Wiener-Hopf factorization and non-Hermitian topology for Amoeba formulation in one-dimensional multiband systems

TL;DR

This paper develops a mathematical framework using Wiener-Hopf factorization to analyze non-Hermitian topological effects in one-dimensional multiband systems, extending the Amoeba formulation beyond single-band cases.

Contribution

It establishes the Wiener-Hopf factorization as a rigorous foundation for Amoeba analysis in multiband systems and clarifies the applicability of the Szeg"o limit theorem in this context.

Findings

Wiener-Hopf factorization provides a unified framework for multiband systems.

The approach rigorously proves the generalized Szeg"o limit theorem for symmetry class AII$^ op$.

The method elucidates criteria for applying the Szeg"o limit theorem in complex non-Hermitian topologies.

Abstract

The non-Hermitian skin effect (NHSE), characterized by the extensive localization of bulk modes at the boundaries, has attracted significant attention as a hallmark feature of non-Hermitian topology. This localization invalidates the conventional Bloch band theory, necessitating an analysis under open boundary conditions even in the thermodynamic limit. The Amoeba formulation addresses this challenge by computing the spectral potential rather than the spectrum itself. Based on the (strong) Szeg\"o limit theorem and its topological generalization, this approach reduces the evaluation of the potential to an optimization problem involving the Ronkin function. However, while the generalized Szeg\"o limit theorem is formally applicable in arbitrary dimensions, its implementation is limited to single-band systems, and its applicability to multiband systems remains unclear even in one-dimensional systems. In this paper, we establish the Wiener-Hopf factorization (WHF) of the non-Bloch Hamiltonian as a powerful framework, providing a unified and rigorous foundation for Amoeba analysis in one-dimensional multiband systems. By combining the WHF with Hermitian doubling, we first elucidate the applicability criteria for the generalized Szeg\"o limit theorem in multiband systems. We then show that the WHF provides the natural mathematical origin for the symmetry-decomposed Ronkin function in symmetry class AII$^\dagger$, leading to a rigorous proof of the generalized Szeg\"o limit theorem for these systems and opening a path toward systematic generalizations to other symmetry classes.