Consecutive-gap ratio distribution for crossover ensembles

TL;DR

This paper introduces a two-parameter model for the consecutive-gap ratio distribution to describe the transition from GOE to Poisson statistics in disordered quantum spin chains, analyzing the MBL transition and related flow patterns.

Contribution

It proposes a novel surmise expression for the distribution, models the crossover as a flow pattern in parameter space, and analyzes the fixed points and stochastic dynamics of the system.

Findings

The model accurately describes the GOE to Poisson crossover.

The flow pattern analysis reveals fixed points corresponding to different phases.

Finite-size effects are significant in the symmetry-preserving case.

Abstract

The study of spectrum statistics, such as the consecutive-gap ratio distribution, has revealed many interesting properties of many-body complex systems. Here we propose a two-parameter surmise expression for such distribution to describe the crossover between the Gaussian orthogonal ensemble (GOE) and Poisson statistics. This crossover is observed in the isotropic Heisenberg spin-$1/2$ chain with disordered local field, exhibiting the Many-Body Localization (MBL) transition. Inspired by the analysis of stability in dynamical systems, this crossover is presented as a flow pattern in the parameter space, with the Poisson statistics being the fixed point of the system, which represents the MBL phase. We also analyze an isotropic Heisenberg spin-$1/2$ chain with disordered local exchange coupling and a zero magnetic field. In this case, the system never achieves the MBL phase because of the spin rotation symmetry. This case is more sensitive to finite-size effects than the previous one, and thus the flow pattern resembles a two-dimensional random walk close to its fixed point. We propose a system of linearized stochastic differential equations to estimate this fixed point. We study the continuous-state Markov process that governs the probability of finding the system close to this fixed point as the disorder strength increases. In addition, we discuss the conditions under which the stationary probability distribution is given by a bivariate normal distribution.