Theoretical description of a photonic topological insulator based on a cubic lattice of bianisotropic resonators

TL;DR

We develop a theoretical description of a 3D photonic topological insulator based on a simple cubic lattice of bianisotropic resonators. Employing a dyadic Green's function approach, we derive effective Bloch Hamiltonians and real-space tight-binding models that include couplings up to the next-to-next nearest neighbor, enabling full dispersion and Berry-curvature calculations in the Brillouin zone. The introduction of a finite bianisotropic coupling characterized by $|\Omega|$ opens a band gap and supports domain-wall states localized at interfaces, with nontrivial Berry-curvature distributions in Models II and III indicating a weak 3D PTI. The study emphasizes the necessity of longer-range couplings to accurately describe the topological properties and discusses feasible microwave-based experimental implementations using dielectric or metallic resonators.

Abstract

In the present paper, we construct a theoretical description of a three-dimensional photonic topological insulator in the form of a simple cubic lattice of bianisotropic resonators that is based on a dyadic Green's function approach. By considering electric and magnetic dipole modes and the interactions between different numbers of the nearest resonators, we obtain the Bloch Hamiltonians and the corresponding tight-binding models and analyze the band diagrams, spatial structure of the eigenmodes, and their localization, revealing quadratic degeneracies in the vicinity of high-symmetry points in the absence of bianisotropy and the emergence of in-gap states localized at a domain wall upon the introduction of bianisotropy. Finally, we visualize the Berry curvature distributions to study the topological properties of the considered models.

Figures (4)

• Figure 1: (a) The schematic of point electric and magnetic dipoles arranged in a cubic lattice with period $a$. The inset points the orientation of the dipoles $\mathbf{p}$ and $\mathbf{m}$. (b) The example of the model realization as an array of dielectric resonators with inversion symmetry broken shape.
• Figure 2: (a)-(c) Energy bands of the Bloch Hamiltonians for (a) Model I (the nearest neighbors), Eq. \ref{['eq:H_I_spin_basis_component']}; (b) Model II (the next-nearest neighbors), Eq. \ref{['eq:H_II_spin_basis_component']}; and (c) Model III (the next-to-next nearest neighbors), Eq. \ref{['eq:H_III_spin_basis_component']} in the absence of bianisotropy ($\Omega = 0$). (d)-(f) Density of states $\rho(\lambda)$ for tight-binding models corresponding to cubic lattices of $10 \times 20 \times 10$ sites without bianisotropy ($\Omega = 0$) for (d) Model I, (e) Model II, and (f) Model III real-space Hamiltonians plotted as histogram with 150 bins. Panels (g)-(l) are the same as (a)-(f), but for non-zero bianisotropic parameter $\Omega = 7$. The gray shaded areas in dispersion diagrams (g)-(i) highlight the bandgaps.
• Figure 3: (a) Spectrum of eigenvalues $\lambda$ for the system consisting of two domains with $10 \times 10 \times 10$ sites and bianisotropic parameters $\Omega = 7$ and $\Omega = -7$, respectively. Color shows the inverse participation ratio Eq. \ref{['eq:IPR']}. (b)-(i) Eigenfunctions profiles corresponding to the absolute value of pseudospin-up polarization $p_x + m_x$ shown by color, which demonstrate (b),(c) bulk states, (d)-(f) interface states localized at the domain wall in the bandgap, (g) an interface state hybridized with a bulk mode, (h) a bulk mode with dominant surface localization at the boundary, and (i) an edge state in the continuum.
• Figure 4: Numerically evaluated Berry curvature distributions for two pseudospins for the (a)-(f) upper and (g)-(l) lower energy bands of the Hamiltonian Eq. \ref{['eq:H_II_spin_basis_component']} with bianisotropic parameter $\Omega = 7$.