Algebraic non-Hermitian skin effect and generalized Fermi surface formula in arbitrary dimensions

TL;DR

This paper introduces the algebraic non-Hermitian skin effect, a new type of boundary localization with power-law decay in higher dimensions, and develops a generalized Fermi surface framework to describe it across arbitrary dimensions.

Contribution

It presents the first theoretical framework for algebraic skin effects in multi-dimensional non-Hermitian systems, extending beyond exponential localization models.

Findings

Reveals algebraic decay of skin modes in 2D and higher dimensions.

Develops a generalized Fermi surface formula applicable to arbitrary dimensions.

Demonstrates the framework in both tight-binding and continuum models.

Abstract

The non-Hermitian skin effect, characterized by a proliferation of exponentially localized edge modes in open-boundary systems, has led to the discovery of numerous novel physical phenomena that challenge the limits of conventional band theory. In sharp contrast to this familiar exponential localization, we report a distinct phenomenon--the algebraic non-Hermitian skin effect--which arises generically in non-Hermitian systems with two or more spatial dimensions. In such cases, the amplitude of skin modes typically decays from the boundary following a power law, rather than an exponential form--a behavior not captured by existing theoretical frameworks. To bridge this gap and describe the transition in localization from one to higher dimensions, we develop a generalized Fermi surface framework applicable to open-boundary systems in arbitrary dimensions. This framework not only reproduces known results for the exponential skin effect in 1D, but also predicts a new class of skin effects with algebraic decay in 2D and above. We demonstrate this framework in both tight-binding and continuum models in two and three dimensions. This investigation not only unveils a novel category of the non-Hermitian skin effect but also offers a comprehensive theoretical structure that describes skin effects in any non-Hermitian system, irrespective of its spatial dimensionality.