Complex Brillouin Zone for Localised Modes in Hermitian and Non-Hermitian Problems

TL;DR

The work develops a comprehensive framework that unifies the study of evanescent and defect modes in subwavelength band gap materials through the complex Brillouin zone. By combining 1D transfer-matrix theory, decay densities, and generalized capacitance matrices with a 2D band-gap Green's function and a complex Floquet transform, it links decay rates, phase transitions, and localization phenomena to complex quasimomenta. The key contributions include a parametrisation of 1D complex bands, a rigorous treatment of non-Hermitian skin effects, a detailed 2D Green's-function-based analysis of defect modes, and numerically stable methods (including Kummer's transform) for computing complex band structures and defect resonances. Collectively, these results enable accurate prediction of defect-mode decay lengths, reveal phase-transition behavior of defect states inside the band gap, and point to avenues for studying disorder, topology, and higher-decay Bloch branches in periodic systems.

Abstract

We develop a mathematical and numerical framework for studying evanescent waves in subwavelength band gap materials. By establishing a link between the complex Brillouin zone and various Hermitian and non-Hermitian phenomena, including defect localisation in band gap materials and the non-Hermitian skin effect, we provide a unified perspective on these systems. In two-dimensional structures, we develop analytical techniques and numerical methods to study singularities of the complex band structure. This way, we demonstrate that gap functions effectively predict the decay rates of defect states. Furthermore, we present an analysis of the Floquet transform with respect to complex quasimomenta. Based on this, we show that evanescent waves may undergo a phase transition, where local oscillations drastically depend on the location of corresponding frequency inside the band gap.

Figures (22)

• Figure 2.1: Unit cell of an infinite chain of subwavelength resonators, with lengths $(\ell_i)_{1\leq i\leq N}$ and spacings $(s_{i})_{1\leq i\leq N+1}$.
• Figure 2.2: The complex quasimomentum (red line) given by \ref{['eq: Decay condition']} accurately predicts the exponential decay rates of the eigenmodes superposed onto each other (black lines). In each case, there is a single outlier arising from the fact that the one-dimensional capacitance matrix has a one-dimensional kernel with distinct decay properties.
• Figure 2.3: In contrast to Figure \ref{['Fig: Skin effect']}, when the gauge potential lacks lattice periodicity, the eigenmodes exhibit algebraic decay at a rate given in \ref{['eq: algebraic decay rate']}. This is because the system is no longer strictly periodic and there is no complex band structure promoting the exponential decay of gap modes.
• Figure 2.4: Resonator chain with random gauge potential. $60$ monomers with $s_1 = \ell_1 =1$ and gauge potential with expectation $\mathbb{E}[\gamma_{i_j}] = 0.25.$
• Figure 2.5: The non-Hermitian skin effect with a random gauge potential applied to the resonators. The exponential decay rate given by \ref{['eq: expected decay rate skin effect']} (red) accurately forecasts the exponential decay rate of the eigenmodes (black) that were superimposed onto each other. In each case, there is a single outlier arising from the fact that the one-dimensional capacitance matrix has a one-dimensional kernel.

Theorems & Definitions (43)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 2.9
• Theorem 2.10: Bloch’s Theorem