  Symmetry plays an important role in the topological band theory to remedy the
eigenstates' gauge obstruction at the cost of a symmetry anomaly and
zero-energy boundary modes. One can also make use of the symmetry to enumerate
the topological invariants - giving a symmetry classification table. Here we
consider various topological phases protected by different symmetries, and
examine how the corresponding topological invariants evolve once the protecting
symmetry is spontaneously lost. To our surprise, we find that the topological
invariants and edge states can sometimes be robust to symmetry breaking quantum
orders. This topological robustness persists as long as the mean-field
Hamiltonian in a symmetry breaking ordered phase maintains its adiabatic
continuity to the non-interacting Hamiltonian. For example, for a time-reversal
symmetric topological phase in 2+1D, we show that the Z_2 time-reversal
polarization continues to be a good topological invariant even after including
distinct time-reversal breaking order parameters. Similar conclusions are drawn
for various other symmetry breaking cases. Finally, we discuss that the change
in the internal symmetry associated with the spontaneous symmetry breaking has
to be accounted for to reinstate the topological invariants into the expected
classification table.
