Non-Bloch band theory of nonlinear eigenvalue problems

TL;DR

This paper develops a non-Bloch band theory for nonlinear eigenvalue problems, enabling accurate spectrum calculation and revealing unique nonlinear phenomena, including topological bulk-boundary correspondence.

Contribution

It introduces a non-Bloch framework for nonlinear systems, extending topological analysis to nonlinear eigenvalue problems with boundary sensitivity.

Findings

Successfully reproduces spectra of nonlinear systems with open boundaries.

Reveals phenomena unique to nonlinear eigenvalue problems.

Extends topological bulk-boundary correspondence to nonlinear Chern insulators.

Abstract

Nonlinear eigenvalue problems arise in a wide range of physical systems, in which system parameters depend on the eigenvalue. Such systems have been proposed to exhibit an extreme sensitivity of their spectra to boundary conditions, which leads to the breakdown of conventional topological characterizations. In this work, we establish a non-Bloch framework for calculating continuum bands that reproduce the spectra of the nonlinear system with open boundary conditions. This non-Bloch band theory enables us not only to calculate the eigenvalues but also to reveal phenomena unique to the nonlinear system. We further investigate the topological bulk-boundary correspondence in a nonlinear Chern insulator within an extended version of this framework.