Topological invariant of S4 protected high-order topological insulator.

The point group symmetry S4 protects a high-order topological state that is gapped on all its 2D surfaces but gapless on a 1D hinge. In this talk, I present a bulk topological invariant of this S4 topology in terms of Wilson loop of Wilson loop. The nonlocal Wilson loop operator provides an effective Hamiltonian on the surface, the Wilson loop of which provides an intuitive understanding of the hinge state. Based on this picture, the breaking of Kirchhoff's law on surfaces are discussed.