Green's Function Approach to Interacting Higher-order Topological Insulators

The Bloch wave functions have been playing a crucial role in the diagnosis of topological phases in non-interacting systems. However, the Bloch waves are no longer applicable in the presence of finite Coulomb interaction and alternative approaches are needed to identify the topological indices. In this paper, we focus on three-dimensional higher-order topological insulators protected by $C_4T$ symmetry and show that the topological index can be computed through eigenstates of inverse Green's function at zero frequency. If there is an additional $S_4$ rotoinversion symmetry, the topological index $P_3$ can be determined by eigenvalues of $S_4$ at high symmetry momenta, similar to the Fu-Kane parity criterion. We verify this method using many-body exact diagonalization in higher-order topological insulators with interaction. We also discuss the realization of this higher-order topological phase in tetragonal lattice structure with $C_4T$-preserving magnetic order. Finally, we discuss the boundary conditions necessary for the hinge states to emerge and show that these hinge states exist even when the boundary is smooth and without a sharp hinge.