Spectra and pseudo-spectra of tridiagonal $k$-Toeplitz matrices and the topological origin of the non-Hermitian skin effect

Abstract

We establish new results on the spectra and pseudo-spectra of tridiagonal $k$-Toeplitz operators and matrices. In particular, we prove the connection between the winding number of the eigenvalues of the symbol function and the exponential decay of the associated eigenvectors (or pseudo-eigenvectors). Our results elucidate the topological origin of the non-Hermitian skin effect in general one-dimensional polymer systems of subwavelength resonators with imaginary gauge potentials, proving the observation and conjecture in arXiv:2307.13551. We also numerically verify our theory for these systems.

Figures (3)

• Figure 1: A chain of $N$ times periodically repeated $k$ subwavelength resonators, with lengths $(\ell_i)_{1\leq i\leq N}$ and spacings $(s_{i})_{1\leq i\leq N-1}$.
• Figure 2: The region of $\lambda$ so that $\sum_{j=1}^k\operatorname{wind}(\lambda_j(\mathbb T), \lambda)\neq 0$ and the localization of the eigenvectors. Computation performed for $s_1 = 1, s_2 = 2,$ and $N = 50$.
• Figure 3: Figures A and C show the spectrum of the operator. The green regions consist of all the eigenvalues $\lambda$ that satisfy $\sum_{j=1}^3 \operatorname{wind}(\lambda_j(\mathbb T), \lambda)\neq 0$. The black dots along the real line denote the spectrum of the gauge capacitance matrix $C^\gamma$ and the solid blue and orange lines around the spectrum are the $\varepsilon$-pseudospectra for $\varepsilon = 10^{k}$ and $k = -5, -2$. Figures B and D show the eigenvectors of $C^{\gamma}$.

Theorems & Definitions (17)

• Definition 2.1
• Theorem 2.2: F. and M. Riesz
• Theorem 2.3: Coburn's lemma; tridiagonal $k$-Toeplitz version
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Remark 2.7
• Theorem 2.8
• Theorem 2.9
• Remark 2.10