Is there slow particle transport in the MBL phase?
IIs there slow particle transport in the MBL phase?
David J. Luitz and Yevgeny Bar Lev Max Planck Institute for the Physics of Complex Systems, Noethnitzer Str. 38, Dresden, Germany ∗ Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel † (Dated: July 29, 2020)We analyze the saturation value of the bipartite entanglement and number entropy starting froma random product state deep in the MBL phase. By studying the probability distributions of theseentropies we find that the growth of the saturation value of the entanglement entropy stems from asignificant reshuffling of the weight in the probability distributions from the bulk to the exponentialtails. In contrast, the probability distributions of the saturation value of the number entropy areconverged with system size, and exhibit a sharp cut-off for values of the number entropy whichcorrespond to one particle fluctuating across the boundary between the two halves of the system.Our results therefore rule out slow particle transport deep in the MBL phase and confirm that theslow entanglement entropy production stems uniquely from configurational entanglement. Introduction .—Generic interacting quantum many-body systems are expected to thermalize after a quenchby virtue of the eigenstate thermalization hypothesis [1–7]. However the addition of sufficiently strong quencheddisorder allows such systems to avoid thermalization [8–13], a phenomenon which is called many-body localiza-tion (MBL). For one dimensional systems the existenceof the MBL phase at strong disorder is now firmly es-tablished [14], but the situation for higher dimensionalsystems is still an open question [15–18]. When all many-body eigenstates are localized, the phenomenology ofMBL is understood by the emergence of local conservedquantities called the l-bits [9, 19, 20]. The existence ofl-bits predicts the absence of particle transport and ther-malization in the MBL phase, but also an unboundedlogarithmic growth of the bipartite entanglement entropy[9, 19, 21–25] up to a nonthermal, extensive, saturationvalue. This behavior is in stark contrast with the ther-mal phase, where the entanglement entropy grows as apower-law in time [26, 27]. Without disorder, a lineargrowth of the entanglement entropy is typically observed[28].Within the l-bit model the production of entanglementdeep in the MBL phase does not rely on particle trans-port, since due to the emergent conservation laws theparticle number in any part of the system is essentiallyconstant for all times [29]. This view was challenged veryrecently in a study of the entropy of the subsystem parti-cle number distribution [30], where a long growth regimeof the number entropy was observed, which was argued tocontinue indefinitely in the thermodynamic limit in theMBL phase, suggesting that there is very slow transportin the MBL phase.In the present work, we address this question by a de-tailed statistical analysis of the behavior of the saturationvalues of the entanglement and the entropy of the sub-system particle number distribution (number entropy).We find that the fluctuations of the particle number arestrictly limited at strong disorder and preclude an indef-inite growth of the number entropy.
Model and method .—We consider the standardmodel of many-body localization, an open spin chain withrandom fields:ˆ H = J L − (cid:88) i =1 S i · S i +1 + L (cid:88) i =1 h i ˆ S zi , (1)where J corresponds to the interaction between the spinsand the random fields are drawn from a box distributionwith h i ∈ [ − W, W ]. Without loss of generality we set J = 1, throughout this work. Using the Jordan-Wignertransformation, this model maps exactly to a model ofinteracting spinless fermions,ˆ H = J (cid:16) ˆ c † i ˆ c i +1 + ˆ c † i +1 ˆ c i (cid:17) + J L − (cid:88) i =1 (cid:18) ˆ n i − (cid:19) (cid:18) ˆ n i +1 − (cid:19) + L (cid:88) i =1 h i (cid:18) ˆ n i − (cid:19) , (2)where ˆ c † i creates a fermion at site i and ˆ n i = ˆ c † i ˆ c i . Themodel conserves the total magnetization (respectively,the particle number), and throughout this work we fix (cid:80) i ˆ S zi = 0, (respectively, half-filling). While the model(1) has been studied in great detail, the critical disorderis only known with a large margin of error, W c = 3 . ± | σ , σ , . . . , σ L (cid:105) in time. Here, σ i = ± are the eigenvalues of the corresponding localˆ S zi operators. The initial states have a definite number ofup spins (i.e. σ i = +1 /
2) in any subsystem, which corre-sponds to a definite number of particles in the equivalentspinless fermion model (2).
Results .— We consider a quench from a product state | ψ (cid:105) = | σ , σ , . . . σ L (cid:105) in the ˆ S z basis, and cut the sys-tem into two subsystems of equal size, A and B , where a r X i v : . [ c ond - m a t . d i s - nn ] J u l L . . . . . m e a n [ S ∞ E ] W = 4 W = 5 W = 6 W = 8 W = 10 Figure 1. Disorder average of the long time-averaged entan-glement entropy S ∞ E as a function of system size L for differentdisorder strengths W . spins i = 1 , . . . , L/ A and spins L/ , . . . , L are in subsystem B . Due to the con-servation law (cid:80) i ˆ S zi = 0, the reduced density matrixˆ ρ A ( t ) = Tr B | ψ ( t ) (cid:105) (cid:104) ψ ( t ) | of the subsystem A is blockdiagonal with blocks labeled by the number of up spins n A in the subsystem. The probability p ( n A ) to find n A up spins (corresponding to particles in the spinless sys-tem) in subsystem A is given by the trace of the reduceddensity matrix ˆ ρ A in this block.The entanglement entropy is given by, S E = − Tr (ˆ ρ A ln ˆ ρ A ) (3)and the number entropy S N is the Shannon entropy ofthe number distribution p ( n A ) [38] S N = − (cid:88) n A p ( n A ) ln p ( n A ) . (4)In the context of MBL it is useful to split the contribu-tions to entanglement from particle number fluctuationsand the configurations of the particles, by introducingthe configurational entropy S C , which is the difference S E − S N [30, 38].The wavefunction | ψ ( t ) (cid:105) of the system evolves accord-ing to, | ψ ( t ) (cid:105) = (cid:88) n e − i E n t (cid:104) n | ψ (0) (cid:105) | n (cid:105) , (5)where E n and | n (cid:105) are the eigenvalues and eigenstates ofthe Hamiltonian. At any point in time we can calcu-late the entanglement entropy and the number entropy S N/E ( t ), using (3), (4) and the definition of ˆ ρ A . Sincein this work we are interested in the saturation values ofthe entropies, we consider the infinite time-average S ∞ N/E = lim T →∞ T T (cid:90) d t S N/E ( t ) , (6) L . . . . . . m e a n [ S ∞ N ] W = 4 W = 5 W = 6 W = 8 W = 10 Figure 2. Full lines: Disorder average of the time-averagednumber entropy S ∞ N as a function of system size L for differ-ent disorder strengths W . Dashed lines: Disorder average ofthe entropy of the infinite time averaged subsystem numberdistribution S [ p ∞ ( n A )], which is an upper bound of S ∞ N . which is estimated numerically by averaging the entropiesover 40 time points at very late times to represent thesaturation value of the entropy for finite systems. Prac-tically, we have checked that the saturation values arerobustly reached at times as late as t ∈ (cid:2) , (cid:3) forall sizes and disorder strengths which we consider in thiswork.The entropies exhibit significant temporal fluctuationsat late times (cf. Appendix) , therefore in order to geta more robust insight of the late time behavior, we alsoconsider the infinite time-average of the number distri-bution p ∞ ( n A ) = lim T →∞ T T (cid:90) d t p ( n A , t ) , (7)which unlike S [ p ∞ ( n A )] is not affected by temporal fluc-tuations, since it can be calculated numerically exactly.It is easy to show that the entropy S [ p ∞ ( n A )] boundsfrom above the infinite time-averaged number entropy, S ∞ N , noting that the entropy is a concave function of p ( n )and using Jensen’s inequality.In this work we do not study the temporal dependence,but focus directly on the saturation values of the en-tropies. If the saturation value grows with system size,then the dynamical growth regime continues indefinitelyin the thermodynamic limit, and if is independent of thesystem size, the temporal growth regime is transient.In Fig. 1, we show the disorder averaged saturationvalue of the entanglement entropy S ∞ E , obtained by timeevolution of the wavefunction to very long times t ≥ and averaging over 40 time points for each of the 50 ran-dom initial product states in addition to averaging overthe disorder realizations. We note that it is important toaverage over a very large number ( n = 50 000) of disorderrealizations to obtain converged statistical averages. − − p ( S ∞ E ) W = 3 W = 410 − − p ( S ∞ E ) W = 5 W = 60 2 4 S ∞ E − − p ( S ∞ E ) W = 8 0 2 4 S ∞ E W = 10 L = 4 L = 6 L = 8 L = 10 L = 12 L = 14 L = 16 Figure 3. Distribution of the time averaged entanglement en-tropy S ∞ E for disorder strengths W = 3 , , , , ,
10 and dif-ferent system sizes L . The distributions are taken over 50 000disorder realizations and 50 initial random product states.For each disorder realization the time average is computed bysampling 40 time points for t ≥ . The black dashed lineshows the entropy S E = ln(2) and the colored ticks on thehorizontal axis mark the corresponding mean of S ∞ E . It is clearly visible that the saturation value of theentanglement entropy grows with system size. Interest-ingly, we observe a significant upturn of the curves evenfor very strong disorder, which is only weakly visiblein previous data obtained at weak interaction strengths[23, 39].The full lines in Fig. 2 show the time averaged num-ber entropy S ∞ N as a function of system size L and fordifferent disorder strengths W , spanning both the criti-cal and MBL regimes. While for W ≤ S ∞ N still growsslightly with system size, we observe a saturation andeven a weak decrease for strong disorders ( W ≥ S [ p ∞ ( n A )],which satisfies S ∞ N ≤ S [ p ∞ ( n A )] and is plotted as thedashed lines in Fig. 2. We see that S [ p ∞ ( n A )] is sat-urated for W = 6 and slightly decreases with systemsize for stronger disorders. Since S ∞ N can not exceed S [ p ∞ ( n A )], we conclude that S ∞ N is independent of thesystem size for strong disorder.As the mean contains only limited information about − − p ( S ∞ N ) W = 3 W = 410 − − p ( S ∞ N ) W = 5 W = 60 . . . . S ∞ N − − p ( S ∞ N ) W = 8 0 . . . . S ∞ N W = 10 Figure 4. Similarly to Fig. 3, but for the time averaged num-ber entropy S ∞ N . The red dashed line shows the entropy S = ln(3). the probability distribution, we study the full distribu-tions of S ∞ E/N . Fig. 3 shows the distribution of S ∞ E over50 random initial product states and 50 000 disorder re-alizations. For intermediate disorder strengths, in thecritical regime W = 3 , , S ∞ E is significantly differ-ent from the distribution of the entanglement entropy ofeigenstates (cf. e.g. Fig. 10 c) and d) in Ref. [40]). Whilethe mean entanglement of eigenstates does not dependon the system size [41], S ∞ E grows with size as shown inFig. 1, due to dephasing between the various eigenstateswhich are spanning the initial state.At very strong disorder, the weight around S E = 0is visibly decreasing with increasing systems size, whichleads to a corresponding increase in the mean. For highentropies, the distribution exhibits a long, seemingly ex-ponential tail, with a negligible contribution to the mean.In Fig. 4 we consider the distribution of the timeaveraged number entropy S ∞ N . While in the criticalregime ( W = 3 , , − − p [ S ( p ∞ N A ) ] W = 3 W = 410 − − p [ S ( p ∞ N A ) ] W = 5 W = 60 . . . . S ( p ∞ NA )10 − − p [ S ( p ∞ N A ) ] W = 8 0 . . . . S ( p ∞ NA ) W = 10 Figure 5. Similarly to Fig. 4, but for the entropy of the timeaveraged subsystem particle number distribution S [ p ∞ ( n A )]. system size is increased, the entire distribution seem-ingly converges to a limiting distribution at large sizesand strong disorderes W = 8 ,
10. This is accompaniedwith an effective independence of the mean of the dis-tribution on the system size, as is also shown in Fig. 2.For strong disorder, the distributions exhibit a secondarypeak for
S < ln (2), which is reminiscent of the ln(2)peak in the distributions of the eigenstate entanglemententropy [40, 42, 43]. Here this peak is broadened andstays strictly below ln (2) (black dashed horizontal line)for all considered system sizes. The observed probabil-ity distribution seems to decay significantly for entropieslarger than ln (3) (red dashed horizontal line) as we willdiscuss in more detail below.For a better, quantitative understanding of the highentropy part of the distribution of S ∞ N , we consider nextthe distributions of S [ p ∞ ( n A )], which is cleaner due tothe fact that the infinite time-average can be calculatedexactly and since it provides an upper bound to S ∞ N .Fig. 5 shows the distribution of S [ p ∞ ( n A )], which ex-hibit a sharp secondary peak at ln (2), corresponding toan equal probability to have n A , n A + 1 or n A, n A − n A , n A +1 or n A − n A is the particle number in the initial state. This means that in this rare state oneparticle has crossed the boundary between the two sub-systems.For strong disorder ( W = 8 , Discussion .—In this work we presented a detailedstudy of the saturation value of entanglement and num-ber entropies including the probability distributions overthe initial product states and disorder realizations. Wehave shown that at strong disorder the mean of the sat-uration value of the entanglement entropy grows withsystem size, which is consistent with previous literature.The distributions of these quantities are quite broad andrequire a very large number of disorder realizations tobe sampled precisely. We identify that the growth of themean with system size results from a reshuffling of weightfrom low to high entropies.The entanglement entropy can be decomposed into asum of the number entropy and the configurational en-tropy. We show that at strong disorder W = 8 ,
10 themean of the saturation value of the number entropy doesnot grow with system sizes, moreover, for large systemsthe entire probability distribution converges to a limit-ing distribution. We further study the entropy of theinfinite-time averaged number distributions, which al-lows us to show that the number fluctuations are typ-ically bounded by a change of one particle across the twohalves of the system. This leads to a sharp cut-off in theprobability distribution of the number entropy at ln(3).Even for the very large number of disorder realizationsused in this work, we did not observe realizations withnumber entropies which exceed this limit, which leads usto conclude that there is no particle transport for suffi-ciently strong disorder and the observed growth of thenumber entropy in time in Ref. [30] is pertinent to thecritical regime and likely disappears for stronger disorderor larger system sizes.We are grateful to Achilleas Lazarides and RoderichMoessner for useful comments and thank Jesko Sirkerfor discussions. We acknowledge financial support fromthe Deutsche Forschungsgemeinschaft through SFB 1143(project-id 247310070). YBL acknowledges support bythe Israel Science Foundation (grants No. 527/19 and218/19). ∗ [email protected] † [email protected][1] Mario Feingold, Nimrod Moiseyev, and Asher Peres,“Ergodicity and mixing in quantum theory. II,” Phys.Rev. A , 509–511 (1984)[2] J. M. Deutsch, “Quantum statistical mechanics in aclosed system,” Phys. Rev. A , 2046–2049 (1991) [3] Mark Srednicki, “Chaos and quantum thermalization,”Phys. Rev. E , 888–901 (1994)[4] Mark Srednicki, “The approach to thermal equilibriumin quantized chaotic systems,” J. Phys. A: Math. 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Sheng, “Many-body localization andtransition by density matrix renormalization group andexact diagonalization studies,” Phys. Rev. B , 045111(2016) Appendix A: Distribution of the configurationalentropy − − p ( S ∞ C ) W = 3 W = 410 − − p ( S ∞ C ) W = 5 W = 60 1 2 3 S ∞ C − − p ( S ∞ C ) W = 8 0 1 2 3 S ∞ C W = 10 Figure 6. Distribution of the saturation value of the configu-rational entropy for disorder strengths W = 3 , , , , ,
10 anddifferent system sizes L . Distributions are taken over 50 000disorder realizations and 50 initial random product states. Ineach disorder realization, the time-average is sampled over 40time points for t ≥ . In this appendix, we provide additional data for thedistribution of saturation value of the configurational en-tropy S ∞ C , which is defined as, S ∞ C = S ∞ E − S ∞ N . (A1) We show the full probability distribution (with itsmean indicated by colored ticks at the top and bottomof each panel) in Fig. 6. The mean of the distributiongrows with system size for all disorder strengths with aslower growth at strong disorder. This growth stems fromreshuffling of the weight from low to high entropies.The distribution of the configurational entropy is quitesimilar to that of the entanglement entropy and exhibitsan exponential tail as well as a peak at zero entropy whichdecreases with system size. The sharp peak located at S ∞ E < ln(2) for small systems in the entanglement en-tropy distribution is strongly suppressed in the numberentropy distribution, indicating that it stems from thenumber entropy. Appendix B: Temporal fluctuations of the saturationvalue of the entanglement and number entropies
Besides the time average of the late time entropies,we consider also the temporal fluctuations around thesaturation value of the entropies. L . . . . . s t d [ S ( t →∞ ) ] S E LS N W = 4 W = 5 W = 6 W = 8 W = 10 Figure 7. Temporal fluctuations of the entanglement entropy S E (left) and the number entropy S N (right), as a function ofthe system size and for various disorder strengths.(right), as a function ofthe system size and for various disorder strengths.