Exact formulas of the end-to-end Green's functions in non-Hermitian systems

TL;DR

This paper derives exact formulas for end-to-end Green's functions in non-Hermitian systems, enabling precise theoretical and experimental comparisons and revealing the relationship with the non-Hermitian skin effect.

Contribution

The authors develop exact formulas based on Widom's formula for single-band non-Hermitian systems, depending on algebraic roots, advancing analytical tools in the field.

Findings

Exact formulas depend on roots of algebraic equations

Bulk Green's functions approach the integral formula exponentially fast

Green's functions are linked to the non-Hermitian skin effect

Abstract

Green's function in non-Hermitian systems has recently been revealed to be capable of directional amplification in some cases. The exact formulas for end-to-end Green's functions are significantly important for studies of both non-Hermitian systems and their applications. In this work, based on the Widom's formula, we derive exact formulas for the end-to-end Green's functions of single-band systems which depend on the roots of a simple algebraic equation. These exact formulas allow direct and accurate comparisons between theoretical results and experimentally measured quantities. In addition, we verify the prior established integral formula in the bulk region to agree with the result in our framework. We also find that the speed at which the Green's functions in the bulk region approach the prior established integral formula is not slower than an exponential decay as the system size increases. The correspondence between the signal amplification and the non-Hermitian skin effect is confirmed.