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

TL;DR

This work tackles the computation of surface density of states (SDOS) for semi-infinite topological photonic and acoustic systems, addressing the inefficiency and surface-discrimination limitations of the conventional supercell approach. It introduces two complementary algorithms, the cyclic reduction method (CRM) and the transfer matrix method (TMM), both exploiting the block-Toeplitz structure of the system operator to obtain the surface Green's function and hence the SDOS with far reduced resources. Numerical results across photonic Chern insulators, valley photonic crystals, and acoustic topological insulators demonstrate accurate, surface-specific spectra and reveal topological edge modes, slow light, and valley/pseudospin features, with SDOS directly amenable to near-field experiments. The methods offer substantial practical impact for designing topological devices and enabling direct comparisons with experiments, while extending to complex scenarios such as coatings, heterostructures, and layered interfaces, and potentially to non-Hermitian or higher-order topological systems.

Abstract

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.

Figures (8)

• Figure 1: Various photonic or acoustic crystal structures which support topological surface states. The structure is divided into layers along the direction perpendicular to the surface indicated by an index $m$. ${\bm{Z}_{m,m}}$ is the intra-coupling matrix within a single layer, ${\bm{Z}_{m,m+1}}$ and ${\bm{Z}_{m+1,m}}$ are the inter-coupling matrices between two nearest-neighbor layers. $\mkern 2mu\overline{\mkern-2mu\bm{Z}\mkern-2mu}\mkern 2mu$ represents the coupling matrix in the opposite direction. (a) A bare semi-infinite crystal. (b) A semi-infinite coated crystal. (c) Two different semi-infinite crystals interfaced with each other. (d) Two different semi-infinite crystals separated by another crystal slab.
• Figure 2: Topological surface states of a bare semi-infinite photonic Chern insulator terminated by a PEC. (a) Mode profile calculated at a normalized frequency of $0.64 c/a$, which corresponds to the red circle in (b). (b) Band structure of a 12-cell photonic Chern insulator slab with two PEC boundaries. (c) SDOS spectrum of a semi-infinite photonic Chern insulator with a PEC boundary. The SDOS spectrum clearly shows that only a unidirectional chiral edge state can be excited at the PEC-crystal interface, whereas the eigenvalue spectrum presents chiral edge states at opposite boundaries simultaneously.
• Figure 3: Topological surface states of a semi-infinite photonic Chern insulator coated with a crystal slab and terminated by a PEC. (a) Mode profile calculated at a normalized frequency of $0.61 c/a$, which corresponds to the red circle in (b). (b) Band structure of a 13-cell photonic Chern insulator slab with two PEC boundaries. (c) SDOS spectrum of a semi-infinite photonic Chern insulator with a PEC boundary. The SDOS spectrum clearly shows that the modified photonic crystal boundary can support a helical topological slow light, while the eigenvalue spectrum redundantly exhibits other surface states on the opposite boundary.
• Figure 4: Topological surface states of a valley photonic crystal formed by two semi-infinite crystals face to face. (a) Mode profile calculated at a normalized frequency of $0.45 c/a$, which corresponds to the red circle in (b). (b) Band structure of a 24-cell valley photonic crystal slab with two PEC boundaries. (c) SDOS spectrum of a valley photonic crystal extending infinitely on both sides. The SDOS spectrum clearly shows that a pair of pseudospin-valley locked topological states are equally exited at the interface of the photonic crystal heterostructure, while the eigenvalue spectrum redundantly presents other trivial states (at the bottom of b) on the supercell boundaries.
• Figure 5: Topological surface states of a valley photonic crystal formed by two semi-infinite crystals separated by another crystal slab. (a) Mode profile calculated at a normalized frequency of $0.26 c/a$, which corresponds to the red circle in (b). (b) Band structure of a 25-cell valley photonic crystal slab with two PEC boundaries. (c) SDOS spectrum of a valley photonic crystal extending infinitely on both sides. The SDOS spectrum clearly shows that the modified interface can support a pair of topological slow light in different valleys, whereas the eigenvalue spectrum redundantly shows other trivial states on the supercell boundaries.