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

TL;DR

This work introduces a non-Bloch band theory for time-modulated discrete mechanical systems by combining temporal Floquet analysis with a generalized Brillouin zone (GBZ). It demonstrates that conventional Bloch band theory fails to predict spectra under open boundaries due to non-Hermiticity and nonreciprocity, revealing non-Hermitian skin modes. The GBZ-based method yields spectra that align with long finite chains and explains transient nonreciprocal wave propagation and edge localization. These insights extend non-Hermitian band theory to time-varying mechanical lattices and open avenues for higher-dimensional and forced-response analyses.

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.

Figures (6)

• Figure 1: Chain of spring-mass pairs.
• Figure 2: Verification of the Floquet theory for a time-modulated system. (a) Two mass-spring pairs. (b), (c) Trajectory of eigenvalues $\omega$ as the modulation amplitude $\delta$ varies from $0$ to $0.99$. (d) Displacements $u^1$ and $u^2$ as a function of time $t$ when the initial conditions $u^1(0)=u_0$ and $u^2(0)=\partial_t u^1(0) = \partial_t u^2(0)=0$ are given.
• Figure 3: Spectrum of $\mathcal{H}$ for a long-chain of mass-spring pairs. (a) A long chain consists of spatially periodic units. Its unit comprises three mass-spring pairs. (b) Time modulation of $\mu_1$, $\mu_2$, and $\mu_3$. (c) Distribution of eigenvalues $\omega$ for the system (a). Some representatives are denoted by the symbols A, B, C, D, and E with their conjugate pairs. (d) Eigenmodes corresponding to the representatives.
• Figure 4: Bloch band $\sigma_\mathrm{BZ}$ and OBC spectrum $\sigma_\mathrm{OBZ}$. (a), (b) Periodic chain of mass-spring pairs. The unit of the chain is the same as in \ref{['fig:result-finite-spectrum_result']} (a), (b). (c), (d) $\sigma(k)$ as a function of $k\in[-\pi,\pi]$ (Bloch band diagram). (e) Comparison between $\sigma_\mathrm{BZ}$ and $\sigma_\mathrm{OBZ}$ in \ref{['fig:result-finite-spectrum_result']}.
• Figure 5: Non-Bloch band theory for the periodic chain shown in \ref{['fig:result-periodic-spectrum_result']} (a). (a) Eigenvalues $\beta(\omega)$ of \ref{['eq:periodic-beta']} as a function of $\omega\in\mathbb{C}$. The contour $|\beta^{(1)}(\omega)|=|\beta^{(2)}(\omega)|$ is shown in the complex $\omega$-plane. (b) Comparison between the GBZ spectrum $\sigma_\mathrm{GBZ}$ and $\sigma_\mathrm{OBC}$ shown in \ref{['fig:result-finite-spectrum_result']}. (c) $C_\mathrm{GBZ}$ in the complex $\beta$-plane. The red, green, and black lines represent $|\beta|>1$, $|\beta|<1$, and $|\beta|=1$, respectively.