Dynamics and Asymptotics of Correlations in a Many-Body Localized System
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Dynamics and asymptotics of correlations in a many-body localized system
Steve Campbell, , Matthew J. M. Power, and Gabriele De Chiara Centre for Theoretical Atomic, Molecular and Optical Physics, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom Istituto Nazionale di Fisica Nucleare, Sezione di Milano, & Dipartimento di Fisica,Universit`a degli Studi di Milano, Via Celoria 16, 20133 Milan, Italy (Dated: May 6, 2019)We examine the dynamics of nearest-neighbor bipartite concurrence and total correlations in the spin-1 / XXZ model with random fields. We show, starting from factorized random initial states, that the concurrencecan su ff er entanglement sudden death in the long time limit and therefore may not be a useful indicator of theproperties of the system. In contrast, we show that the total correlations capture the dynamics more succinctly,and further reveal a fundamental di ff erence in the dynamics governed by the ergodic versus many-body localizedphases, with the latter exhibiting dynamical oscillations. Finally, we consider an initial state composed of severalsinglet pairs and show that by fixing the correlation properties, while the dynamics do not reveal noticeabledi ff erences between the phases, the long-time values of the correlation measures appear to indicate the criticalregion. I. INTRODUCTION
The study of entanglement in strongly correlated systemsfocused initially on the critical properties of the ground stateof one-dimensional spin chains close to a quantum phase tran-sition [1, 2] (see Ref. [3] for a comprehensive review). Af-ter these early studies, much interest has been devoted tothe study of the time evolution of entanglement after a sud-den quench or a continuous change of the Hamiltonian [4–6].Of particular interest are those works dealing with disorderedsystems and the possibility of inducing, in the presence of in-teractions, many-body localisation (MBL) [7], see the recentreviews Refs. [8–10] and references therein. In contrast toAnderson localization, in MBL systems in one dimension, lo-calization does not occur for an infinitesimal disorder but fora non-zero value.In the last few years, interest on the MBL phase has grownremarkably fast. It is now well established that this phase ischaracterised by the absence of thermalisation, notwithstand-ing the presence of interactions, due to the emergence of localconservation laws similarly to integrable systems. Such con-siderations have helped to develop useful tools for studyingthe MBL phase using local probes [11–14]. Recent studieshave also shown the use of quantities such as quantum mu-tual information and entanglement are useful for examiningthe transition to the MBL phase [15–18]. Energy eigenstatesin the middle of the spectrum of an MBL Hamiltonian fulfilthe entanglement area-law and gives rise to a slow logarith-mic growth of entanglement after a sudden quench. MBL hasbeen recently observed in experiments with ultracold atoms[19–21] and trapped ions [22].While often block entanglement entropy is the focus, inthis work we consider the dynamical onset of the MBL phaseand study the dynamics of the nearest-neighbor concurrence, afaithful measure of two-spin entanglement, and the total corre-lations, which measure all the correlations, classical and quan-tum, shared by all the spins in the chain. Although these twoquantities have been analysed for the centre of the spectrumof an interacting many-body Hamiltonian [16, 17] (see alsoRef. [18] where more general pairwise correlations are con-sidered), the study of the evolution of these quantities and the corresponding asymptotic properties is still missing.To this end and following Ref. [23], we fix the initial staterather than focusing on a particular energy band in the spec-trum [16–18, 24]. To begin we will focus on a random, pure,separable state analogous to the situation in Ref. [23]. Ourinitial states thus uniformly sample the full spectrum of thesystem, i.e. the initial energy distribution forms a Gaussiancentred around zero. This is equivalent to exploring a hightemperature region of the energy spectrum. We also consideran initial state composed of tensor products of singlets. Thisstate, similarly to MBL states, is locally entangled but doesnot have long range entanglement. This ensures the state ini-tially has entanglement localized between certain spin pairsand fixes the marginal probability distributions. As we willsee both settings reveal interesting features of the nature ofthe ergodic-MBL transition.
II. PRELIMINARIES
We consider the spin-1 / XXZ model with periodic bound-ary conditions subject to random disorder (longitudinalfields), h i , applied to each spin. The Hamiltonian is givenby H = L − X i = (cid:16) σ ix ⊗ σ i + x + σ iy ⊗ σ i + y + ∆ σ iz ⊗ σ i + z (cid:17) + L X i = h i σ iz . (1)The random fields h i are uniformly chosen from the interval (cid:2) − η, η (cid:3) . For ∆ > η . In what followswe will consider ∆ =
1, unless otherwise stated. For thisinteraction strength the current best estimates for the criticaldisorder strength is predicted to occur at η c ≈ . η c ≈ . η < η c [17, 27]. Weremark that a recent study has shown a closely related modelwhere the system is quasi-periodic rather than random appearsto be in a distinct universality class [28].We will focus on two figures of merit in particular: the con-currence and the total correlations. Concurrence is a measureof entanglement valid for arbitrary states of two qubits. Itis defined in terms of the eigenvalues λ ≥ λ , , of the spin-flipped density matrix ρ ( σ y ⊗ σ y ) ρ ∗ ( σ y ⊗ σ y ) as C = max , p λ − X i = p λ i . (2)Therefore, when evaluating the entanglement we will focus onthe first two spins of the chain, i.e. ρ = Tr i , , (cid:2) ρ (cid:3) . We re-mark the relationship between concurrence and MBL was re-cently explored in the high energy region of the spectrum [16].Total correlations are defined as the information shared be-tween all constituents of the state. As such, and unlike entan-glement, the total correlation encompasses both classical andquantum natures. We define the total correlations as I = X i S ( ρ i ) − S ( ρ ) (3)where S ( · ) denotes the von Neumann entropy, ρ is the totaldensity matrix of the system and ρ i is the reduced density ma-trix of spin i . For two-spins this is equivalent to the mutualinformation shared between them. Furthermore, since in whatfollows the state is always pure and therefore S ( ρ ) = I issimply the sum of the von Neumann entropy of the marginals.In Ref. [17] this figure of merit was used to explore theergodic-MBL transition, again in the high energy region. III. RESULTSA. Random initial states
As an initial state, each spin at site i is prepared in a purestate | ψ i i = cos (cid:18) θ i (cid:19) | i i + e i φ i sin (cid:18) θ i (cid:19) | i i , (4)where cos( θ i ) is chosen randomly to be ± v and φ i is chosenrandomly from [0 , π ), see Ref. [23] Fig. 1. We remark thissampling means that each spin is at a fixed angle above orbelow the equatorial plane of the Bloch sphere, pointing ina random direction and therefore we are not considering ran-dom states in the typical sense sampled according to the Haarmeasure. Our initial state is then | ψ i = L O i = | ψ i i . (5)We evolve this state for many realizations of the disorder, witheach one starting from a di ff erent | ψ i . We perform at least1000 simulations for each value of η in order to ensure goodconvergence. (a) (d) -4 -3 -2 -1 -2 -1 L =12 C t -4 -3 -2 -1 -2 -1 L =12 C t h =0.5 h =1.0 h =1.5 h =2.0 h =2.5 h =3.0 h =3.5 h =4.0 h =4.5 h =5.0 (b) (e) -4 -3 -2 -1 -2 -1 L =16 C t -4 -3 -2 -1 -2 -1 L =16 C t (c) (f) C h L =8 L =12 L =16 C hn C FIG. 1: (Color online) (a-b-d-e)
Dynamics of nearest neighbor con-currence fixing ∆ = η = . v = (a) L =
12 and (b) L =
16. In the rightcolumn we fix v = . (d) L =
12 and (e) L = (c) Asymp-totic value of the nearest neighbor concurrence against disorder, η ,for v = (f) Asymptotic value of the nearest neighbor concurrenceagainst disorder, η , and v for L =
12. The color-coding for the panels (a-b-e) is the same as in panel (d) . In Fig. 1 we examine the dynamics of the nearest neighborconcurrence. In panels (a) and (b) we take v =
1, this corre-sponds to the situation in which each individual spin, i.e. itsBloch vector, in Eq. (5) is randomly chosen to point along the ± z -axis. We see the initial dynamics are insensitive to themagnitude of the disorder. However, after t ∼ − . and as η is increased, the amount of nearest neighbor concurrence isalso increased. Furthermore, small η witnesses a sharp dropin the amount of entanglement shared between the two spinsbefore settling into its long-time value, while for larger val-ues of η , entering the MBL phase, the system takes longer tosettle. Such a behavior is consistent with the slow growth ofblock entropy [23]. Comparing panels (a) and (b) in Fig. 1we see that qualitatively these features persist regardless ofthe system size L . Panel (c) shows the asymptotic values for L = ,
12 and 16. These results are in agreement with thosereported in Ref. [16] where the entanglement properties ofstates in the middle of the spectrum of the Hamiltonian wereexamined, similarly showing that the (total) nearest neighborconcurrence grows from zero in the ergodic phase to compar-atively large values when the system transitions into the MBL (a) (d) -2 -1 L =12 I t -2 -1 L =12 I t (b) (e) -2 -1 L =16 I t -2 -1 L =16 I t (c) (f) I / L h L =8 L =12 L =16 0.860.880.900.920.940.960.981.00 0 1 2 3 4 5 I / L h L =8 L =12 L =16 FIG. 2: (Color online) (a) - (c) Dynamics of the total correlationsfixing ∆ = η = . v = (a) L =
12 and (b) L =
16. In theright column we fix v = . (d) L =
12 and (e) L =
16. Alsoshown is the asymptotic value of the total correlations (rescaled with L ) against disorder, η , for (c) v = (f) v = .
5. The color-codingfor the panels (a-b-d-e) is the same as in Fig. 1. phase.We perform the same simulations only altering the initialstate such that v = .
5. In this case the short-time dynamicsare qualitatively the same as before. Once again, while ini-tially all values of η present the same dynamics, as we increasethe disorder strength the systems evolving within the MBLphase settle slower than in the ergodic phase. However, animportant di ff erence arises: now the asymptotic value of thenearest neighbor concurrence tends to zero. Panel (f) showsthat the long-time behaviour of the concurrence is stronglya ff ected by the choice of initial state. Taking v = v ∈ (0 , .
84) (although we remark this studyfocused on the MBL phase with η ≥ (f) this range corresponds to nearest-neighbor concur-rence being zero, and therefore the invariance reported maybedue to such pathological features. We recall that the approach employed here is expected tomodel the high temperature behaviour in the long-time limit.Similarly, directly accessing the middle of the spectrum, asdone in Ref. [16], is also expected to reproduce the same hightemperature features. We have checked that the initial energydistribution obtained by taking the class of states (4) is qualita-tively similar (although generally broader) to the one obtainedby taking a few tens of states in the middle of the spectrum asin Ref. [16]. Here, we have shown that great care must betaken when considering entanglement measures such as theconcurrence. Changing the initial state can lead to seeminglycontradictory conclusions, stemming from the (in this case)pathological occurrence of ESD. We therefore seek to employa di ff erent figure of merit to alleviate this problem.In Fig. 2 we examine the dynamics and asymptotic valuesfor the total correlations, Eq. (3). While all of the main qual-itative features persist we can now more clearly identify therole v plays. In panels (a-b-c) , v = η , with the total correlationsdecreasing as η grows. This is in agreement with Fig. 1 (b) and (c) for the same‘favorable’ value of v , i.e. one that main-tains a non-zero value of concurrence in the long time limitand thus does not exhibit ESD. We remark that it is intuitivethat a measure of total correlations, that encompasses all clas-sical and quantum aspects, should decrease when the bipartiteentanglement grows since, due to the monogamy properties ofthe entanglement, larger bipartite entanglement generally ne-cessitates a reduction in the the total quantum correlations. Inpanels (d-e-f) we fix v = .
5. In this case a remarkable featureemerges that was not immediately evident when studying theconcurrence. In the ergodic phase I grows monotonically un-til settling to its long time value. As the disorder is increased,and we enter the MBL phase, we see the emergence of oscilla-tions. These oscillations persist for a significant time, gradu-ally dissipating until the system settles to its asymptotic value.Indeed, in the MBL phase we expect the system to store somememory of the initial state in the dynamics, and this wouldappear to be evidenced by these oscillations. Additionally, inpanel (f) we now see that the asymptotic values for the totalcorrelations reflect the changes in the disorder strength. Fur-thermore, the magnitude of this e ff ect is significantly largerthan compared to block entropy [23]. B. Singlet pairs
We next turn our attention to a di ff erent initial configura-tion for the chain. Starting from a singlet (cid:12)(cid:12)(cid:12) Ψ − i E = √ ( | i −| i ) (2 i − , i , we take our initial state to be | ψ i = L / O i = (cid:12)(cid:12)(cid:12) Ψ − i (cid:11) . (6)The initial energy is now fixed regardless of the random fields h i and it is negative. We evolve Eq. (6) in precisely the samemanner as done previously and in what follows we study thedynamical properties and asymptotic values of our figures ofmerit. (a) (b) -3 -2 -1 L =12 C t -2 -1 L =16 C t (c) (d) -2 -1 L =12 I t -2 -1 L =16 I t FIG. 3: (Color online) (a-b) : Dynamics of nearest neighbour con-currence and (c-d) : dynamics of total correlations for the initial statecomposed of singlet pairs, Eq. (6). We fix ∆ = η ∈ [0 . ,
5] in steps of 0.5. (a-c) L =
12 and (b-d) L =
16. Thecolor-coding for the panels is the same as in panel Fig. 1 (a) . In Fig. 3 we show the concurrence and total correlations.For both, the initially localized correlation is frozen for a shorttime window, which is then followed by an exponential decaywith minimal dynamical fluctuations before settling to a non-zero long-time value, a trend followed regardless of the lengthconsidered and consistent for all disorder strengths. Here wefind that the long-time value of both quantifiers is stronglya ff ected by the disorder strength. In Fig. 4 we explore thebehavior of these asymptotic values against disorder strengthmore closely. Panel (a) shows the total correlations, rescaledby L , for several chain lengths. We see the curves remainclose to one another for increasing lengths, confirming the ex-tensive nature of I . Additionally, notice that the total correla-tions increase slightly as we increase system size. The concur-rence shown in panel (b) exhibits a similar qualitative behav-ior, however with one important di ff erence, we now see thatthe asymptotic value decreases as the system is enlarged. In-terestingly, as we increase η moving from the ergodic to MBLphases, this value decreases. In the MBL phase, it reachesa minimum value after which it starts to grow to its large η value.Another evidence of the ergodic-MBL transition is pro-vided by the long-time distribution of concurrence as shownin Fig. 5. In the plots we have excluded the values C = η <
2, i.e.in the ergodic phase, the distribution for low values of C hasa peak at a non zero value of the concurrence and then de-cays rapidly to zero for large values of C . For η ≥ η some nearest-neighbor entanglement is re-tained leading to a large mean concurrence as in Fig. 4. Forlarger values of η , although the mean value is comparable, (a) (b) I / L h L =8 L =12 L =16 0.000.050.100.150.200.250.300.35 0 1 2 3 4 5 C h L =8 L =12 L =16 FIG. 4: (Color online) Asymptotic value of the (a) total correlations(rescaled with L ) and (b) nearest neighbor concurrence against thedisorder strength η . (a) (b) C P ( C ) C P ( C ) (c) (d) C P ( C ) C P ( C ) FIG. 5: (Color online) Long-time distribution of the concurrence.The histograms show the probability of observing a value of concur-rence in each interval. The panels are for (a) η = . (b) η = (c) η = (d) η =
4. We have excluded the data values C = L = the distribution is completely di ff erent. We would like to addthat similar results hold for the long-time distribution of thetotal correlations. Although the e ff ect is not as strong as forthe concurrence, the probability distribution of the total cor-relations change from a skewed distribution away from theergodic-MBL transition to an approximately Gaussian distri-bution near the predicted transition point (results not shown).We remark that the distribution of block entanglement is ex-amined in Ref. [30]. IV. CONCLUSIONS
We have examined the dynamics of correlations, encom-passing both quantum and classical natures, in a many-bodylocalized system. Using random, factorized initial states wehave shown that care must be taken regarding the choice ofinitial state and correlation measure. In particular, despite ef-fectively modelling the high-temperature behavior of the sys-tem, we have shown that concurrence can exhibit markedlydi ff erent behaviors depending on the initial state. This is inlarge part due to the occurrence of (pathological) entangle-ment sudden death. By employing a global measure of corre-lations that encompasses both classical and quantum natures,we have shown that such issues can be neatly alleviated. Wetherefore argue that the total correlations serve as a more use-ful indicator in studying the dynamics. Furthermore, the totalcorrelations highlight a clear change in the nature of the dy-namics when the system is quenched into the ergodic or theMBL phase, with the latter showing oscillations in the corre-lations, likely related to memory e ff ects. Finally, we assesseda di ff erent initial state composed of tensor products of singletpairs. Our results provide important insight into the nature ofthe ergodic-MBL transition and highlight the care that mustbe taken in choosing suitable figures of merit to assess suchsystems. Such an observation is particularly important in thecontext of the recent experimental [22] and theoretical [31]developments in studying MBL systems where the initial state is fixed.It is important to stress that the correlation measures weconsider in this paper can be measured in experiments withultracold atoms, trapped ions and solid state implementationsof spin chains. In fact the concurrence only requires two-spincorrelations while, at zero temperature, total correlations re-quire only the single spin density matrix that can be deter-mined from the single-spin polarisation. Acknowledgments
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