# Statistics of correlation functions in the random Heisenberg chain

## TL;DR

This paper investigates the probability distributions of eigenstate correlation functions in the random-field Heisenberg chain, revealing distinct statistical features across ergodic, thermal, and many-body localized phases, and providing theoretical explanations for these behaviors.

## Contribution

It provides a comprehensive analysis of correlation function distributions in the MBL transition, including a perturbation theory that explains the observed distribution features.

## Key findings

- Gaussian distributions at weak disorder consistent with ETH
- Heavy tails indicating anomalous thermalization in the thermal phase
- Asymmetry and multimodality in MBL phase correlation distributions

## Abstract

Ergodic quantum many-body systems satisfy the eigenstate thermalization hypothesis (ETH). However, strong disorder can destroy ergodicity through many-body localization (MBL) -- at least in one dimensional systems -- leading to a clear signal of the MBL transition in the probability distributions of energy eigenstate expectation values of local operators. For a paradigmatic model of MBL, namely the random-field Heisenberg spin chain, we consider the full probability distribution of eigenstate correlation functions across the entire phase diagram. We find gaussian distributions at weak disorder, as predicted by pure ETH. At intermediate disorder -- in the thermal phase -- we find further evidence for anomalous thermalization in the form of heavy tails of the distributions. In the MBL phase, we observe peculiar features of the correlator distributions: a strong asymmetry in $S_i^z S_{i+r}^z$ correlators skewed towards negative values; and a multimodal distribution for spin-flip correlators. A quantitative quasi-degenerate perturbation theory calculation of these correlators yields a surprising agreement of the full distribution with the exact results, revealing, in particular, the origin of the multiple peaks in the spin-flip correlator distribution as arising from the resonant and off-resonant admixture of spin configurations. The distribution of the $S_i^zS_{i+r}^z$ correlator exhibits striking differences between the MBL and Anderson insulator cases.