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

Topological photonics and acoustics have recently garnered wide research interests for their topological ability to manipulate the light and sound at surfaces. Conventionally, the supercell technique is the standard approach to calculating these boundary effects, whereas it consumes increasingly large computational resources as the supercell size grows. Additionally, it falls short in differentiating the surface states at opposite boundaries and from bulk states due to the finite size of systems. To overcome the limitations, here we provide two complementary efficient methods for obtaining the ideal topological surface states of a semi-infinite system. The first one 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 the eigenanalysis of a transfer matrix for a pair of unit cells. Benchmarks show that, compared to the traditional supercell method, the cyclic reduction method can reduce both memory and time consumption by two orders of magnitude; the transfer matrix method can reduce memory by an order of magnitude, take less than half the time, and achieve high accuracy. Our methods are applicable to more complex scenarios, such as coated structures, heterostructures, and sandwiched structures. As examples, the surface-density-of-states spectra of photonic Chern insulators, valley photonic crystals, and acoustic topological insulators are demonstrated. Our computational schemes enable direct comparisons with near-field scanning measurements and expedite the exploration of topological artificial materials and the design of topological devices.