Analytical solutions of topological surface states in a series of lattice models

We derive the analytical solutions of surface states in a series of lattice models for three-dimensional topological insulators and their nontopological counterparts based on an ansatz. A restriction on the spin-flip matrices in nearest-neighbor hopping characterizes the series. This restriction affords the ansatz and favors analytical solvability of surface-state eigenvectors. Despite the restriction, the series retains sufficient designability to describe various types of surface states. We also describe how it can serve as a tractable tool for elucidating unique phenomena on topological surfaces.