Multifractal Analysis of the Non-Hermitian Skin Effect: From Many-Body to Tree Models

TL;DR

This paper reviews the multifractal properties of the non-Hermitian skin effect across various models, highlighting differences in localization, spectral statistics, and ergodicity in open quantum systems.

Contribution

It introduces a unified perspective on multifractality in the non-Hermitian skin effect, including an analytically solvable tree model for many-body Hilbert space.

Findings

Many-body skin effect exhibits multifractality in Hilbert space.

Coexistence of many-body skin effect with random-matrix spectral statistics.

Analytical multifractal dimensions derived from a Cayley tree model.

Abstract

The non-Hermitian skin effect is an anomalous localization phenomenon induced by nonreciprocal dissipation and has attracted considerable attention in recent years both theoretically and experimentally. In this article, we review the multifractal aspects of the non-Hermitian skin effect. In particular, we discuss how the many-body skin effect exhibits multifractality in many-body Hilbert space, unlike the trivial Hilbert-space occupation of the single-particle skin effect on crystalline lattices. We further highlight that the many-body skin effect can coexist with random-matrix spectral statistics, in contrast to the multifractality associated with many-body localization, which typically accompanies the absence of ergodicity. We also introduce a solvable model on a Cayley tree as an effective description of the many-body Hilbert space, in which the multifractal dimensions can be obtained analytically. This review provides a unified perspective on multifractal structures in the non-Hermitian skin effect across single-particle, many-body, and tree models, and clarifies their distinctive relation to ergodicity in open quantum systems.