On intermediate statistics across many-body localization transition

Abstract

The level statistics in the transition between delocalized and localized {phases of} many body interacting systems is {considered}. We recall the joint probability distribution for eigenvalues resulting from the statistical mechanics for energy level dynamics as introduced by Pechukas and Yukawa. The resulting single parameter analytic distribution is probed numerically {via Monte Carlo method}. The resulting higher order spacing ratios are compared with data coming from different {quantum many body systems}. It is found that this Pechukas-Yukawa distribution compares favorably with {$\beta$--Gaussian ensemble -- a single parameter model of level statistics proposed recently in the context of disordered many-body systems.} {Moreover, the Pechukas-Yukawa distribution is also} only slightly inferior to the two-parameter $\beta$-h ansatz shown {earlier} to reproduce {level statistics of} physical systems remarkably well.

Quantum Chaos and Level Dynamics

“…The interested reader should consult the Ref. [ 7 ] for details, but it suffices to say here that the single-parameter Yukawa-like model described above compared favorably with other single-parameter models and quite faithfully reproduced the disordered spin data for the MBL–ergodic crossover.…”

“…The first term in ( 14 ) represents the pairwise interaction between the particles and the exponential term provides the harmonic binding of the eigenvalues. The resulting distribution, obtained using Monte-Carlo sampling for different p , was shown to faithfully reproduce statistics of eigenvalues on the transition between ergodic and many body localized situations [ 7 ].…”

“…Comparison of some of these models with numerics for interacting disordered spin systems modelling ergodic to the MBL transition is given in the Ref. [ 7 ]. The interested reader should consult the Ref.…”

“…We begin with defining the notation for level dynamics and the corresponding statistical mechanics picture in Section 2 showing how standard results from this approach provide a prediction for level statistics in a chaotic integrable transition. The resulting distribution seems to work surprisingly well for the data in many-body localization crossovers [ 7 ]. To somehow complete the picture, we review in the next Section several other models for the statistics in the transition, notably the banded matrix model, developed by Casati and coworkers [ 8 , 9 , 10 , 11 , 12 ].…”

Universal scaling of higher-order spacing ratios in Gaussian random matrices

“…This relation has been employed successfully to various physical systems like chaotic billiards, Floquet systems, circular ensembles, spin chains, observed stock market, etc. [9,28,[55][56][57][58][59] , to estimate the number of symmetries in complex physical systems [44,45]. It should be noted that, a similar scaling relation between the higherorder and NN spacing distributions has be proposed earlier in Refs.…”