The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought

revolutionary ideas to physics. Recently, through the discovery of symmetry protected topological phases, it is realized that continuous quantum phase transition can also occur between states with the same symmetry but different topology. The subject of this dissertation is the phase transition between these symmetry protected topological states (SPTs). There are two main parts in this dissertation.

In the first part we consider spatial dimension $d$ and symmetry group $G$ so that the cohomology group, $H^{d+1}(G,U(1))$, contains at least one $Z_{2n}$ or $Z$ factor. We show that the phase transition between the trivial SPT and the root states that generate the $ Z_{2n} $ or $Z$ groups can be induced on the boundary of a d+1 dimensional $G\times Z_2^T$-symmetric SPT by a $Z_2^T$ symmetry breaking field. Moreover we show these boundary phase transitions can be ``transplanted'' to d dimensions and realized in lattice models as a function of a tuning parameter. The price one pays is for the critical value of the tuning parameter there is an extra non-local (duality-like) symmetry. In the case where the phase transition is continuous, our theory predicts the presence of unusual (sometimes fractionalized) excitations corresponding to delocalized boundary excitations of the non-trivial SPT on one side of the transition. This theory also predicts other phase transition scenarios including first order transition and transition via an intermediate symmetry breaking phase.

In the second part, we study the phase transition between bosonic topological phases protected by $Z_n\times Z_n$ in 1+1 dimensions. We find a direct transition occurs when $ n=2,3,4 $ and in all cases the critical point possesses two gap opening relevant operators: one leads to a Landau-forbidden symmetry breaking phase transition and the other to the topological phase transition. We also obtained a constraint($ c\geq1 $) on the central charge for general phase transitions between symmetry protected bosonic topological phases in 1+1D.

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.