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Complex Band Structure and localisation transition for tridiagonal non-Hermitian k-Toeplitz operators with defects

Yannick De Bruijn, Erik Orvehed Hiltunen

TL;DR

The paper introduces the complex band structure for tridiagonal $k$-Toeplitz operators as a quantitative alternative to winding-based diagnostics, enabling explicit decay bounds for eigenvectors through $oldsymbol{\varepsilon}$-pseudoeigenvectors. By treating the symbol $f$ and complex Floquet parameters $(\alpha,\beta)$, it derives the open-limit spectrum $\boldsymbol{\sigma_{open}}$ and characterises eigenvectors in the open and winding regions, including edge- and bulk-localised states. This framework is then applied to subwavelength non-Hermitian resonator chains via the gauge capacitance matrix, revealing skin-to-bulk localisation transitions and defect-mode formation, with decay rates determined by the complex band structure (e.g., $\beta=\tfrac{1}{2}\log\prod b_i/c_i$ and $r=\tfrac{1}{2}\log\prod b_i/c_i$). The theory is extended to finite chains, monomer and polymer configurations, and a non-Hermitian tight-binding Hamiltonian, demonstrating broad applicability to photonic/phononic metamaterials and quantum-style lattice models; all numerical data and code are publicly available. The results provide precise, computable predictions for localisation lengths and defect-induced modes, offering a unified lens on non-Hermitian localisation phenomena across discrete and continuum settings.

Abstract

Using the Bloch-Floquet theory, we propose an innovative technique to obtain the eigenvectors of tridiagonal k-Toeplitz operators. This method offers a more extensive and quantitative basis for describing localised eigenvectors beyond the non-trivial winding zone, yielding sharp decay bounds. The validity of our results is confirmed numerically in one-dimensional resonator chains, showcasing non-Hermitian skin localisation, bulk localisation, and tunnelling effects. We conclude the paper by analysing non-Hermitian tight binding Hamiltonians, illustrating the broad applicability of the complex band structure.

Complex Band Structure and localisation transition for tridiagonal non-Hermitian k-Toeplitz operators with defects

TL;DR

The paper introduces the complex band structure for tridiagonal -Toeplitz operators as a quantitative alternative to winding-based diagnostics, enabling explicit decay bounds for eigenvectors through -pseudoeigenvectors. By treating the symbol and complex Floquet parameters , it derives the open-limit spectrum and characterises eigenvectors in the open and winding regions, including edge- and bulk-localised states. This framework is then applied to subwavelength non-Hermitian resonator chains via the gauge capacitance matrix, revealing skin-to-bulk localisation transitions and defect-mode formation, with decay rates determined by the complex band structure (e.g., and ). The theory is extended to finite chains, monomer and polymer configurations, and a non-Hermitian tight-binding Hamiltonian, demonstrating broad applicability to photonic/phononic metamaterials and quantum-style lattice models; all numerical data and code are publicly available. The results provide precise, computable predictions for localisation lengths and defect-induced modes, offering a unified lens on non-Hermitian localisation phenomena across discrete and continuum settings.

Abstract

Using the Bloch-Floquet theory, we propose an innovative technique to obtain the eigenvectors of tridiagonal k-Toeplitz operators. This method offers a more extensive and quantitative basis for describing localised eigenvectors beyond the non-trivial winding zone, yielding sharp decay bounds. The validity of our results is confirmed numerically in one-dimensional resonator chains, showcasing non-Hermitian skin localisation, bulk localisation, and tunnelling effects. We conclude the paper by analysing non-Hermitian tight binding Hamiltonians, illustrating the broad applicability of the complex band structure.

Paper Structure

This paper contains 15 sections, 20 theorems, 98 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Complex Band structure for tridiagonal $k$-Toeplitz Operators
  3. Complex band structure
  4. Eigenvectors of tridiagonal $k$-Toeplitz and $k$-Laurent matrices
  5. Subwavelength Localisation in Defected non-Hermitian Resonator Chains
  6. Setting and Problem Formulation
  7. Subwavelength Regime
  8. Defected capacitance matrix
  9. Finite resonator chains.
  10. 1-periodic Toeplitz matrix
  11. Polymer chains
  12. Non-Hermitian Tight-Binding Hamiltonian
  13. Concluding remarks
  14. Data availability
  15. Integral Evaluation

Key Result

Theorem 2.1

ammari2024generalisedbrillouinzonenonreciprocal Let $\boldsymbol{\mathbf{T}}(f)$ be the tridiagonal Toeplitz operator with symbol $f(z) \in \mathbb{R}^{k \times k}$ and $b_ic_i > 0$ for all $1 \leq i \leq k$. We then have where $r = \frac{1}{2}\log \Bigl(\prod_{j = 1}^k \frac{b_j}{c_j}\Bigr)$.

Figures (11)

  • Figure 2.1: The light blue area, labelled $\sigma_{\mathrm{wind}}$, depicts the positive winding zone where eigenvectors are exponentially localized on one side of the system, although they may exhibit varying decay rates. Within the fully encompassed red area, called the open limit spectrum $\sigma_{\mathrm{open}}$, eigenmodes are uniformly localised on one side with consistent decay. The green region, $\sigma_{\mathrm{wind}}^\mathsf{c}$, signifies zones where defect eigenmodes are exponentially localized within the bulk, displaying differing localisation strengths on either side of the defect. The calculations are performed based on the Capacitance Toeplitz matrix detailed in Section \ref{['sec: quasiperiodic capacitance matrix']}
  • Figure 2.2: Truncated eigenvectors produce $\varepsilon_N$-pseudoeigenvectors. We examine the analytical convergence bounds of these pseudoeigenvectors, represented by the solid red line, against numerical convergence rates across various band gap frequencies of $\lVert (\boldsymbol{\mathbf{A}}_N-\lambda_N)\boldsymbol{\mathbf{v}}_{0,N}\rVert \leq e^{-B(\lambda)N}=: \varepsilon_N$ marked by the blue crosses. The close agreement observed signifies that the presented bounds are sharp estimates.
  • Figure 3.1: Unit cell $Y$ of an infinite chain of subwavelength resonators, with lengths $(\ell_i)_{1\leq i\leq k}$ spacings $(s_{i})_{1\leq i\leq k+1}$ spanned by the lattice vector $\Lambda$ of length $L$.
  • Figure 3.2: The band function $\omega^{\alpha, \beta}$ as given by Proposition \ref{['Prop: Gauge Capacitance Thm']} is plotted as a function of $(\alpha, \beta)$. The colouring in the surface plot illustrates the imaginary part of the band functions $\mathfrak{Im}(\omega^{\alpha, \beta})$. The black lines illustrate the surface plot where $\mathfrak{Im}(\omega^{\alpha, \beta}) = 0$, which represent admissible resonances to our scattering problem \ref{['eq: wave equation skin effect']}. Following Theorem \ref{['Thm: alpha and beta fixed']} it holds for either $\alpha \in \{0, \pi\}$ or $\beta = \frac{\gamma}{2}\sum_{i=1}^k \ell_i$.
  • Figure 3.3: The band and gap functions plotted as characterised by Proposition \ref{['Prop: Gauge Capacitance Thm']}. Figure (A) is exactly the same plot as in Figure \ref{['fig: Monomer Band function surface']}, but with the admissible bands projected onto the same axis. Qualitatively, the non-Hermitian nature of the problem shifts the origin of $\beta$ by the rate $r$ of non-reciprocity. Another difference to Hermitian systems is the emergence of a band gap below the first spectral gap. Computation performed for $\gamma = 3$ and $\delta = 10^{-3}$.
  • ...and 6 more figures

Theorems & Definitions (43)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • proof
  • Lemma 2.7
  • ...and 33 more