Breakdown of the topological classification Z for gapped phases of noninteracting fermions by quartic interactions

TL;DR

The paper systematically demonstrates that the noninteracting tenfold classification of gapped fermionic phases is generically unstable to quartic interactions in odd dimensions, while remaining robust in even dimensions for the appropriate symmetry classes. By mapping boundary gapless modes to dynamical Dirac masses and integrating out the fermions, the authors obtain quantum nonlinear sigma models whose topological terms determine whether interactions gap the edge states; classifications are thus reduced to ZN patterns dictated by homotopy data of classifying spaces from K-theory. Key outcomes include Z2 stability across all dimensions, Z stability in even dimensions, and explicit ZN reductions in odd dimensions, with SnTe as a concrete TCI example showing Z → Z8. The approach provides a unified framework for interacting SRE phases and crystalline topological phases, with implications for experimental realizations and the design of interacting topological materials. All results are formulated within a K-theory-based, boundary-action formalism that connects Dirac mass spaces, QNLSMs, and topological terms to the stability of topological phases under interactions.

Abstract

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, (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 AII$+R$, for which we establish the reduction $\mathbb{Z}\to\mathbb{Z}^{\,}_{8}$ of the noninteracting topological classification.