Breakdown of the topological classification $\mathbb{Z}$ for gapped phases of noninteracting fermions by quartic interactions

The conditions for both the stability and the breakdown of the topological classification of gapped ground states of noninteracting fermions, the tenfold way, in the presence of quartic fermion-fermion interactions are given for any dimension of space. This is achieved by encoding the effects of interactions on the boundary gapless modes in terms of boundary dynamical masses. Breakdown of the noninteracting topological classification occurs when the quantum nonlinear $\sigma$ models for the boundary dynamical masses favor quantum disordered phases. For the tenfold way, we find that (i) the noninteracting topological classification ${\mathbb{Z}}_2^{}$ is always stable, (ii) the noninteracting topological classification $\mathbb{Z}$ in even dimensions is always stable, and (iii) the noninteracting topological classification $\mathbb{Z}$ in odd dimensions is unstable and reduces to ${\mathbb{Z}}_N^{}$ that can be identified explicitly for any dimension and any defining symmetries. We also apply our method to the three-dimensional topological crystalline insulator SnTe from the symmetry class $\mathrm{AII}+R$, for which we establish the reduction $\mathbb{Z}\to {\mathbb{Z}}_8^{}$ of the noninteracting topological classification.