to the level of an extensive subsystem even though the full system is completelyisolated. Acknowledgements – I thank D. Braun, B. Georgeot, T. Lahaye, I. Nechita, P. Pujol,N. Regnault, M. Rigol, G. Roux and K. Ueda for interesting discussions, IDRIS (Orsay,France) for CPU time on the NEC-SX8 supercomputer and the French Research Council(ANR) for funding.
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