Surface density of states on semi-infinite topological photonic and acoustic crystals

The iterative Green's function, based on a cyclic reduction of block-tridiagonal matrices, has been the ideal algorithm, through tight-binding models, to compute the surface density of states of semi-infinite topological electronic materials. In this paper, we apply this method to photonic and acoustic crystals, using finite-element discretizations and a generalized eigenvalue formulation, to calculate the local density of states on a single surface of semi-infinite lattices. Three-dimensional examples of gapless helicoidal surface states in Weyl and Dirac crystals are shown and the computational cost, convergence, and accuracy are analyzed.

Figure 1

The iterative Green's function method. (a) Schematic of a semi-infinite crystal. Each cell represents a crystal layer and the top gray surface is the surface boundary. The surface cell can be different from the rest. (b) Iterative procedure for solving the surface Green's function. The block elements ($\mathit{\alpha }, \mathit{\beta }$, and $\mathit{\gamma }$) are relabeled from ${\mathit{M}}_{i,j}$ in Eqs. (4) and (5b). After each iteration, half the number of blocks is eliminated and the matrix size is halved. When the off-diagonal blocks are sufficiently small, after enough iterations, the Green's function can be obtained by inverting the remaining diagonal blocks.

Figure 2

Surface DOS on a semi-infinite Weyl photonic crystal. (a) Geometry of semi-infinite dielectric photonic crystals with a relative permittivity of 16. The top gray surface is the perfect electric conductor, which satisfies $\widehat{\mathit{n}}\times \mathit{E}=\mathit{0}$ and $\widehat{\mathit{n}}$ is the surface normal. Here, we use cylinders with a radius of $0.10a$ to approximate the double-gyroid structures and an air cylinder with a height of $0.07a$ to break the inversion symmetry, where $a$ is the lattice constant. (b) Band structure of the 12-cell photonic crystal slab with perfect electric conductors on both surfaces. The Weyl points are at the normalized frequency 0.55. (c) Surface DOS on semi-infinite bulk cells. (d)–(f) The isofrequency cuts at normalized frequencies 0.567, 0.552, and 0.537.

Figure 3

Surface DOS on a semi-infinite Dirac acoustic crystal. (a) Geometry of the semi-infinite blue-phase-I acoustic crystal. The material is taken as hard wall boundaries in numerics, which satisfies $\widehat{\mathit{n}}·\mathbf{\nabla }p=\mathit{0}$ and $\widehat{\mathit{n}}$ is the surface normal. (b) Band structure of 12-cell acoustic crystal slab with two hard wall boundaries. The Dirac points are at the normalized frequency 1.30. (c) Surface DOS on the semi-infinite bulk cells. (d)–(f) The isofrequency cuts at normalized frequencies 1.276, 1.221, and 1.166.

Figure 4

Numerical convergence and accuracy of the acoustic example. (a) Convergence variance with different imaginary frequencies. The residual is defined as $\parallel {\mathit{\gamma }}_i^s\left(\eta \right)-{\mathit{\gamma }}_{i-1}^s{\left(\eta \right)\parallel }_F/{\parallel {\mathit{\gamma }}_{i-1}^s\left(\eta \right)\parallel }_F$, where $i$ is the iteration step and ${\parallel \parallel }_F$ represents the Frobenius norm of a matrix. (b) Accuracy dependence on different imaginary frequencies. The error is defined as ${lim}_{i\to \infty }\parallel {\mathit{\gamma }}_i^s\left(\eta \right)-{\mathit{\gamma }}_i^s\left(0^{+}\right){\parallel }_F/{\parallel {\mathit{\gamma }}_i^s\left(0^{+}\right)\parallel }_F$.