Two Universality Classes for the Many-Body Localization Transition

Abstract

We provide a systematic comparison of the many-body localization transition in spin chains with nonrandom quasiperiodic vs. random fields. We find evidence suggesting that these belong to two separate universality classes: the first dominated by "intrinsic" intra-sample randomness, and the second dominated by external inter -sample quenched randomness. We show that the effects of inter-sample quenched randomness are strongly growing, but not yet dominant, at the system sizes probed by exact-diagonalization studies on random models. Thus, the observed finite-size critical scaling collapses in such studies appear to be in a preasymptotic regime near the nonrandom universality class, but showing signs of the initial crossover towards the external-randomness-dominated universality class. Our results provide an explanation for why exact-diagonalization studies on random models both see an apparent scaling near the transition while also obtaining finite-size scaling exponents that strongly violate Harris/Chayes bounds that apply to disorder-driven transitions. We also show that the MBL phase is more stable for the quasiperiodic model as compared to the random one, and the transition in the quasiperiodic model suffers less from certain finite-size effects.Many-body localization (MBL) generalizes the phenomenon of Anderson localization to the interacting setting [1][2][3][4][5][6]. The dynamics in an MBL system fails to establish local thermal equilibrium, and even highly excited states can retain local memory of their initial conditions for arbitrarily late times. The transition between an MBL phase and a "thermalizing" one is not a thermodynamic phase transition and lies outside the framework of equilibrium statistical mechanics. Instead it is a novel eigenstate phase transition [7,8] across which thermal and "volume-law" entangled many-body eigenstates obeying the eigenstate thermalization hypothesis (ETH) [9][10][11] change in a singular way to non-thermal and arealaw entangled eigenstates in the MBL phase.Although the MBL transition has attracted much recent interest [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], very little is definitively known about its properties. Phenomenological renormalization group (RG) treatments of the transition are approximate but can probe large system sizes, and such studies [26][27][28][29] find a continuous transition in one dimension with a finite-size critical scaling exponent ν F S ∼ 3 satisfying rigorous Harris/CCFS/CLO scaling bounds [30-32] which require ν F S ≥ 2/d for transitions in d dimensions in the presence of quenched randomness. On the other hand, most other studies of the transition use numerical exact diagonalization (ED) of spin-chains which is limited to system sizes L ≤ 22. These ED studies observe an apparent scaling collapse near the transition, but with scaling exponents ν F S ∼ 1 violating the CCFS/CLO bound [12,13]. Strikingly, some aspects of this transition even look first-order-like in that quantities like the eigenstate entangle...

Many-body localization transition: Schmidt gap, entanglement length, and scaling

“…3(a) we show the optimal data collapse of for various L's, which is found to occur for h c = 5.06 ± 0.09 and ν = 2.35 ± 0.21. Remarkably, this value for ν is consistent with the CCFS/CLO bound, in contrast to finite-size scaling analyses for S and r, which previous studies [27,30,54] have generally shown to yield values of ν ∼ 1-a finding also reproduced in our analyses (data not shown). We show the quality of collapse Q for all possible combinations of h c and ν in Fig.…”

“…In order to estimate the exponent ν, finite-size scaling analysis [53] has previously been employed for various quantities, including the entanglement entropy S. These analyses, based on exact numerical methods, find ν ∼ 1 [27,30,54], contradicting the CCFS/CLO bound. A recently proposed explanation [30] suggests that there are two universality classes at play here with that of intersample randomness not yet dominant for the system sizes studied.…”

“…[27] in which * john.gray.14@ucl.ac.uk finite-size scaling analysis throughout the spectrum allows the observation of a mobility edge. In fact, it is now commonplace to diagnose the MBLT with the mean energy level statistics and the block entanglement entropy [6,10,[27][28][29][30][31]. These works are largely based on the assumption that the MBLT is continuous, and their exact numerical analyses have consistently found ν ∼ 1.…”

“…This is in striking contrast with analytic results, found by Chayes-Chayes-Fisher-Spencer (CCFS) [32] and Chandran-Laumann-Oganesyan (CLO) [33], which would demand ν 2/d for system dimension d (the CCFS/CLO bound). A recent explanation [30] posits that, at the finite system sizes available for exact studies, the fluctuations in these quantities are not yet dominated by the true disorder. Thus it is highly desirable to use a new quantity better able to capture the real disorder-induced transition properties.…”

“…For reference, we consider the normalized half chain entropy, widely employed to herald the MBLT [6,10,[27][28][29][30]. The von Neumann entropy of subsystem A is defined as the disorder-averaged S = S vn k /S P .…”