Emerging ergodic behavior within many-body localized states
AAn emerging many-body ergodic state between two classes of many-body localizedstates
Wai Pang Sze and Tai Kai Ng
Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
In this paper, we report our numerical analysis of energy level spacing statistics for one-dimensional spin-1 / B i . We con-centrate on the strong disorder limit where the system is expected to be in either a paramagnetic many-body localized (MBL) state or a spin-glass MBL state. By analyzing the energy-level spacingstatistics as a function of strength of random magnetic field h/J ⊥ ( − h ≤ B i ≤ h ), interactionstrength J z /J ⊥ , energy of the many-body state E and the number of spin- ↑ particles in the system M = (cid:80) i ( s zi + ), we show that there exists a small region between the paramagnetic and spin-glassMBL phases where an ergodic phase emerges. The ergodic phase emerges at the middle of themany-body energy spectrum when M ∼ N and is not adiabatically connected to the ergodic phasethat occurs in the weak-disorder, weak-interaction limit. Introduction
In recent years there have been growinginterests in the study of localization in interacting many-particle systems with strong disorder [1–3]. In contrast tousual man-body systems that are chaotic and irreversible(i.e. obey thermodynamics description), it is observedthat there exist many-body localized (MBL) phases thatare non-ergodic and reversible[4]. MBL phases are ofinterest to the scientific community because of their po-tential application in manipulating quantum informationwithout dissipation. The problem is difficult theoreti-cally because of its intrinsic nature (strong interaction+ disorder), and most of the existing theoretical resultsare based on numerical studies of one-dimensional (1D)systems.It is generally accepted that in the ergodic, irreversiblephase, the many-body state energy eigenvalues follow thestatistics of Random Matrix Theory[5–8] and the energylevel spacing s n = E n − E n − obeys Wigner-Dyson lawfor distribution. On the other hand, s n follows a Poissondistribution P ( s ) = exp ( − s/λ ) in the MBL phase whereeigenstates are localized randomly and are uncorrelated, λ is the mean-energy-level spacing which is in generala (smooth) function of E n . The difference between thetwo types of distribution can be measured by the ratio ofconsecutive level spacings r n = min( s n ; s n − )max( s n ; sn − introducedby Oganesyan and Huse [9]. Its average value in the er-godic phase (GOE ensemble) is (cid:104) r (cid:105) GOE ∼ . (cid:104) r (cid:105) P oisson = 2 ln − ∼ .
386 in MBL phase. Thus (cid:104) r (cid:105) provides a convenient tool that gives an overall estimateof whether the many-body states are localized (MBL) orextended (ergodic). Since then, more sophisticated toolshave been developed to study the MBL phase, includingenergy-resolved (cid:104) r ( E ) (cid:105) that computes the average valueof r n over a narrow energy window E n ∼ E ± δE [3],the entanglement entropy [10–15] and non-equilibrium(quench) dynamics [16–19], etc. A Fermi-liquid typephenomenology has also been developed that providesa physical picture of the eigenstates in the MBL phase.In this description, the many-body states can be thought of as adiabatically connected to a set of localized single-particle orbital [20, 21] or local integrals of motion ( liom )[22–24]. The main differences between MBL and Fermiliquid states are that (i) the one-to-one correspondencebetween bare- and quasi- particles in Fermi liquids are,strictly speaking, restricted only to ground and very low-energy states but there seems to be no such restrictionin the MBL states. However (ii), whereas the correspon-dence between bare- and quasi- particle states are notaffected by the presence of other quasi-particles in Fermiliquid theory, the liom states may depend strongly on theoccupations of other liom states when the interaction be-tween particles is strong.In this paper, we shall study the 1D spin-1 / B i . The Hamiltonian is H XXZ = N (cid:88) i =1 (cid:0) J ⊥ ( S xi .S xi +1 + S yi S yi +1 ) + J z S zi S zi +1 + B i S zi (cid:1) , (1)where S αi is the spin(-1 /
2) operator at direction α =ˆ x, ˆ y, ˆ z at site i and B i is a random magnetic field at z -direction with magnitude | B i | < h . We note thatthe model can be mapped onto an interacting spin-less fermion model in a random potential via a Jordan-Wigner transformation. The Hamiltonian has been stud-ied extensively in the weak/intermediate disorder regimeto illustrate the transition between the ergodic and MBLphases[3, 9, 16, 25, 26]. We shall study the strongly dis-ordered regime h >> J ⊥ in this paper.We note that the total magnetization in z -direction, S ztot = (cid:80) i S zi is a conserved quantity in the above Hamil-tonian and the eigenstates of the system can be classifiedinto sectors with different values of M = (cid:80) i ( s zi + )which measures the total number of spin- ↑ particles. Thesystem has also a spin-inversion symmetry which mapsthe system with M spin- ↑ particles to the system with N − M spin- ↑ particles.To understand the properties of the model in the strongdisorder limit we start with considering the limit J ⊥ = 0. a r X i v : . [ c ond - m a t . d i s - nn ] J un In this case, the system becomes classical and the eigen-states of the system are all localized. The system has twoMBL phases: (1) the paramagnetic phase which occursin the limit h >> J z . In this case, the spin S zi ’s takerandom values S zi = ± and there is no correlation be-tween spins on different sites. (2) In the opposite limit h << J z the system resides in the spin-glass phase char-acterized by a spin-glass order parameter defined for anyeigenstate | n (cid:105) of the system, SG ( E n ) = lim | i − j |→∞ (cid:104) n | S i S j | n (cid:105) (cid:54) = 0 . (2)For a finite system with open-boundary condition, thespin-glass order parameter [27] can be measured with i, j being the two endpoints of the spin chain.To understand the spin-glass order we first considerthe limit h = 0. In this case, the ground state of thesystem is anti-ferromagnetically ordered and SG ( E ) = − × .
25 for a spin chain with even (odd) numberof sites. The first excited states of the system have en-ergy E + J z . corresponding to exciting one domain wallin the system and SG ( E ) = 1( − × .
25, independentof where the domain wall locates. Generalizing the ar-gument, we find that the n th excited states of the sys-tem have n domain walls with energy E n ∼ nJ z + E and SG ( E n ) ∼ ( − n +1 [( − n ] × .
25 for chains witheven (odd) number of sites. The degenerate energy lev-els E n split into distributions with width ∼ √ N h cen-tered around E n when the random magnetic field h (cid:54) = 0is added. SG ( E n ) decreases when h increases and goesto zero when h >> J z where the spin configuration be-comes completely randomized. Upon ensemble averagingthe two MBL phases are separated by a phase transitionthat occurs at some critical values h = h c .We shall study numerically in this paper what hap-pens when a small J ⊥ << h is added to the systemby studying the energy level spacing statistics. Beforepresenting our results we provide some details of our nu-merical analysis here. We consider spin-1 / N ≤
16 and Exact Diagonalization(ED) is used to obtain all eigenstates of our models. Weemploy open boundary condition in oiur study. We donot consider
N >
16 systems in our study because of thelimitation of computer power. For each set of parame-ters characterizing the system, we generate 100 randommagnetic field samples to perform disorder averaging ex-cept in the J ⊥ =0 limit where the system becomes classicaland a much larger sampling size can be employed (see be-low). We shall be interested in energy resolved propertiesof the system in this paper, and for a given energy E , wecompute the expectation value of a variable (cid:104) ˆ O ( E ) (cid:105) byaveraging over N M eigen-states with energies closest to E . The disorder averaging is performed afterward, i.e. (cid:104) ˆ O ( E ) (cid:105) = (cid:104) N M (cid:88) n ; E n ∼ E (cid:104) n | ˆ O | n (cid:105) (cid:105) disorder . where we took N M =4 = 10 , N M =6 , = 100. When com-paring between the model with different sets of parame-ters, it is convenient to introduce the renormalized energy (cid:15) = ( E − E min ) / ( E max − E min ), where E max and E min are the highest and lowest eigen-energy of the system,respectively. We note that the total number of statesin a system with M spin- ↑ particles in N lattice sitesis C NM and changes rapidly with M . In particular, C NM is small for M ∼ M until M = N/
2. For this reason, we shall restrict our calcula-tions to 3 ≤ M ≤ N = 16) to ensure betterstatistics. The J ⊥ = 0 limit We consider first the J ⊥ = 0 limit where we studynumerically the order parameter (cid:104) SG ( (cid:15) ) (cid:105) to confirm theexistence and transition between the paramagnetic andspin-glass phases. In this limit the XXZ model becomesthe classical Ising model with random magnetic field on z − direction where a much larger number of random fieldconfigurations can be studied. We fix h = 1 and con-sider different values of J z and M = 8 , , (cid:104) SG ( (cid:15) ) (cid:105) by averaging over ∼ disorder samples in ourstudy. (a) J z = 1 .
25 (b) J z = 1 . J z = 2 .
25 (d) J z = 5 FIG. 1. (cid:104) SG ( (cid:15) ) (cid:105) for the random field ising model with systemsize N = 16 , M = 8 and h = 1. The increase in magnitude ofoscillations for increasing J z = 1 . , . , . , Our results for M = 8 , J z = 1 . , . , . , (cid:104) SG ( (cid:15) ) (cid:105) de-pends on the density of states at energy (cid:15) . In the large J z limit, the oscillation in (cid:104) SG ( (cid:15) ) (cid:105) with changing energy (cid:15) is clear. The oscillation vanishes for small J z .The magnitude of oscillation M SG = difference be-tween maximum and minimum of (cid:104) SG ( (cid:15) ) (cid:105) ) indicates thestrength of spin-glass order and is used to estimate thetransition point between the paramagentic and spin-glassphases. We show M SG for different values of M and J z ≥ . (cid:104) SG ( (cid:15) ) (cid:105) >
0) in Fig.(2). M SG is fittedto a function M SG = A ( J z − J z,Critical ) β (3)to determine the critical point of transition J z,Critical .We find J z,Critical ∼ . ± .
05 for M = 8. Similaranalysis is also carried out for M = 4 ,
6. The spin-glassorder is calculated using (cid:104) S z S zN (cid:105) − (cid:104) S z (cid:105)(cid:104) S zN (cid:105) as (cid:104) S zi (cid:105) (cid:54) = 0in these cases where the system is magnetized. We find J z,Critical ∼ . , .
33 for M = 6 ,
4, respectively.
FIG. 2. The magnitude of oscillation (cid:104) SG ( (cid:15) ) (cid:105) as a function of J z /h for M = 8 , , N = 16). We include only J z > . (cid:104) SG ( (cid:15) ) (cid:105) is found to be nonzero. We note that the critical point J z,critical is increasingwhen the system moves away from half-filling, indicatingthat the effect of J z (or interaction) on spin-glass order ismost prominent at M = 8 and weakens when the systemmoves towards smaller value of M . We emphasize oncemore that the many-body states are localized in bothcases, i.e., the paramagnetic and spin-glass phases belongto two different classes of MBL states. J ⊥ (cid:54) = 0 - existence of an ergodic state between the twoMBL phases We next consider the XXZ model with non-zero J ⊥ .We first consider J ⊥ = 0 .
25 ( h = 1) and shall perform ourcalculations on a system with size N = 16 for differentvalues of M = 3 , , ..., J z .Generally speaking J ⊥ drives the system away fromlocalization into ergodic phases. The exchange of spinsleads to delocalization (+ creation and destruction) ofdomain walls leading to the weakening of spin-glass or-der in the spin-glass phase and to delocalization of spinsdirectly in the paramagnetic phase. Numerically, we con-firm that spin-glass order at the strong interaction andstrong disorder J z , h >> J ⊥ limit is much weakened withthe introduction of J ⊥ (cid:54) = 0. (see Fig.(3)). However, wehave difficulty locating the point of transition betweenthe paramagnetic and spin-glass phases because we can-not arrive at enough numerical accuracy with the smallnumber of disorder samples we considered.We next examine the energy-level statistics. We firstconsider weak interaction J z = 0 .
5. In Fig.(4), we plot (cid:104) r ( (cid:15) ) (cid:105) for different values of (cid:15) and M . We note thatsince both J ⊥ and J z are much smaller than h we ex-pect that the system should remain at the paramag- FIG. 3. The oscillation of spin-glass order for a system with N = 16 , M = 8 , J z = 5 , J ⊥ = 0 .
25 and h = 1. (150 ensem-bles) netic MBL state where all states are uncorrelated (i.e. (cid:104) r ( (cid:15) ) (cid:105) ∼ .
38) through out this region. Instead, we findsignificant derivation from Poisson statistics behavior inFig.(4) with (cid:104) r ( (cid:15) ) (cid:105) ∼ .
47 centered around the region M = 8 and (cid:15) = 0 .
5, indicating that correlations betweenclose-by energy levels is builing up around this region.
FIG. 4. The ratio of consecutive level spacing (cid:104) r ( (cid:15) ) (cid:105) for vari-ous (cid:15) and M with N = 16 , J z = 0 . , J ⊥ = 0 . , h = 1 To understand this behavior better we focus ourselvesat (cid:15) = 0 . (cid:104) r ( (cid:15) ) (cid:105) for different values of M and J z at the fixed energy window (cid:15) = 0 . J ⊥ = 0 .
25 and h = 1. FIG. 5. The ratio of consecutive level spacing (cid:104) r ( (cid:15) ) (cid:105) with N = 16 , J ⊥ = 0 . , h = 1 and (cid:15) = 0 .
5. The black dash lineindicates J z,critical ’s determined in the J ⊥ = 0 limit. Interestingly, we find that similar behavior exists, withthe derivation of energy levels from uncorrelated (Pois-son) behavior strongest at the point J z ∼ h , M = 8,the energy levels go back to uncorrelated behavior bothfor stronger and weaker values of interaction J z andwhen M deviates from 8. The corresponding value of (cid:104) r ( (cid:15) ) (cid:105) is around 0 .
48 at the strongest non-Poisson regime M = 8 , J z ∼ h . The black dash line indicates J z,critical ’sdetermined in the J ⊥ = 0 limit.To further understand this behavior we repeat the cal-culations with different values of J ⊥ = 0 . . J ⊥ increases, with (cid:104) r ( (cid:15) ) (cid:105) moving to-wards 0 .
53 at M = 8 , J z ∼ h for J ⊥ = 0 . J ⊥ decreases. Thecritical region is close to the phase transition line separat-ing the paramagnetic and spin-glass MBL phases deter-mined in the J ⊥ = 0 limit, suggesting that a new ergodicphase is emerging at the critical region between the twodifferent MBL phases when J ⊥ becomes nonzero. Weshall call it a Many-Body Ergodic phase as it is not an-alytically connected to the trivial ergodic phase at weakdisorder, weak interaction limit (where M is arbitraryand h (cid:54) = J z in general) and appears only at around aregion with J z ∼ h and M ∼ N/ (a) J ⊥ = 0 . J ⊥ = 0 . FIG. 6. The ratio of consecutive level spacing (cid:104) r ( (cid:15) ) (cid:105) with N = 16 , J ⊥ = 0 . , . , h = 1 and (cid:15) = 0 .
5. The black dashline indicates J z,critical ’s determined in the J ⊥ = 0 limit. To see whether this critical regime is really approach-ing an ergodic phase, we also examine the EntanglementEntropy (EE). In the 1D XXZ model we consider here,it is expected that in the limit of large system size N, EEshould scale as N in the ergodic phase, and is independentof N in the MBL phase. In Fig.(7), we show the EE fordifferent values of J z for system size N = 8 , , , , M = N/ h = 1 in our calculation. Foreach random field configuration, the EE is computed for P eigenstates with energy closest to the middle of theenergy spectrum. The result is then averaged over the P eigenstates and sampled over Q random field configu-rations. The values of P and Q are determined by theconvergence of the calculated value of EE and their val-ues for different system size N are shown in Table.I. We note from Fig.(7) that the entanglement entropy reachesmaximum around J z ∼ h and becomes smaller when J z moves away from h , consistent with our energy level spac-ing analysis that an ergodic state is approached when J z ∼ h . However, we are not able to obtain the expectedscaling behaviours both in the ergodic and MBL phases,suggesting that the sizes of the system we considered arestill too small [3]. N P
20 50 50 100 500 Q
500 500 500 100 10TABLE I. The number of states P and random field configu-rations Q used in calculation of EE for each system size N .FIG. 7. The log-log plot of the Average Entanglement En-tropy of (cid:15) ∼ . J z . We fix J ⊥ = 0 . , h = 1 Discussion
We study in this paper the strongly-disordered regime of the 1D spin-1 / z -direction with an aim toinvestigate the transition between the paramagnetic andspin-class phases. Surprisingly, we find that an ergodicphase emerges at a narrow region between the param-agnetic and spin-glass phases at half-filling ( M = N/ (cid:15) = 0 . J z ∼ h .We call it a Many-Body Ergodic phase as it is not adi-abatically connected to the ”trivial” ergodic phase inthe weak-disorder, weak-interaction limit. This is, toour knowledge, the first time where an emergent ergodicphase between two MBL phases is discovered in exactdiagonalization study. It should be emphasized that ourfinding should be considered as preliminary at this stage.The limitation of our computer power has forced us towork on relatively small size systems ( N ≤
16) where wewere not able to perform a convincing scaling analysis tothe N → ∞ limit. As a result we are not able to con-clude with certainty whether the ergodic phase survivesin the thermodynamic limit and we are also not able topin down precisely whether the ergodic phase is a crit-ical phenomenon that occurs at the boundary betweenthe paramagnetic and spin-glass MBL phases. Large sizecomputer simulations are needed to provide better an-swers to these questions.Theoretically, it should be pointed out that renormal-ization group analysis has suggested that there were noergodic phase in the critical region between the para-magnetic and spin-glass phases on the disordered 1Dtransverse-field Ising model [27, 28]. However, the pos-sibility of an emerging ergodic phase between differentMBL phases has not been ruled out, in particular whenthe MBL phases belong to different topological classes[2, 29–31] as in the present case where the liom in theparamagnetic phase and spin-glass phase are topolog-ically different (single spins and domain walls, respec-tively). The prevailing view of a MBL → ergodic phasetransition is that it is driven by the emergence of reso-nant clusters. As the strength of randomness is reduced,resonant clusters start to occur more frequently in thesystem. At the critical disorder strength, a critical clus-ter grows to encompass the entire system, driving thesystem into an ergodic phase. Our finding, if correct,suggests a different mechanism of forming ergodic phasewhen the system is close to the phase boundary betweendifferent MBL phases that belong to different topologocalclasses, even in the strong disorder limit. Further theo-retical analysis is needed to understand this phenomenon. acknowledgement This work is supported by Hong Kong RGC throughgrant HKUST3/CRF/13G. [1] R. Nandkishore and D. A. Huse, Annual Review of Con-densed Matter Physics , 15 (2015).[2] D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev.Mod. Phys. , 021001 (2019).[3] D. J. Luitz, N. Laflorencie, and F. Alet, Phys. Rev. B , 081103 (2015).[4] D. A. Huse, R. Nandkishore, and V. Oganesyan, Phys.Rev. B , 174202 (2014).[5] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux,Phys. Rev. Lett. , 084101 (2013). [6] Y. Avishai, J. Richert, and R. Berkovits, Phys. Rev. B , 052416 (2002).[7] K. Kudo and T. Deguchi, Phys. Rev. B , 132404(2004).[8] B. Georgeot and D. L. Shepelyansky, Phys. Rev. Lett. , 5129 (1998).[9] V. Oganesyan and D. A. Huse, Phys. Rev. B , 155111(2007).[10] B. Bauer and C. Nayak, Journal of Statistical Mechanics:Theory and Experiment , P09005 (2013).[11] J. A. Kj¨all, J. H. Bardarson, and F. Pollmann, Phys.Rev. Lett. , 107204 (2014).[12] D. N. Page, Phys. Rev. Lett. , 1291 (1993).[13] E. Altman and R. Vosk, Annual Review of CondensedMatter Physics , 383 (2015).[14] S. P. Lim and D. N. Sheng, Phys. Rev. B , 045111(2016).[15] V. Khemani, S. P. Lim, D. N. Sheng, and D. A. Huse,Phys. Rev. X , 021013 (2017).[16] M. ˇZnidariˇc, T. c. v. Prosen, and P. Prelovˇsek, Phys.Rev. B , 064426 (2008).[17] M. Serbyn, Z. Papi´c, and D. A. Abanin, Phys. Rev. Lett. , 260601 (2013).[18] R. Vosk and E. Altman, Phys. Rev. Lett. , 067204(2013).[19] J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys.Rev. Lett. , 017202 (2012).[20] T. L. M. Lezama, S. Bera, H. Schomerus, F. Heidrich-Meisner, and J. H. Bardarson, Phys. Rev. B , 060202(2017).[21] S. Bera, T. Martynec, H. Schomerus, F. Heidrich-Meisner, and J. H. Bardarson, Annalen der Physik ,1600356 (2017).[22] J. Z. Imbrie, V. Ros, and A. Scardicchio, Annalen derPhysik , 1600278 (2017).[23] L. Rademaker, M. Ortuo, and A. M. Somoza, Annalender Physik , 1600322 (2017).[24] S. Bera, H. Schomerus, F. Heidrich-Meisner, and J. H.Bardarson, Phys. Rev. Lett. , 046603 (2015).[25] A. Pal and D. A. Huse, Phys. Rev. B , 174411 (2010).[26] M. Serbyn, Z. Papi´c, and D. A. Abanin, Phys. Rev. X , 041047 (2015).[27] D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal, andS. L. Sondhi, Phys. Rev. B , 014206 (2013).[28] D. Pekker, G. Refael, E. Altman, E. Demler, andV. Oganesyan, Phys. Rev. X , 011052 (2014).[29] T. B. Wahl and B. Bri, (2020), arXiv:2001.03167.[30] Y. Bahri, R. Vosk, E. Altman, and A. Vishwanath, Na-ture Communications , 7341 (2015).[31] S. A. Parameswaran and R. Vasseur, Reports on Progressin Physics81