Unveiling Non-Hermitian Spectral Topology in Hyperbolic Lattices with Non-Abelian Translation Symmetry

TL;DR

This paper develops a novel method to determine spectra of non-Hermitian hyperbolic lattices with non-Abelian symmetry, revealing higher-dimensional skin effects and topological phase transitions, advancing understanding of non-Hermitian topological physics in non-Euclidean spaces.

Contribution

It introduces a reciprocal space approach using supercells and point gap topology to analyze non-Hermitian spectra in hyperbolic lattices with non-Abelian translation symmetry, addressing the breakdown of Bloch theorem.

Findings

Revealed higher-dimensional skin effects in non-Hermitian hyperbolic lattices

Identified topological phase transitions in non-Abelian semimetal models

Established a universal spectral range under open boundary conditions

Abstract

The hyperbolic lattice (HBL) has emerged as a compelling platform for exploring matter in non-Euclidean space. Among its notable features, the breakdown of the conventional Bloch theorem stands out, prompting a reexamination of band theory, with the determination of spectra for non-Hermitian systems being a prominent example. Here, we develop an approach to determining the spectra under open boundary conditions (OBCs), one of the foundations in non-Hermitian lattices, from the reciprocal space of HBLs. By introducing supercells to encompass states that are allowed by non-Abelian translational groups, we perform analytic continuation and base on the point gap topology to acquire uniform spectra, the universal OBC spectral range. Applying this method to a single-band nonreciprocal model and a reciprocal non-Abelian semimetal model, we reveal higher-dimensional skin effects and topological phase transitions, respectively, demonstrating the feasibility of our method in predicting spectral topology and investigating non-Hermitian physics in HBLs.