Many-body localization transition with power-law interactions: Statistics of eigenstates

TL;DR

This paper investigates the many-body localization transition in systems with power-law decaying interactions, analyzing spectral statistics and wavefunction properties to understand the nature of the transition and its scaling behavior.

Contribution

It refines the understanding of the localization transition with long-range interactions, deriving new scaling laws and analyzing wavefunction fractality at criticality.

Findings

Localization transition depends on rescaled disorder W*

Wavefunctions exhibit fractal behavior at criticality

Scaling laws for critical disorder and IPR are supported by numerical data

Abstract

We study spectral and wavefunction statistics for many-body localization transition in systems with long-range interactions decaying as $1/r^\alpha$ with an exponent $\alpha$ satisfying $ d \le \alpha \le 2d$, where $d$ is the spatial dimensionality. We refine earlier arguments and show that the system undergoes a localization transition as a function of the rescaled disorder $W^* = W / L^{2d-\alpha} \ln L$, where $W$ is the disorder strength and $L$ the system size. This transition has much in common with that on random regular graphs. We further perform a detailed analysis of the inverse participation ratio (IPR) of many-body wavefunctions, exploring how ergodic behavior in the delocalized phase switches to fractal one at the critical point and on the localized side of the transition. Our analytical results for the scaling of the critical disorder $W$ with the system size $L$ and for the scaling of IPR in the delocalized and localized phases are supported and corroborated by exact diagonalization of spin chains.