UC Berkeley

In this dissertation, we discuss the non-regularizability and the bosonization of massless fermions with relativistic dispersion in 1,2,3 spatial dimensions. The non-regularizability is the root of various quantum anomalies and plays a central role in the physics of symmetry-protected topological phases. We generalized the Nielsen-Ninomiya theorem to all minimal nodal free fermion field theories protected by the time reversal, charge conservation, and charge conjugation symmetries. We prove that these massless field theories cannot be regularized on a lattice while respecting the protection symmetries. However, they can be realized on the boundaries of symmetry-protected topological phases in one higher dimension. We then generalize Witten's non-abelian bosonization of massless free fermion theories in one spatial dimension to two and three spatial dimensions. We shown the resulting boson theories share the same emergent symmetries and anomalies with the fermion theories. Moreover, we also show the boson theories possess fermion degrees of freedom, namely solitons. These bosonized models are non-linear sigma models with level-1 Wess-Zumino-Witten terms. As applications, we apply the bosonization results to the SU(2) gauge theory of the Mott insulating phase of nearest-neighbor hopping Hubbard model, ``bipartite-Mott insulators'' in 1,2,3 spatial dimensions and twisted bilayer graphene.