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 noninteracting systems. However, waves are no longer applicable presence finite Coulomb interaction and alternative approaches needed to identify indices. In this paper, we focus on three-dimensional higher-order insulators protected by ${C}_{4}T$ symmetry show that index can be computed through eigenstates inverse Green's function at zero frequency. If there is an additional ${S}_{4}$ rotoinversion symmetry, ${P}_{3}$ determined eigenvalues high-symmetry momenta, similar Fu-Kane parity criterion. We verify method using many-body exact diagonalization with interaction. also discuss realization phase tetragonal lattice structure ${C}_{4}T$-preserving magnetic order. Finally, boundary conditions necessary for hinge states emerge these exist even when smooth without sharp hinge.