Symmetric Mass Generation of K\"ahler-Dirac Fermions from the Perspective of Symmetry-Protected Topological Phases

The K\"ahler-Dirac fermion, recognized as an elegant geometric approach, offers an alternative to traditional representations of relativistic fermions. Recent studies have demonstrated that symmetric mass generation (SMG) can precisely occur with two copies of K\"ahler-Dirac fermions across any spacetime dimensions. This conclusion stems from the study of anomaly cancellation within the fermion system. Our research provides an alternative understanding of this phenomenon from a condensed matter perspective, by associating the interacting K\"ahler-Dirac fermion with the boundary of bosonic symmetry-protected topological (SPT) phases. We show that the low-energy bosonic fluctuations in a single copy of the K\"ahler-Dirac fermion can be mapped to the boundary modes of a $\mathbb{Z}_2$-classified bosonic SPT state, protected by an inversion symmetry universally across all dimensions. This implies that two copies of K\"ahler-Dirac fermions can always undergo SMG through interactions mediated by these bosonic modes. This picture aids in systematically designing SMG interactions for K\"ahler-Dirac fermions in any dimension. We present the exact lattice Hamiltonian of these interactions and validate their efficacy in driving SMG.