Inverse participation ratios in the XXZ spin chain

TL;DR

This paper numerically studies inverse participation ratios in the XXZ spin chain, revealing exponential growth in the gapped phase and linear scaling in the gapless phase, with effects of non-integrability also examined.

Contribution

It provides a detailed analysis of how inverse participation ratios behave in different phases of the XXZ model, including effects of non-integrability.

Findings

Exponential growth of T in the gapped phase.

Linear scaling of T in the gapless phase.

Saturation of T in the non-integrable gapless phase.

Abstract

We investigate numerically the inverse participation ratios in a spin-1/2 XXZ chain, computed in the "Ising" basis (i.e., eigenstates of $\sigma^z_i$). We consider in particular a quantity $T$, defined by summing the inverse participation ratios of all the eigenstates in the zero magnetization sector of a finite chain of length $N$, with open boundary conditions. From a dynamical point of view, $T$ is proportional to the stationary return probability to an initial basis state, averaged over all the basis states (initial conditions). We find that $T$ exhibits an exponential growth, $T\sim\exp(aN)$, in the gapped phase of the model and a linear scaling, $T\sim N$, in the gapless phase. These two different behaviors are analyzed in terms of the distribution of the participation ratios of individual eigenstates. We also investigate the effect of next-nearest-neighbor interactions, which break the integrability of the model. Although the massive phase of the non-integrable model also has $T\sim\exp(aN)$, in the gapless phase $T$ appears to saturate to a constant value.