Local markers for crystalline topology

Abstract

Over the last few years, crystalline topology has been used in photonic crystals to realize edge- and corner-localized states that enhance light-matter interactions for potential device applications. However, the band-theoretic approaches currently used to classify bulk topological crystalline phases cannot predict the existence, localization, or spectral isolation of any resulting boundary-localized modes. While interfaces between materials in different crystalline phases must have topological states at some energy, these states need not appear within the band gap, and thus may not be useful for applications. Here, we derive a class of local markers for identifying material topology due to crystalline symmetries, as well as a corresponding measure of topological protection. As our real-space-based approach is inherently local, it immediately reveals the existence and robustness of topological boundary-localized states, yielding a predictive framework for designing topological crystalline heterostructures. Beyond enabling the optimization of device geometries, we anticipate that our framework will also provide a route forward to deriving local markers for other classes of topology that are reliant upon spatial symmetries.

Figures (2)

• Figure 1: (a) Diagram of a 2D photonic structure with a $120^\circ$ corner between crystals formed from expanded (purple) and contracted (blue) hexagonal clusters, bounded by a perfect electric conductor. The high dielectric rods $\varepsilon = 11.7$ embedded in air have radius $r = a/9$ and are offset from being a honeycomb lattice by $\pm 0.06a$, where $a$ is the lattice constant. (b) Bulk TM band structure for the expanded (purple) and contracted (blue) photonic crystals. (c) Density of states for the finite system in (a). (d) Local gap $(\mu_{(0,\omega^2)})^{1/2}$ in units of $2\pi c/a$ and local index $\zeta_{\omega^2}^{\mathcal{R}_y}$ calculated using $\kappa = 0.01(2\pi c)^2/a^3$. Note, $(\mu_{(0,\omega^2)})^{1/2}$ has units of frequency, enabling direct comparison against the system's band structure. In (b)-(d) the shaded regions demarcate those frequencies where bulk states exist. (e) Local density of states (LDOS) at the frequency of the local gap closing and real part of the $E_z$ field for the nearest mode of the system. Orange corresponds to $\omega = 0.515(2\pi c/a)$, and red to $\omega = 0.480(2\pi c/a)$.
• Figure 2: (a),(b) Zoomed in diagram of the perturbed rods (red) in the photonic crystal corner heterostructure from Fig. \ref{['fig:corner']}(a) tailored to the lower-frequency (a) and higher-frequency (b) corner states. (c) Local gap $(\mu_{(0,\omega^2)})^{(1/2)}$ in units of $2\pi c/a$ and local index $\zeta_{\omega^2}^{\mathcal{R}_y}$ calculated using $\kappa = 0.01(2\pi c)^2/a^3$ for the lower-frequency perturbation with $\delta \varepsilon = \varepsilon_{\textrm{red}} - \varepsilon = 1.17$. (d) Similar to (c), except using the higher-frequency perturbation with $\delta \varepsilon = -4.28$. The shaded regions in (c),(d) demarcate those frequencies within the bulk bands.