Enforced symmetry breaking by invertible topological order

Abstract

It is well known that two-dimensional fermionic systems with a nonzero Chern number must break the time-reversal symmetry, manifested by the appearance of chiral edge modes on an open boundary. Such an incompatibility between topology and symmetry can occur more generally. We will refer to this phenomenon as enforced symmetry breaking (ESB) by topological orders. In this work, we systematically study ESB of a finite symmetry group G_{f} by fermionic invertible topological orders. Mathematically, the group G_{f} is a central extension over a bosonic symmetry group G by the fermion parity group Z_{2}^{f}, characterized by a 2-cocycle λ∈H^{2}(G,Z_{2}). For given G and λ, we are able to obtain a series of criteria on the existence or nonexistence of ESB by the corresponding fermionic invertible topological orders. For 2D systems, we define a set of physical quantities to describe symmetry-enriched invertible topological orders and derive obstruction functions using both fermionic and bosonic languages. The study in the bosonic language is performed after gauging the fermion parity, and we find that some obstruction functions are consequences of conditional anomalies of the bosonic symmetry-enriched topological states, with the conditions inherited from the original fermionic system. The obstruction functions are crucial components of the ESB criteria that we derive. With these criteria, we discover many new ESB examples, e.g., we find that the quaternion group Q_{8} is incompatible with two copies of p+ip superconductors. We also obtain explicit results on the ESB phenomena of the continuous group SU_{f}(N) by 2D invertible topological orders through a different argument.