Efficient algorithms for the surface density of states in topological photonic and acoustic systems

Topological photonics and acoustics have attracted wide research interest for their ability to manipulate light and sound at surfaces. The supercell technique is the conventional standard approach used to calculate these boundary effects, but, as the supercell grows in size, this method requires increasingly large computational resources. Additionally, it falls short in differentiating the surface states at opposite boundaries and, due to finite-size effects, from bulk states. Here, to overcome these limitations, we provide two complementary efficient methods for obtaining the ideal topological surface states of semi-infinite systems of diverse surface configurations. The first is the cyclic reduction method, which is based on iteratively inverting the Hamiltonian for a single unit cell, and the other is the transfer matrix method, which relies on eigenanalysis of a transfer matrix for a pair of unit cells. Numerical benchmarks, including gyromagnetic photonic crystals, valley photonic crystals, spin-Hall acoustic crystals and quadrupole photonic crystals, jointly show that both methods can effectively sort out the boundary modes via the surface density of states, at reduced computational cost and increased speed. Our computational schemes enable direct comparisons with near-field scanning measurements, thereby expediting the exploration of topological artificial materials and the design of topological devices.