This dissertation investigates the boson condensation of topological phases and the entan glement entropies of exactly solvable models.
First, the bosons in a "parent" (2+1)D topological phase can be condensed to obtain a "child" topological phase. We prove that the boson condensation formalism necessarily has a pair of modular matrix conditions: the modular matrices of the parent and the child topological phases are connected by an integer matrix. These two modular matrix conditions serve as a numerical tool to search for all possible boson condensation transitions from the parent topological phase, and predict the child topological phases. As applications of the modular matrix conditions, (1) we recover the Kitaev's 16-fold way, which classifies 16 different chiral superconductors in (2+1)D; (2) we prove that in any layers of topological theories S0(3)kwith odd k, there do not exist condensable bosons.
Second, an Abelian boson is always condensable. The condensation formalism in this scenario can be easily implemented by introducing higher form gauge symmetry. As an application in (2+1)D, the higher form gauge symmetry formalism recovers the same results of previous studies: bosons and only bosons can be condensed in an Abelian topological phase, and the deconfined particles braid trivially with the condensed bosons while the confined ones braid nontrivially. We emphasize again that the there exist non-Abelian bosons that cannot be condensed.
Third, the ground states of stabilizer codes can be written as tensor network states. The entanglement entropy of such tensor network states can be calculated exactly. The 3D fracton models, as exotic stabilizer codes, are known to have several features which exceed the 3D topological phases: (i) the ground state degeneracy generally increase with the system size; (ii) the gapped excitations are immobile or only mobile in certain sub manifolds. In our work, we calculate, for the first time, the entanglement entropy for the fracton models, and show that the entanglement entropy has a topological term linear to the subregions' sizes, whereas the topological phases only have constant topological entanglement entropies.