Exact solutions for topological surface states of three-dimensional lattice models

In this work, we employ a generalized transfer matrix method that provides exact analytical and numerical solutions for lattice versions of topological models with surface termination in one direction. We construct a generalized eigenvalue equation, equivalent to the conventional transfer matrix, which neither suffers from nor requires singular (non-invertible) inter-layer hopping matrices. This contrasts with some previous approaches that use the transfer matrix technique to obtain analytical expressions for surface states, which only focus on singular cases. We then apply this formalism to derive, with exactness, the topological surface states and Fermi arc states in two prototypical topological models: the 3D Bernevig–Hughes–Zhang model and a lattice model exhibiting Weyl semimetal behavior. Our results show that the surface states and bulk bands, across the projected two-dimensional Brillouin zone, agree perfectly with those obtained through direct numerical diagonalization of the corresponding Hamiltonians in a slab geometry. This highlights that the generalized transfer matrix method is not only a powerful tool but also a highly efficient alternative to fully numerical methods for investigating surface physics and interfaces in topological systems, particularly when it is required to go beyond low-energy effective descriptions.