# Nonreciprocal response theory of nonhermitian mechanical metamaterials: response phase transition from the skin effect of zero modes

## TL;DR

This paper develops a response theory for nonreciprocal nonhermitian mechanical metamaterials, revealing a phase transition from localized skin modes to an extended phase with diverging response, based on biorthogonal eigenmodes.

## Contribution

It introduces a response-based framework that uncovers the biorthogonality and phase transition in nonreciprocal mechanical metamaterials, extending understanding beyond eigenmode analysis.

## Key findings

- Response reveals biorthogonality in the system.
- Identification of a phase transition from skin effect to extended phase.
- Prediction of diverging response in large systems.

## Abstract

Nonreciprocal nonhermitian systems provide an unconventional localization mechanism of topological zero modes via the nonhermitian skin effect. While fundamental theoretical characterizations of this effect involve the biorthogonal system of right and left eigenmodes, the recent demonstration of this effect for a zero mode in a robotic metamaterial (Ghatak et al., arXiv:1907.11619) is based on the direct experimental observation of the conventional right eigenvectors. Here I show that such nonreciprocal mechanical metamaterials reveal their underlying biorthogonality in the directly observable response of the system to external excitation. Applied to the ground-breaking experiment, this nonreciprocal response theory predicts that the zero-mode skin effect goes along an extended phase where the system is highly sensitive to physical perturbations, leading to a diverging response in the limit of a large system.