Eigenstate Thermalization Hypothesis (ETH) for off-diagonal matrix elements in integrable spin chains

TL;DR

This paper studies off-diagonal matrix elements in integrable spin chains, revealing exponential decay behaviors and distribution patterns, advancing understanding of ETH applicability in such systems.

Contribution

It provides a detailed numerical analysis of off-diagonal matrix elements in the Heisenberg chain using Bethe Ansatz, highlighting decay rates and distribution types.

Findings

Matrix elements decay exponentially with system size in the same macrostate.

Distribution functions of matrix elements follow Gumbel distributions.

Different macrostate pairs exhibit faster decay and similar distribution patterns.

Abstract

We investigate off-diagonal matrix elements of local operators in integrable spin chains, focusing on the isotropic spin-$1/2$ Heisenberg chain ($XXX$ chain). We employ state-of-the-art Algebraic Bethe Ansatz results, which allow us to efficiently compute matrix elements of operators with support up to two sites between generic energy eigenstates. We consider both matrix elements between eigenstates that are in the same thermodynamic macrostate, as well as eigenstates that belong to different macrostates. In the former case, focusing on thermal states we numerically show that matrix elements are compatible with the exponential decay as $\exp(-L |{M}^{\scriptscriptstyle{\mathcal{O}}}_{ij}|)$. The probability distribution functions of ${M}_{ij}^{\scriptscriptstyle{\mathcal{O}}}$ depend on the observable and on the macrostate, and are well described by Gumbel distributions. On the other hand, matrix elements between eigenstates in different macrostates decay faster as $\exp(-|{M'}_{ij}^{\scriptscriptstyle{\mathcal{O}}}|L^2)$, with ${M'}_{ij}^{\scriptscriptstyle \mathcal{O}}$, again, compatible with a Gumbel distribution.