Rigidity of topological invariants to symmetry breakings

TL;DR

This paper investigates how topological invariants and edge states in topological phases can remain robust even after spontaneous symmetry breaking, provided the Hamiltonian remains adiabatically connected to the original phase.

Contribution

It demonstrates the persistence of topological invariants under symmetry breaking, expanding understanding of topological phase stability beyond symmetry-preserving conditions.

Findings

Topological invariants can be robust to symmetry breaking.

Edge states may persist despite symmetry loss.

Topological classification must consider symmetry changes.

Abstract

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.