Multifractal statistics of non-Hermitian skin effect on the Cayley tree

TL;DR

This paper reveals that the non-Hermitian skin effect on the Cayley tree exhibits multifractal properties, contrasting with conventional systems, and introduces a new mechanism for multifractality in open quantum systems without disorder.

Contribution

It analytically demonstrates multifractal statistics for skin states on the Cayley tree, highlighting a unique feature of non-Hermitian systems in this geometry.

Findings

Multifractal dimensions derived analytically for Cayley tree skin states.

Contrasts with the absence of multifractality in crystalline lattice skin effects.

Provides a new mechanism for multifractality in open quantum systems without disorder.

Abstract

Multifractal analysis is a powerful tool for characterizing the localization properties of wave functions. Despite its utility, this tool has been predominantly applied to disordered Hermitian systems. Multifractal statistics associated with the non-Hermitian skin effect remain largely unexplored. Here, we demonstrate that the tree geometry induces multifractal statistics for the single-particle skin states on the Cayley tree by deriving the analytical expression of multifractal dimensions. This sharply contrasts with the absence of multifractal properties for conventional single-particle skin effects in crystalline lattices. Our work uncovers the unique feature of the skin effect on the Cayley tree and provides a novel mechanism for inducing multifractality in open quantum systems without disorder.