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

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 gapped surface states 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.