Finite-Size scaling analysis of many-body localization transition in quasi-periodic spin chains

Abstract

We analyze the finite-size scaling of the average gap-ratio and the entanglement entropy across the many-body localization (MBL) transition in one dimensional Heisenberg spin-chain with quasiperiodic (QP) potential. By using the recently introduced cost-function approach, we compare different scenarios for the transition using exact diagonalization of systems up to 22 lattice sites. Our findings suggest that the MBL transition in the QP Heisenberg chain belongs to the class of Berezinskii-Kosterlitz-Thouless (BKT) transition, the same as in the case of uniformly disordered systems as advocated in recent studies. Moreover, we observe that the critical disorder strength shows a clear sub-linear drift with the system-size as compared to the linear drift seen in random disordered models, suggesting that the finite-size effects in the MBL transition for the QP systems are less severe than that in the random disordered scenario. Moreover, deep in the ergodic regime, we find an unexpected double-peak structure of distribution of on-site magnetizations that can be traced back to the strong correlations present in the QP potential.

Local Integrals of Motion in Quasiperiodic Many-Body Localized Systems

“…[39] and found broadly consistent results, however it would be interesting to revisit this assumption in light of the recent results of Ref. [43]. We further suggest a connection to the work of Ref.…”

“…In the late stages of completion of this work, we became aware of a recent preprint (Ref. [43]) which conducted a careful finite-size scaling analysis of the MBL transition in a quasiperiodic system and concluded that, for system sizes accessible to ED, the best data collapse was achieved with the assumption of a Berezinskii-Kosterlitz-Thouless (BKT) scaling form, as also seen in recent numerical studies on random systems. This result is a surprise in a quasiperiodic system, as the rare ergodic seed regions which lead to avalanaches and a BKT-like transition are not currently known to exist in deterministic potentials, and is in contrast with Refs.…”

“…Furthermore we show that the richer structure of the tensor flow equation and of its LIOM description allows us to build up a diagnostic for the MBL phase transition, so far out of reach for previous implementations of the method, that we estimate here for the quasiperiodic problem using finite-size scaling. We end by discussing our results in the contexts of other recent related works [38,39,43].…”

“…The slow drift of critical disorder strength with increasing system size interpreted in Ref. [43] as evidence of a BKT-like transition provides further support for the idea that finite-size effects are much weaker in quasiperiodic potentials -in this respect, further studies on large systems will need to be done in order to ascertain whether the most accurate scaling form is the logarithmic drift of system size indicative of a BKT transition, or a more generic system-size-dependent critical disorder which is also shown in Ref. [43] to result in a good collapse of the data.…”

“…[43] as evidence of a BKT-like transition provides further support for the idea that finite-size effects are much weaker in quasiperiodic potentials -in this respect, further studies on large systems will need to be done in order to ascertain whether the most accurate scaling form is the logarithmic drift of system size indicative of a BKT transition, or a more generic system-size-dependent critical disorder which is also shown in Ref. [43] to result in a good collapse of the data. In the present work, we interpret the weak dependence on system size for increasingly large systems as a reflection the slow growth of the number of single-particle resonances at distances r = F n (where F n is the nth Fibonacci number for φ equal to the golden ratio) with system size.…”