Critical Level Statistics at the Many-Body Localization Transition Region
CCritical Level Statistics at the Many-Body Localization Transition Region
Wen-Jia Rao ∗ School of Science, Hangzhou Dianzi University, Hangzhou 310027, China. (Dated: March 2, 2021)We study the critical level statistics at the many-body localization (MBL) transition region in random spinsystems. By employing the inter-sample randomness as indicator, we manage to locate the MBL transition pointin both orthogonal and unitary models. We further count the n -th order gap ratio distributions at the transitionregion up to n = 4 , and find they fit well with the short-range plasma model (SRPM) with inverse temperature β = 1 for orthogonal model and β = 2 for unitary. These critical level statistics are argued to be universalby comparing results from systems both with and without total S z conservation. We also point out that thesecritical distributions can emerge from the spectrum of a Poisson ensemble, which indicates the thermal-MBLtransition point is more affected by the MBL phase rather than thermal phase. I. INTRODUCTION
The non-equilibrium phases of matter in isolated quantumsystems is a focus of modern condensed matter physics, itis now well-established the existence of two generic phases:a thermal phase and a many-body localized (MBL) phase .Physically, a thermal phase is ergodic with extended and fea-tureless eigenstate wavefunctions, which results in a corre-lated eigenvalue spectrum with level repulsion. In contrary, inMBL phase localization persists in the presence of weak inter-actions. Modern understanding about these two phases relieson quantum entanglement. In thermal phase, the system actsas the heat bath for its subsystem, hence the entanglement isextensive and exhibits ballistic (linear in time) spreading afterquantum quench. In contrast, the absence of thermalization inMBL phase leads to small (area-law) entanglement and slow(logarithmic) entanglement spreading. The qualitative differ-ence in the scaling of quantum entanglement and its dynamicsafter quantum quench are widely used in the study of thermal-MBL transition .More traditionally, the thermal phase and MBL phase isdistinguished by their eigenvalue statistics, whose theoret-ical foundation is provided by the random matrix theory(RMT) . RMT is a powerful mathematical tool that de-scribes the universal properties of a complex system thatdepend only on its symmetry while independent of micro-scopic details. Specifically, the Gaussian orthogonal ensem-ble (GOE) describes systems with spin rotational and time re-versal symmetry; the Gaussian unitary ensemble (GUE) cor-responds to those with spin rotational invariance and brokentime reversal symmetry; and Gaussian symplectic ensemble(GSE) refers to systems which conserve time reversal sym-metry while break spin rotational invariance. It is well estab-lished that in the thermal phase with correlated eigenvalues,the distribution of nearest level spacings { s i = E i +1 − E i } will follow a Wigner-Dyson distribution with Dyson index β = 1 , , for GOE,GUE,GSE respectively. On the otherhand, in MBL phase with uncorrelated eigenvalues, P ( s ) isexpected to follow Poison distribution. The difference in thelevel spacing distribution is also widely-used in the study ofMBL systems .Compared to the properties of each phase, the nature of the thermal-MBL transition is much less understood. Manyworks on one-dimensional MBL system indicate the existenceof Griffiths regime near the transition point, where the sys-tem becomes an inhomogeneous mixture of locally thermaland localized regions. Consequently, the system’s dynamicsbecome anomalously slow and eigenstates exhibits multifrac-tality. However, this regime is not free of uncertainties, and aunified theory has not been established by now .Despite the lack of understanding about the thermal-MBLtransition, there are a number of effective models proposed forthe critical level statistics at the transition point. For example,the Rosenzweig-Porter model , mean field plasma model ,short-range plasma models (SRPM) and its generalization –the weighed SRPM , Gaussian β ensemble and the gener-alized β − h model , and others . In this work, we willfocus on the SRPM, whose formal definition will be given inSec. II. Historically, SRPM is introduced as a RMT modelthat holds the semi-Poisson level statistics, which is an inter-mediate statistics between GOE and Poisson that close to theone found numerically at the critical point of Anderson metal-insulator transition . As for the MBL transition, SRPM withinverse temperature β = 1 has been shown to describe thenearest level spacing distribution at critical region well, whileits effectiveness in describing long-range level correlations isdebated . In this work, we will study the higher-order levelspacings that incorporate level correlations on longer ranges,and show the SRPM is indeed a good effective model for thecritical region, at least when level correlations on moderateranges are concerned.Besides, current works on the thermal-MBL transition aremostly dealing with orthogonal systems, whose correspond-ing RMT description is GOE to Poisson. It is natural to askwhat’s the critical spacing distributions in a unitary system,and what’s the corresponding effective model. Given the RMTdescription for MBL transition in a unitary system is GUE toPoisson, a natural candidate for the effective model would bethe SRPM with inverse temperature β = 2 . It is the secondpurpose of current work to verify this guess.In this paper, we study the level statistics in the thermal-MBL transition region of 1D random spin systems, our anal-ysis relies solely on the energy spectrum. By using the inter-sample randomness as the “order parameter”, we quantita-tively locate the transition points, which are in well-agreementwith previous results based on eigenstate properties. We fur- a r X i v : . [ c ond - m a t . d i s - nn ] F e b ther count the n -th order level correlations in the transition re-gion up to n = 4 , and verify they fit well with those of SRPMwith inverse temperature β = 1 for orthogonal model and β = 2 for unitary, and these critical behaviors are expected tobe universal by comparing results from models both with andwithout total S z conservation. We also discuss how the SRPMcan emerge from the eigenvalue spectrum of the MBL phase,indicating the thermal-MBL transition point is more affectedby the MBL phase rather than thermal phase.This paper is organized as follows. In Sec. II we introducethe spin model and SRPM. In Sec. III we focus on the or-thogonal models, and unitary models are studied in Sec. IV.Conclusion and discussion come in Sec. V. II. MODEL AND METHOD
We will study the “standard model” for MBL physics, i.e.,the anti-ferromagnetic Heisenberg model with random exter-nal fields, whose Hamiltonian is H = L (cid:88) i =1 S i · S i +1 + L (cid:88) i =1 (cid:88) α = x,y,z h α ε αi S αi , (1)where S i is spin- / operators. The anti-ferromagnetic cou-pling strength is set to be , and periodic boundary conditionis assumed in the Heisenberg term. The ε αi s are random vari-ables within range [ − , , and h α is referred as the random-ness strength. This Hamiltonian’s property depends on theexternal fields: when they are non-zero in only one or twospin directions, the model is orthogonal; while when all ofthem are non-zero, the model is unitary. In all cases, the sys-tem will undergo a thermal-MBL transition with increasingrandomness, and the corresponding RMT description is GOE(GUE) to Poisson in the orthogonal (unitary) case.To describe the level statistics, we choose to study the dis-tributions of the nearest gap ratios, whose definition is r i = s i +1 s i ≡ E i +2 − E i +1 E i +1 − E i . (2)Compared to the more traditional quantity of level spacings { s i = E i +1 − E i } , the gap ratios { r i } have two major advan-tages: (i) unlike level spacings, P ( r ) is independent of den-sity of states (DOS), hence requires no unfolding procedure,which is non-unique and may raise subtle misleading signa-tures when studying the long-range level correlations in somesystems ; (ii) counting P ( s ) requires an additional normal-ization for (cid:104) s (cid:105) , while counting P ( r ) does not. Actually, themean value (cid:104) r (cid:105) can be a measure to distinguish phases, as hasbeen adopted in many recent works.The analytical form of P ( r ) for the thermal phase has beenderived in Ref. [38] using a Wigner-like surmise, which gives P ( β, r ) = Z β (cid:0) r + r (cid:1) β (1 + r + r ) β/ , (3)where the Dyson index β = 1 , , stands for GOE,GUE,GSErespectively, and Z β is a normalization factor determined by (cid:82) ∞ P ( β, r ) dr = 1 . The gap ratio can be generalized tohigher order to describe level correlations on longer ranges,whose definition is r ( n ) i = E i +2 n − E i + n E i + n − E i , (4)and the corresponding distribution is P (cid:16) β, r ( n ) = r (cid:17) = P ( γ, r ) , (5) γ = n ( n + 1)2 β + n − .On the other hand, for the MBL phase with uncorrelated en-ergy spectrum, we have P (cid:16) r ( n ) = r (cid:17) = r n − (1 + r ) n . (6)As for the spectral statistics at the thermal-MBL transitionregion, a number of effective models have been proposed, andin this work we will focus on the short-range plasma model(SRPM). The SRPM describes the eigenvalues of a randommatrix ensemble as an ensemble of one-dimensional systemof classical particles with two-body repulsive interactions,whose distribution can be written into a canonical ensembleform P β ( { E i } ) = Z − β e − βH ( { E i } ) , (7) H ( { E i } ) = (cid:88) i U ( E i ) + (cid:88) | i − j |≤ k V ( | E i − E j | ) , (8)where U ( E i ) ∝ E i is the trapping potential and the Dysonindex β is interpreted as the inverse temperature. Thetwo-body interaction takes the logarithmic form V ( x ) = − log | x | , and k is the interaction range. It’s easy to see the k → ∞ limit corresponds to the standard Gaussian ensem-bles for thermal phase; while in k → limit no interaction ispresent, which corresponds to the Poisson ensemble with nolevel correlation; the thermal-MBL transition is thus reflectedby the evolution of the interaction range k . Unlike the mean-field plasma model, which is also suggested to describe thecritical spectral statistics , it is the interaction range ratherthan the interaction form that changes during the thermal-MBL transition.One major advantage of SRPM is that it is exactly solv-able, and the general form of n -th order level spacing distri-bution has been derived in Ref. [30]. Notably, for the simplestcase with β = k = 1 , the nearest level spacings { s i } fol-lows the semi-Poisson distribution, which is close to the onefound numerically at the MBL transition region in an orthog-onal spin model . In this work, we will proceed to studythe higher-order gap ratios (cid:110) r ( n ) i (cid:111) in SRPM that incorporatelevel correlations on longer ranges. Unlike the more tradi-tional quantities such as number variance Σ , the higher-ordergap ratios are numerically easier to obtain and require no un-folding procedure hence avoid the potential ambiguity raisedby concrete unfolding strategy .First of all, we need to get the expression of P (cid:0) r ( n ) (cid:1) forthe SRPM, which is not an easy task since a Wigner-like sur-mise is not applicable due to the limited interaction range inEq. (8). However, we can make use of an elegant correspon-dence between the SRPM and the “reduced energy spectrum”of Poisson ensemble, whose idea goes as follows.Formally, a r -th order reduced energy spectrum (cid:110) E ( r ) i (cid:111) iscomprised of every ( r + 1) -th level of the original spectrum { E i } , which is mathematically achieved by tracing out every r levels in between. This construction is very similar to thatof the reduced density matrix where we trace out the degreesof freedom in a subsystem, hence we suggest to call (cid:110) E ( r ) i (cid:111) the “reduced energy spectrum” . It is proved in Ref. [43] thatthe energy spectrum of SRPM with k = 1 and inverse temper-ature β has the same structure as the β -th order reduced en-ergy spectrum of a Poisson ensemble (which is named “Daisymodel” by the authors). By this mapping, the n -th eigen-value in the SRPM with inverse temperature β becomes the n ( β + 1) -th level in the Poisson ensemble, and the n -th or-der gap ratio in the former is mapped to the n ( β + 1) -th or-der counterpart in the latter, whose distribution is then easilywritten down according to Eq. (6), that is P (cid:16) β, r ( n ) = r (cid:17) = r γ − (1 + r ) γ , γ = n ( β + 1) . (9)In the next sections, we will use Eq. (9) with β = 1 ( ) to fitthe critical level statistics in orthogonal (unitary) model. Be-sides, by comparing results from models both with and with-out total S Z conservation, we argue that this effective modelis universal that independent of microscopic details. III. ORTHOGONAL MODELS
We start by studying the orthogonal models in Eq. (1). Wefirst consider the case that h x = h z = h (cid:54) = 0 and h y = 0 . Thischoice breaks total S Z conservation and makes the eigenstatesin thermal phase fully featureless, hence is less affected byfinite-size effect. The MBL transition point is, according toprevious studies , h c (cid:39) . Note that, although the pureHeisenberg chain has different energy spectrums in systemsof even and odd lengths, the difference is wiped out by therandom external fields. In this work, we study up to L = 13 system with Hilbert space dimension N = 2 = 8192 .To get an intuitive picture of the gap ratio’s evolution, wenumerically simulate Eq. (1) in a L = 13 system in the ran-domness range h ∈ [1 , , with samples taken at eachrandomness strength. For each energy spectrum sample, weselect eigenvalues in the middle to determine P ( r ) , andthe results are displayed in Fig. 1(a). As we can see, whenthe randomness is small ( h = 1 ), P ( r ) meets perfectly withthe prediction for GOE; when randomness increases, P ( r ) starts to deform, and finally reach to the Poisson distributionfor MBL phase ( h = 5 ). We note the fittings for h = 5 hasminor deviations from ideal Poisson, this is due to finite-size r = s i +1 / s i P ( r ) (a) Orth., L=13 GOEPoissonh=1h=2h=3h=4h=5 h V S ×10 (b) L=11L=12L=130 1 2 3 4 r ( n ) P ( r ( n ) ) (c)L=13, h=3.2 =2=4=6=8 r (1) r (2) r (3) r (4) r ( n ) P ( r ( n ) ) (d)L=13, h=3 =2=4=6=8 r (1) r (2) r (3) r (4) FIG. 1. (a) The evolution of gap ratio distribution in an L = 13 orthogonal system, P ( r ) evolves from GOE to Poisson when thesystem is evolved from thermal ( h = 1 ) to MBL phase ( h = 5 ).(b) The evolution of inter-sample variance V S as a function of h , insystems with different sizes. Both the peak position and value of V S are larger in larger system, indicating a larger Giffiths regime.(c) P (cid:16) r ( n ) (cid:17) at the estimated transition point h = 3 . in an L =13 system, non-negligible deviations from SRPM are found due toresidue correlations raised by finite-size effect. (d) P (cid:16) r ( n ) (cid:17) at L =13 and h = 3 , perfect matches with SRPM ( β = 1 ) are observed. effect, since in a finite system there will always remain expo-nentially decaying but finite correlations between eigenstates.From the evolution of P ( r ) in Fig. 1(a), we can have a qual-itative estimation about the location of MBL transition point.To be specific, take a closer look at P ( r ) at h = 3 (the greendots in Fig. 1(a)), we see it lies roughly at the middle betweenGOE and Poisson, which indicates h = 3 is close to the tran-sition point.To more quantitatively locate the transition point, we adopta variant definition of gap ratio, which is t i = min { s i +1 , s i } max { s i +1 , s i } (10)where s i = E i +1 − E i is the i -th energy gap. This isactually the original definition of gap ratio introduced byOganesyan and Huse . Compared to r i , t i takes valuesin the range (0 , , and their distributions are related by P ( t ) = 2 P ( r ) Θ (1 − r ) . The mean value of gap ratio t can be easily calculated from Eq. (3), namely t GOE = 0 . , t GUE = 0 . and t Poisson = 0 . . Technically, the calcu-lation of t has two steps: first we calculate the mean gap ra-tio value in one sample , which gives t S = (cid:104) t i (cid:105) samp , then weaverage t S over an ensemble of samples to get t = (cid:104) r S (cid:105) en .These two steps give two types of variance, the first one is V S = (cid:104) t S − t (cid:105) en , i.e. the variance of sample-averaged gap ra-tio over ensemble, which measures the inter-sample random-ness ; another one is V I = (cid:104) v I (cid:105) en where v I = (cid:104) t i − t S (cid:105) samp ,which is the ensemble-averaged gap ratio variance and mea-sures the intrinsic intra-sample randomness . In a systemdriven by pure random disorder (that is, opposite to the onesinduced by quasi-periodic potential ), the distribution of t S near the transition region will exhibit strong deviation from aGaussian type – a manifestation of Griffiths region – whichresults in a peak value of V S at the transition point . There-fore, for our model Eq. (1), we can calculate the evolution of V S to locate the transition point.Strictly speaking, the transition point identified by V S andother quantities based on quantum entanglement may not al-ways coincide in a finite system , meanwhile, V S in essence de-scribes a qualitative structural change in the energy spectrum,hence is more suitable for our purpose to study the criticallevel statistics.In Fig. 1(b) we draw the evolution of V S in systemswith different lengths, where the number of samples are , , for L = 11 , , , respectively. In allcases, expected peaks of V S appear. We see that, in general,both the detected transition point and peak value V S are largerin larger system, which indicates a larger Griffiths regime,in consistence with the results in orthogonal model with S Z conservation .Now we are ready to count the level statistics at the tran-sition region. In a finite system, what we observe is alwaysa combination of universal part and non-universal (model de-pendent) part, we therefore choose the largest system we canreach to minimize the finite-size effects. That is, L = 13 for systems without S Z conservation, and L = 16 for thosewith S Z conservation. As for the present model with L = 13 ,the detected transition point is, according to Fig. 1(b), h c (cid:39) . ± . .First we take out the samples at the identified transitionpoint h = 3 . , and determine the corresponding gap ratio dis-tributions P (cid:0) r ( n ) (cid:1) up to n = 4 , the results are displayed inFig. 1(c), where the reference curves are the ones for SRPMin Eq. (9) with β = 1 . As we can see, the fittings have non-negligible deviations. We attribute this to the finite systemsize we are studying. That is, in a finite system, the eigen-states even in the MBL phase remain an exponentially decay-ing but finite correlations, hence the randomness required todrive the phase transition is slightly larger than it really needsin thermodynamic limit, which is in agreement with our anal-ysis for Fig. 1(a). Therefore, the true critical level statistics isexpected to occur in a point slightly smaller than h = 3 . . Tothis end, we take out the samples from h = 3 and count thecorresponding P (cid:0) r ( n ) (cid:1) , the results are in Fig. 1(d). As can beseen, they fit quite well with the SRPM, confirming the SRPMis indeed a good effective model, at least when level correla-tions on moderate ranges (up to levels) are concerned. To becomplete, we have checked the same situations happen in the L = 12 system.To show this critical distribution is universal, we consideranother orthogonal model, that is, the one with h z = h (cid:54) =0 , h x = h y = 0 in Eq. (1). This is actually the one mostwidely studied in the literature since it preserves total S z , andallows one to reach to larger system size by focusing on onesector, which is commonly chosen to be the one with S TZ = 0 .Technically, this also requires the number of spins L to beeven. However, eigenstates in this sector share one common feature, i.e. S TZ = 0 , which violates the featureless prop-erty of a fully thermalized state. Therefore, the eigenstates inthis sector is easier to be localized, which results in a largelarge finite-size effect, and the estimated transition point ismuch less smaller than the interpolated value in thermody-namic limit. Actually, it’s widely accepted the transition pointis around h c (cid:39) . for the middle part of energy spectrum,while in a finite system, say L = 16 , the detected transitionpoint is shifted to h c (cid:39) . .In our study, we take the system size to be L = 16 andfocus on the S TZ = 0 sector, whose Hilbert space’s dimensionis N = C = = 12870 . Like before, we first presenta qualitative picture for the gap ratio’s evolution in the range h ∈ [1 , , with samples taken at each point, the resultsare displayed in Fig. 2(a), we see a GOE-Poisson evolutionas expected. Then we numerically determine the evolution ofinter-sample randomness V S , which is presented in Fig. 2(b).The observed peak indicates a transition at h = 2 . ± . , inwell accordance with the previous studies in this system size.Next we consider the critical statistics. With the same reasonas for previous model, we take the samples from h = 2 . ,slightly smaller than the estimated one, and the corresponding P (cid:0) r ( n ) (cid:1) are displayed in Fig. 2(c). As can be seen, they fitquite good with the prediction of SRPM with β = 1 .Up to now, we have confirmed the SRPM with k = β = 1 isa quite good effective model for the critical spectral statisticsin an orthogonal model, not only for nearest-neighbor gap ra-tios, but also for several higher-order ones that describe levelcorrelations on longer ranges, and this model is expected to beuniversal that independent of microscopic details. In the nextsection, we will proceed to study the unitary model. IV. UNITARY MODELS
Now we study the critical level statistics in unitary mod-els, we will show it is well described by SRPM with k = 1 and β = 2 . Like before, we first consider the case without S TZ conservation, that is, the model Eq. (1) with h x = h y = h z = h (cid:54) = 0 . Likewise, we work on an L = 13 system, thequalitative evolution of P ( r ) is given in Fig. 3(a), a GUE-Poisson evolution is observed when increasing randomness asexpected. From the evolution, we can qualitatively see thetransition point lies at somewhere between h = 2 and h = 3 .Next, we calculate the evolution of inter-sample randomness V S , the result is given in Fig. 3(b). We see an expected peakindicating the transition point is h c (cid:39) . ± . , close to h c (cid:39) . got by previous studies .Next, we are considering the critical level statistics. Withthe same reason as in previous section, we take a point slightlyleft to the estimated one, that is h = 2 . , and the correspond-ing P (cid:0) r ( n ) (cid:1) are presented in Fig. 3(c). As can been seen,they fit very well with the predictions of SRPM with k = 1 and β = 2 , which provides a strong evidence that the SRPMis a good effective model. r = s i + 1 / s i P ( r ) (a) Orth. S TZ = 0L=16 GOEPoissonh=1h=2h=3h=4h=5 h V S ×10 (b) r ( n ) P ( r ( n ) ) (c) L=16 S TZ = 0h=2.4 = 2= 4= 6= 8 r (1) r (2) r (3) r (4) FIG. 2. (a) The evolution of gap ratio distribution P ( r ) in the S TZ = 0 sector of the L = 16 orthogonal model, an expected GOE-Poissontransition is found. (b) The evolution of V S , which indicates the transition point is h c = 2 . ± . . (c) P (cid:16) r ( n ) (cid:17) at h = 2 . , a perfect matchwith SRPM ( β = 1 ) is observed, indicating this effective model is universal. r = s i + 1 / s i P ( r ) (a) Unit.L=13 GUEPoissonh=1h=2h=3h=4h=5 h V S ×10 (b) r ( n ) P ( r ( n ) ) (c) L=13h=2.6 = 3= 6= 9= 12 r (1) r (2) r (3) r (4) FIG. 3. (a) The evolution of P ( r ) in an L = 13 unitary system, a GUE-Poisson transition is observed as expected. (b) The evolution ofinter-sample randomness V S , and the shaded area indicates the transition region h c = 2 . ± . . (c) P (cid:16) r ( n ) (cid:17) at h = 2 . , a good match withSRPM ( β = 2 ) is observed. To further show this critical behavior in the unitary modelis also universal, we study a unitary spin model with to-tal S Z conservation, which is constructed by adding a time-reversal breaking next-nearest neighboring interaction term tothe Heisenberg model, the Hamiltonian then reads H = L (cid:88) i =1 [ S i · S i +1 + J S i · ( S i +1 × S i +2 )] + h L (cid:88) i =1 ε zi S zi .(11)This model was introduced to generate the GUE statistics inRef. [15], it was also pointed out the level statistics almost im-mediately changes from GOE to GUE even when J is as smallas . . The thermal-MBL transition point in this modelcertainly depends on J : in general, the larger J , the larger h c will be. In this work, we choose J = 0 . without loss ofgenerality, and focusing on the S TZ = 0 sector in an L = 16 system.The qualitative evolution of P ( r ) is given in Fig. 4(a), aGUE-Poisson evolution when increasing randomness h is ob-served as expected. Next, we calculate the evolution of inter-sample randomness V S , the result is presented in Fig. 4(b).We see an expected peak indicating the transition point is h c (cid:39) . ± . . Interestingly, this coincides with the onein Fig. 3(b), which is purely accidental for the J = 0 . wechoose. Actually, the values of V S in Fig. 4(b) are much larger than those in Fig. 3(b), which means the inter-sample random-ness is generally larger in this model, hence the finite-size ef-fect is expected to be more serious.Next, we are considering the critical statistics. Like before,we take a point slightly smaller than the estimated one, thatis h = 2 . , and the corresponding P (cid:0) r ( n ) (cid:1) are presented inFig. 4(c). As can been seen, they qualitatively meets the pre-dictions of SRPM with k = 1 and β = 2 , but the deviationsare larger than those in Fig. 3(c). We attribute this to the next-nearest interaction term that destroys the integrability of thepure model, which strengthens the thermal phase in the ran-dom model and results in a more serious finite-size effect, inaccordance with our analysis about Fig. 4(b). In fact, if weartificially allow the inverse temperature to be a fraction, wefind the P (cid:0) r ( n ) (cid:1) in Fig. 4(c) can be well fitted into β = 1 . (the dotted lines in Fig. 4(c)), which is close to the expectedvalue β = 2 .To conclude, we have shown the spectral statistics in thetransition region of a unitary model without S TZ conservationis well described by the SRPM with k = 1 and β = 2 , whichis a natural extension of the one for the orthogonal system.The results from unitary model with S TZ conservation suggeststhis critical behavior is also universal for unitary model, al-though the deviations are slightly larger. We suggest a futurework on larger system to confirm this conclusion. r = s i + 1 / s i P ( r ) (a) Unit. S TZ = 0L=16 GUEPoissonh=1h=2h=3h=4h=5 h V S ×10 (b) r ( n ) P ( r ( n ) ) (c) L=16 S TZ = 0h=2.6 = 3= 6= 9= 12 r (1) r (2) r (3) r (4) FIG. 4. (a) The evolution of P ( r ) in the S TZ = 0 sector of the L = 16 unitary model with the next-nearest neighboring interaction strength J = 0 . , an expected GUE-Poisson transition is found. (b) The evolution of V S , which indicates the transition region is h c = 2 . ± . . (c) P (cid:16) r ( n ) (cid:17) at h = 2 . , minor deviations from SRPM with β = 2 are observed, which may be attributed to the large finite-size effect inducedby the next-nearest neighboring interaction that destroys the integrability of pure Hamiltonian. The dotted lines: SRPM with β = 1 . . V. CONCLUSION AND DISCUSSION
We have studied the thermal-MBL transition in both or-thogonal and unitary models in random spin systems. By us-ing the inter-sample randomness as the “order parameter”, wesuccessfully located the transition points, which are in wellagreement with previous studies. We then determine the n -thorder gap ratio distributions up to n = 4 at the critical region,and confirm they fit well with the short-range plasma model(SPRM) with inverse temperature β = 1 for orthogonal modeland β = 2 for unitary. Based on results from models both withand without S TZ conservation, we argue these critical behav-iors are universal that independent of microscopic details.It is worth noting that the level statistics right at the tran-sition points detected by V S show systematic deviations fromSRPM in all cases studied, this is due to the finite size ef-fect. To be precise, in a finite system, there will always re-main exponentially decaying but finite level correlations evendeep in the MBL phase, as can be seen from the fitting re-sults for MBL phases in Fig.1(a),2(a),3(a) and 4(a). There-fore, the randomness strength to drive the phase transition willbe larger to compensate for these residue level correlations.Consequently, the true critical statistics will appear slightlyleft to the detected transition point. The deviations in the uni-tary model with S TZ conservation are larger than the rest mod-els, which can be attributed to the neat-nearest neighboring in-teractions. That is, the next-nearest neighboring term breaksthe integrability of the clean system and stabilizes the ther-mal phase in disordered system, which results in larger residuelevel correlations in a finite system. After all, in all cases, whatwe observe is a combination of universal critical level statis-tics and non-universal (model-dependent) finite-size results,for which a detailed quantification will require a systematicfinite-size scaling study, and is left for a future work.It would be beneficial to compare the SRPM with other pro-posed effective models for the transition region, first of whichis the mean-field plasma model (MFPM), which is proposedby Serbyn and Moore by mapping the thermal-MBL transitioninto a random walk process in the Hilbert space . Mathemat- ically, both SRPM and MFPM describe the energy levels of arandom matrix ensemble as an ensemble of 1D classical parti-cles, however in SRPM the interaction form stays unchangedand interaction range is responsible for the thermal-MBL tran-sition, while the inverse is true for the MFPM. Meanwhile,both models hold the semi-Poisson distribution for the near-est level spacings, hence our results are not controversial tothose in Ref. [29]. In this study, we proceed to consider thehigher-order level correlations, and find good support for theSRPM. In fact, our results suggest the form of local interac-tion between energy levels stays logarithmic during the phasetransition, and the change in interaction range can be revealedby the high-order level correlations. Another proposed effec-tive model is the Gaussian β ensemble, which has the samestructure as the GOE,GUE,GSE but the Dyson index β takesvalue in (0 , ∞ ) . In Ref. [32] the authors showed the Gaussianensemble with non-integer β can describe the lowest-ordergap ratio distribution across the thermal-MBL transition quitewell but the fittings for higher-order ones have large devia-tions. This also suggests the form of interaction between lev-els stays logarithmic but the interaction range changes duringphase transition, hence is also consistent with our results.Another interesting fact to notice is that the SRPM canemerge from the Poisson ensemble, that is, the SRPM with k = 1 and inverse temperature β has the same structure as the β -th order reduced energy spectrum of a Poisson ensemble (which is called “Daisy model” by the authors). This indicatesthe universal lower-order spectral statistics at the transitionregion are secretly hidden in the eigenvalue spectrum of theMBL phase, for which a full physical understanding is lack-ing by now. However, this at least indicates the thermal-MBLtransition point is more affected by the MBL phase ratherthan the thermal phase, a fact that has already been noticedby previous studies based on eigenfunction properties and now appears again by means of the reduced energy spec-trum. On the other hand, in Ref. [43] the authors declare theabsence of a dynamical system that corresponds to the “Daisymodel” with inverse temperature β > , our work thus sug-gests the thermal-MBL transition point in unitary system is anatural candidate for β = 2 .Last but not least, the SRPM was debated for its effective-ness in describing long range level correlations at the MBLtransition , e.g. through the number variance Σ . Unfortu-nately our attempts to fit Σ do not give conclusive results.This may partially due to the intrinsic sensitive dependence of Σ on concrete unfolding procedure , and also may resultsfrom the limited system size we can reach. Nevertheless, ourresults support the SRPM is a good effective model not only for lowest-order level correlations, but also for correlations onmoderate longer ranges. We left an improved study on largersystem size for a future work. ACKNOWLEDGEMENTS
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