Statistics of Matrix Elements of Operators in a Disorder-Free SYK model

TL;DR

This paper studies the statistical distribution of matrix elements of operators in a disorder-free SYK model, revealing a generalized inverse Gaussian distribution for off-diagonal elements, advancing understanding of ETH in solvable models.

Contribution

It extends the analysis of matrix element statistics to a disorder-free SYK model, showing a different distribution form than in previous models, thus broadening ETH insights.

Findings

Off-diagonal matrix elements follow a generalized inverse Gaussian distribution.

The distribution differs from the Fréchet distribution observed in other models.

Results enhance understanding of eigenstate thermalization in solvable models.

Abstract

Recently, studies have explored the statistics of matrix elements of local operators in the Lieb-Liniger model. It was found that the probability distribution function for off-diagonal matrix elements $\langle \boldsymbol{\mu}|\mathcal{O}|\boldsymbol{\lambda} \rangle$ within the same macro-state is well described by the Fr\'{e}chet distributions. This represents a significant development for the Eigenstate Thermalization Hypothesis (ETH). In this paper, we investigate a similar phenomenon in another solvable model: the disorder-free Sachdev-Ye-Kitaev (SYK) model. The Hamiltonian of this model consists of 4-body interactions of Majorana fermions. Unlike the conventional SYK model, the coupling strengths in this model are fixed to a constant, earning it the name ``disorder-free.'' We evaluate the matrix elements of operators constructed from products of $n$ Majorana fermions: $\mathcal{O} = \chi_{a_1}\chi_{a_2}\ldots \chi_{a_n}$. For a general choice of indices and $n \geq 4$, we find that the statistics of the off-diagonal matrix elements are well-fitted by a generalized inverse Gaussian distribution rather than Fr\'{e}chet distributions.