Consecutive-gap ratio distribution for crossover ensembles

TL;DR

The paper introduces a two-parameter surmise P_{β,γ}(r) for the consecutive-gap ratio to describe GOE-to-Poisson crossovers, and validates it by analyzing many-body spectra of disordered Heisenberg spin-1/2 chains. It treats the crossover as a dynamical flow in β–γ space, identifying a Poisson (MBL) fixed point for local-field disorder and a distinct non-ergodic fixed point for exchange-coupling disorder with SU(2) symmetry. The authors deploy exact diagonalization to fit the surmise to ensemble-averaged data, then develop a linear SDE framework to describe parameter fluctuations and a stationary Gaussian distribution around fixed points. This approach provides a compact, quantitative description of finite-size crossover behavior and offers a path to generalize to other crossovers and complex networks.

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.

Figures (4)

• Figure 1: Averaged consecutive-gap ratios (blue histograms) for an $L=18$ Heisenberg spin-$1/2$ chain with disordered local field for various maximal local field strength $h$ (upper panels) and disordered exchange couplings $b$ (lower panels). The surmise distribution, Eq. (\ref{['General_ratio_dist']}), with best-fit parameters $\beta$ and $\gamma$ for each $h$ and $b$, is represented by a solid black line. The GOE [panels (a), (b) and (e)-(h)] and Poisson statistics [(c) and (d)] appear as dashed red lines. Upper panels: as $h$ increases, the distribution gradually changes from GOE [panel (a)] to Poisson statistics [panel (d)], indicating that the system reaches the MBL phase. Lower panels: for small values of $b$, the system is approximately represented by the GOE [panel (e)]. Due to the spin rotation symmetry the system never reaches the MBL phase. This can be seen from panels (f)-(h) as $b$ increases.
• Figure 2: Relative error $|\delta P_{\beta,\gamma} (r)| / P_{\rm hist} (r)$vs.$r$ for $h=1.0$, $2.0$, $3.0$ and $4.0$ (solid curves) and $b=0.5$, $1.0$, $2.0$ and $4.0$ (dashed curves). The relative error between the surmise distribution, Eq. (\ref{['General_ratio_dist']}), fitted with the best-fit parameters, and the histogram is minimal. It is approximately $1\%$ for different disorders, except for $r\rightarrow 0$. Inset: error $\delta P_{\beta,\gamma} (r)$vs.$r$ for the same values of $h$ and $b$ from the main plot.
• Figure 3: Flow patterns of the local field case (blue) and the exchange coupling case (red) in the $\beta\gamma$ space. As the disorder strength $h$ is systematically increased, the best-fit parameters for the local field case trace a trajectory in the $\beta\gamma$ space, converging towards a fixed point at $\vec{x}^{*}_{\rm LF} = (0,1)^\top$ that characterizes the MBL phase. Similarly, as the exchange coupling disorder $b$ increases, the corresponding best-fit parameters appear to be attracted to another fixed point at $\vec{x}^{*}_{\rm EC} \approx (0.7,1.2)^\top$, represented by the intersection of the horizontal and vertical dashed lines. Inset: blow up of the $1.1 \leq b \leq 4.0$ region encircled by a magenta ellipse in the main plot. The solid black line represents the best-fit of the data by Eq. (\ref{['eq:Lin-SDE']}) with $R=0$. The gray curves represent ten realizations of Eq. (\ref{['eq:Lin-SDE']}) with the best-fit parameters and noise intensity $R=0.02$.
• Figure 4: The red ellipse delineates the $95\%$ confidence region corresponding to the stationary probability distribution $P(\vec{X})$. This ellipse is centered at the fixed point $\vec{x}^{*}_{\rm EC} = (\beta^*,\gamma^*)^\top \approx (0.7,1.2)^\top$ and characterized by the covariance matrix $\boldsymbol{\Sigma}^*$ as specified in Eq. (\ref{['Sigma_star']}). In contrast, the blue ellipse depicts the $95\%$ confidence region for a distribution with $\boldsymbol{\Sigma}={\rm diag}(10^{-5},10^{-5})$.