Finite size effects on hinge states in three-dimensional second-order topological insulators

TL;DR

This paper analyzes how finite size influences hinge states in 3D second-order topological insulators with specific symmetries, revealing coupling effects that open energy gaps.

Contribution

It derives effective Hamiltonians for surface and hinge states, showing how finite size causes coupling and gap opening in hinge states.

Findings

Hinge states are derived from surface states using perturbation theory.

Finite size coupling can open an energy gap in hinge states.

Sign alternation of mass terms leads to hinge state formation.

Abstract

We investigate the finite size effects of a three-dimensional second-order topological insulator with fourfold rotational symmetry and time-reversal symmetry. Starting from the effective Hamiltonian of the three-dimensional second-order topological insulator, we derive the effective Hamiltonian of four two-dimensional surface states with gaps derived by perturbative methods. Then, the sign alternation of the mass term of the effective Hamiltonian on the adjacent surface leads to the hinge state. In addition, we obtain the effective Hamiltonian and its wave function of one-dimensional gapless hinge states with semi-infinite boundary conditions based on the effective Hamiltonian of two-dimensional surface states. In particular, we find that the hinge states on the two sides of the same surface can couple to produce a finite energy gap.