Finite-temperature topological invariant for higher-order topological insulators

We investigate the effects of temperature on higher-order topological insulators (HOTIs). The finite-temperature topological invariants for HOTIs can be constructed by generalizing the Resta's polarization for the ground state to the ensemble geometric phase (EGP) for the mixed states, [C.-E. Bardyn, L. Wawer, A. Altland, M. Fleischhauer, and S. Diehl, Phys. Rev. X 8, 011035 (2018)]. The EGP is consistent with the Resta's polarization both at zero temperature and at finite temperatures in the thermodynamic limit. We find that the temperature can change the critical point and thus induce a phase transition from a topologically trivial phase to a nontrivial phase in a finite-size system, manifesting changes in the winding of the EGP.