Building blocks of topological band theory for photonic crystals

TL;DR

This work develops a robust framework to classify topological bands in 3D photonic crystals by regularizing Maxwell's zero-frequency polarization singularity with auxiliary modes and using stable real-space invariants (SRSIs). It defines a formal connection between SRSIs and symmetry-data vectors via BRs, enabling a lattice of allowed invariants Lat_${{\boldsymbol \theta}_{\mathbb{Z}},ph}$ that distinguishes trivial photonic bands from topological ones, including stable equivalence classes. The approach is demonstrated in SG $P4_332$ with explicit ab-initio and tight-binding models, and it reveals polarization-singularity–induced features in Wilson loops, such as windings in cylindrical loops and an Euler-number constraint in spherical loops ($|\mathfrak{e}|=2$). The framework generalizes to all space groups and offers a quantitative, regularization-independent path to compare photonic and electronic bands, with implications for designing photonic crystals and exploring stable topological photonics. Key ideas include the regularization scheme $\mathcal{B}_{phys}=\mathcal{B}_{reg}-\mathcal{B}_{aux}$, the SRSI invariants $\boldsymbol{\theta}_{\mathbb{Z}}$, and the Wilson-loop diagnostics that capture polarization singularity effects.

Abstract

We derive a framework for classifying topological bands in three-dimensional photonic band structures, where the zero frequency polarization singularity implied by Maxwell's equations complicates the direct application of existing symmetry-based approaches. Building on recent advances in the regularization of photonic bands, we use the recently introduced concept of stable real-space invariants (SRSIs) to show how photonic band structures can be unambiguously characterized in terms of equivalence classes of band representations. We classify topologically trivial photonic bands using SRSIs, treating them as the fundamental building blocks of 3D photonic band structures. This means that if certain bands cannot be constructed from these building blocks, they are necessarily topological. Furthermore, we distinguish between photonic and electronic band structures by analyzing which SRSI values are allowed in systems with and without polarization singularity. We also explore the impact of the polarization singularity on the behavior of Wilson loops, providing new insights into the topological classification of 3D photonic systems.

Figures (5)

• Figure 1: Polarization singularity and regularization of photonic bands.a Below the lowest gap, a set of bands $\mathcal{B}_{\rm phys}$ is connected to the transverse ($T$) modes around zero frequency ($\omega=0$) and zero momentum (at $\Gamma$ in the Brillouin zone), where the polarization singularity is located. At high-symmetry momenta, the little-group irreps $\rho^i_{\mathbf K}$ are assigned, except at the singularity. b The band structure can be represented by $\omega({\mathbf k})^2$, the square for frequency. Introducing auxiliary bands $\mathcal{B}_{\rm aux}$ regularizes the singularity and defines the regularized bands $\mathcal{B}_{\rm reg}$. One of $\mathcal{B}_{\rm aux}$ corresponds to the longitudinal ($L$) mode around the singular point. The auxiliary bands have negative $\omega({\mathbf k})^2$, indicating their auxiliary nature. $\mathcal{B}_{\textrm{r}eg,aux}$ can be assigned little-group irreps, $\rho_V$ and $(\rho_L)_\Gamma$, at the singularity. c The physical bands $\mathcal{B}_{\rm phys}$ are considered a formal difference between $\mathcal{B}_{\rm reg}$ and $\mathcal{B}_{\rm aux}$. The physical description of $\mathcal{B}_{\rm phys}$ must remain equivalent regardless of the choice of $\mathcal{B}_{\rm aux}$, including any additional introduction of auxiliary bands $\mathcal{B}_{\rm add}$. d Two topologically distinct regularized bands, $\mathcal{B}_{\rm reg}$ and $\mathcal{B}'_{\rm reg}$, regularize $\mathcal{B}_{\rm phys}$ in a topologically equivalent manner if they can be adiabatically deformed to each other by adding trivial bands, i.e. $\mathcal{B}_{\rm reg} + \mathcal{B}_{\rm add} \simeq \mathcal{B}'_{\rm reg} + \mathcal{B}_{\rm add}$.
• Figure 2: Adiabatic deformation process in SG $P4_332$. The site-symmetry irreps $A_1$ and $A_2$ at WP $4a$ can be adiabatically deformed into the $A_1$ irrep at WP $8c$. Note that $\widetilde{C}_{2,x-y}$ maps ${\mathbf x}_{8c,1}$ to ${\mathbf x}_{8c,2}$, and $x'=-x+1/4$.
• Figure 3: SRSIs of photonic and electronic bands allowed in SG $P4_332$. Three $\mathbb{Z}$-valued SRSIs, ${\boldsymbol \theta}_\mathbb{Z} = (\theta_1, \theta_2, \theta_3)$, are determined by symmetry-data vector, as in Eq. \ref{['eq:sg212_symvec_map']}. The allowed values of ${\boldsymbol \theta}_\mathbb{Z}$ for the number of bands a$\nu=4$ and b$\nu=8$ are shown. Note that $\nu = 4(\theta_1+\theta_2+2\theta_3)$. Gray, red, and blue regions represent $M_{co}$, $M_{ph}$, and $M_{el}$, respectively, defined by inequalities $I_{ph,i}$ and $I_{el,i}$$(i=1,\dots,7)$. Red (blue) dots in $M_{ph}$ ($M_{el}$) are the allowed SRSIs ${\boldsymbol \theta}_\mathbb{Z}$, which give physically allowed, integer-valued symmetry-data vectors. The larger black dots represent SRSIs allowed in both electronic and photonic band structures. Inequalities $I_{{ph},1,4}$ ($I_{{el},1,4}$) hold at the red (blue) lines.
• Figure 4: Ab-initio and tight-binding models in SG $P4_332$.a Frequency spectrum of the SG $P4_332$ photonic crystal built by the MPB. Only the lowest four bands, isolated from higher bands, are shown, as described in the Methods section. The inset displays the unit cell structure of photonic crystal. The box shows the boundary of primitive unit cell $x,y,z \in [-0.5, 0.5]$. b-c Band structures in the tight-binding models. b Energy spectrum for physical and auxiliary bands. Regularized bands $\mathcal{B}_{\rm reg}$ corresponds to the EBR induced from $(E)_a$, while the auxiliary bands $\mathcal{B}_{\rm aux}$ are chosen to have a symmetry-data vector identical to that induced from $(A_1)_a$. All irreps of the auxiliary bands except $\Gamma_1$ have negative $\varepsilon({\mathbf k})$. c Frequency spectrum for physical bands corresponding to $\mathcal{B}_{\rm phys}$ in Eq. \ref{['eq:sg212_irrep_model']}. Note that $\Gamma=(0,0,0)$, $X=(0,\pi,0)$, $M=(\pi,\pi,0)$, and $R=(\pi,\pi,\pi)$.
• Figure 5: Wilson loop spectra and windings. The cylindrical and spherical Wilson loop spectra for the ${\mathbf H}$ fields in the ab-initio model(a-c) and for the tight-binding model (d-f). a,b The spectrum of cylindrical Wilson loop becomes gapless and exhibits helical winding as $k_\rho$ approaches 0. Note that due to the small scale of inversion symmetry breaking in the ab initio structure, the radius of the cylinder for which the the ab-initio cylindrical Wilson loop spectra clearly does not wind is larger than in the tight binding model. We refer the reader to the SM supple for a detailed discussion. c The spectrum of spherical Wilson loop exhibits winding structure corresponding to $|\mathfrak{e}|=2$ for the Euler number $\mathfrak{e}$. d-f We observe qualitatively similar behavior in the Wilson loop spectra for the tight-binding model.