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

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.

💡 Research Summary

In this work the authors present a comprehensive theoretical framework for a three‑dimensional photonic topological insulator (PTI) realized as a simple cubic lattice of bianisotropic resonators. Each lattice site hosts an electric dipole p and a magnetic dipole m confined to the xy‑plane. The electromagnetic response of a resonator is described by an electric polarizability β and a magnetoelectric coupling χ, the latter embodying the bianisotropic (spin‑orbit‑like) interaction. Assuming electromagnetic duality (β = μ) and reciprocity (χ ≠ 0), the authors construct a 2 × 2 effective Hamiltonian in the basis of Pauli matrices.

Using the quasi‑static approximation of the dyadic Green’s function (k₀r ≪ 1), only the near‑field 1/r³ term is retained. Three interaction models are defined: Model I includes only nearest‑neighbor (distance a) couplings; Model II adds next‑nearest neighbors (√2 a); Model III further incorporates third‑shell neighbors (√3 a). Applying Bloch’s theorem yields block‑diagonal Bloch Hamiltonians H↑(k) and H↓(k) for the two pseudospins, with explicit expressions (Eqs. 3‑5) that contain σ₀, σ₁, σ₃, and σ₂ components. The σ₂ term is proportional to the bianisotropic parameter Ω = a³χ/(β² − χ²) and breaks time‑reversal symmetry, while the other terms encode the lattice geometry and range of coupling.

When Ω = 0 the band structures of all three models display quadratic four‑fold degeneracies at the high‑symmetry points Γ, M, Z, and A. These quadratic degeneracies are a hallmark of the C₄ rotational symmetry of the cubic lattice, in contrast to the linear Dirac points found in hexagonal or square lattices. Introducing next‑nearest‑neighbor couplings (Model II) lifts some of the degeneracies (e.g., between M–Γ and A–Z) and generates flat portions of the bands, which manifest as pronounced peaks in the density of states (DOS). Model III shows only minor quantitative changes relative to Model II.

Switching on bianisotropy (Ω ≠ 0) adds the σ₂ term, opening a full band gap whose width scales linearly with Ω. All four‑fold degeneracies split, leaving only Kramers doublets protected by the combined pseudo‑time‑reversal symmetry. The authors then construct a real‑space tight‑binding Hamiltonian (Eq. 7) that incorporates the same near‑field Green’s function couplings and an on‑site Ω term. By assigning opposite signs of Ω to two halves of a finite lattice, a domain wall is created. Numerical diagonalization of a 10 × 20 × 10 lattice reveals in‑gap states localized at the domain wall.

To quantify localization, the inverse participation ratio (IPR) is computed for each eigenmode. Bulk modes exhibit low IPR (delocalized), while domain‑wall modes show high IPR, confirming their confinement. The spatial profiles of the pseudospin‑up component (pₓ + mₓ) are visualized, illustrating bulk, surface, edge, and hybridized states.

Finally, the Berry curvature Ωₙ(k) is evaluated across the Brillouin zone for each band. Non‑zero Ω leads to a non‑trivial distribution of curvature concentrated near the domain wall, indicating a non‑zero Chern‑like invariant for the gapped system. This confirms the topological nature of the photonic crystal and predicts robust, back‑scattering‑immune transport along the engineered interface.

Overall, the paper makes four key contributions: (1) a dyadic‑Green‑function‑based derivation of Bloch Hamiltonians that capture the full Brillouin zone physics; (2) a systematic study of how interaction range influences band dispersion, degeneracies, and flat‑band formation; (3) demonstration of topological band gap opening via bianisotropy and the emergence of domain‑wall localized states, supported by IPR analysis; and (4) explicit calculation of Berry curvature to assign topological invariants. These results provide a solid theoretical foundation for designing and experimentally realizing cubic‑lattice photonic topological insulators, with potential applications in robust waveguiding, topological lasers, and three‑dimensional photonic circuitry.