Optimal spectral transport of non-Hermitian systems

TL;DR

This paper explores how optimal spectral transport, measured by the Wasserstein metric, reveals key features of non-Hermitian systems' spectra as they deform under varying conditions, linking spectral geometry to physical properties.

Contribution

It introduces the use of optimal spectral transport and the Wasserstein metric to analyze and extract physical features of non-Hermitian models, connecting spectral deformation to topological and phase transition properties.

Findings

Spectral deformation connects different boundary condition spectra.

Wasserstein metric reveals topological and phase transition features.

Key spectral features can be extracted from optimal transport analysis.

Abstract

The optimal transport problem seeks to minimize the total transportation cost between two distributions, thus providing a measure of distance between them. In this work, we study the optimal transport of the eigenspectrum of one-dimensional non-Hermitian models as the spectrum deforms on the complex plane under a varying imaginary gauge field. Notably, according to the non-Bloch band theory, the deforming spectrum continuously connects the eigenspectra of the original non-Hermitian model (with vanishing gauge field) under different boundary conditions. It follows that the optimal spectral transport should contain key information of the model. Characterizing the optimal spectral transport through the Wasserstein metric, we show that, indeed, important features of the non-Hermitian model, such as the (auxiliary) generalized Brillouin zone, the non-Bloch exceptional point, and topological phase transition, can be determined from the Wasserstein-metric calculation. We confirm our conclusions using concrete examples. Our work highlights the key role of spectral geometry in non-Hermitian physics, and offers a practical and convenient access to the properties of non-Hermitian models.