This dissertation investigates two aspects of topological phases of matter: 1) the tensor network state (TNS) representations of the ground states as well as their en- tanglement entropies of gapped Hamiltonians in diverse dimensions; 2) the anomalies and dynamics of strongly coupled quantum field theories.
For the first aspect, we first show an efficient method of analytically deriving the translation invariant TNS and matrix product state (MPS) representation for the ground state of translation invariant stabilizer code Hamiltonians in both 1d and higher dimensions. These TNS/MPS states have minimal virtual bond dimension. Using the TNS, we derive the entanglement entropy for a variety of stabilizer codes, including the fracton models the Haah code. We further go beyond the stabilizer codes and study the structure of entanglement entropy for generic 3d gapped Hamiltonians. In particular, an explicit formula for a universal physical observable – topological entanglement entropy (TEE) – has been derived, which sharpens previous results. Our formula shows that the TEE across an arbitrary entanglement surface is linearly proportional to the TEE across a torus.
For the second aspect, we use the global symmetries and their ’t Hooft anomalies of the SU(2) Yang-Mills theory with a theta term to constrain its dynamics. In particular, we point out that there are four different such theories, distinguished by Lorentz symmetry enrichments of the Wilson loops in the SU(2) fundamental representation. We further derive a new mixed anomaly between time reversal and one form symmetry which can only be seen on an unorientable manifold. We further use the anomalies to explore various possible dynamics, such as nontrivial degrees of freedom localized on the domain wall due to spontaneously broken time reversal symmetry, as well as a potentially possible but exotic quantum phase transition — Gauge Enhanced Quantum Critical Point.