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Symmetry plays an important role in the topological band theory to remedy theeigenstates' gauge obstruction at the cost of a symmetry anomaly andzero-energy boundary modes. One can also make use of the symmetry to enumeratethe topological invariants - giving a symmetry classification table. Here weconsider various topological phases protected by different symmetries, andexamine how the corresponding topological invariants evolve once the protectingsymmetry is spontaneously lost. To our surprise, we find that the topologicalinvariants and edge states can sometimes be robust to symmetry breaking quantumorders. This topological robustness persists as long as the mean-fieldHamiltonian in a symmetry breaking ordered phase maintains its adiabaticcontinuity to the non-interacting Hamiltonian. For example, for a time-reversalsymmetric topological phase in 2+1D, we show that the Z_2 time-reversalpolarization continues to be a good topological invariant even after includingdistinct time-reversal breaking order parameters. Similar conclusions are drawnfor various other symmetry breaking cases. Finally, we discuss that the changein the internal symmetry associated with the spontaneous symmetry breaking hasto be accounted for to reinstate the topological invariants into the expectedclassification table.

Rigidity of topological invariants to symmetry breakings