Mathematical Foundation for the Generalised Brillouin zone of m-banded Toeplitz operators

TL;DR

This paper establishes a mathematical foundation for the generalized Brillouin zone in non-Hermitian physics, showing the spectrum's reality under certain conditions and developing methods to analyze and symmetrize banded Toeplitz matrices.

Contribution

It provides a rigorous proof linking the generalized Brillouin zone to the polar curve where the symbol function is real, advancing theoretical understanding in non-Hermitian systems.

Findings

Spectrum of open-boundary banded Toeplitz matrices is real under certain conditions.

Developed analytical and numerical methods for symmetrizing non-Hermitian Toeplitz matrices.

Proved the generalized Brillouin zone coincides with the polar curve of real-valued symbol functions.

Abstract

We show that the spectrum of the open-boundary limit of banded Toeplitz matrices is real whenever the associated symbol function is real-valued along a closed polar curve. Building on this result, we develop both analytical and numerical methods to symmetrise a class of banded non-Hermitian Toeplitz matrices whose asymptotic spectra are real. Finally, we provide a rigorous mathematical foundation for the generalised Brillouin zone, a concept widely used in non-Hermitian physics, by proving that it coincides with the polar curve on which the symbol function takes real values.