Non-Bloch band theory for time-modulated discrete mechanical systems

TL;DR

This paper develops a non-Bloch band theory for time-modulated discrete mechanical systems, revealing unique phenomena like non-reciprocal transmission and skin effects, and providing a new way to analyze their eigenvalues.

Contribution

It introduces a non-Bloch band theory for time-modulated mechanical systems using a generalized Brillouin zone, addressing limitations of standard Bloch theory in such contexts.

Findings

Standard Bloch theory fails for finite, time-modulated chains.

The non-Bloch theory accurately predicts eigenvalue distributions.

Numerical experiments confirm the validity of the proposed theory.

Abstract

This study establishes a non-Bloch band theory for time-modulated discrete mechanical systems. We consider simple mass-spring chains whose stiffness is periodically modulated in time. Using the temporal Floquet theory, the system is characterized by linear algebraic equations in terms of Fourier coefficients. This allows us to employ a standard linear eigenvalue analysis. Unlike non-modulated linear systems, the time modulation makes the coefficient matrix non-Hermitian, which gives rise to, for example, parametric resonance, non-reciprocal wave transmission, and non-Hermitian skin effects. In particular, we study finite-length chains consisting of spatially periodic mass-spring units and show that the standard Bloch band theory is not valid for estimating their eigenvalue distribution. To remedy this, we propose a non-Bloch band theory based on a generalized Brillouin zone. The proposed theory is verified by some numerical experiments.