Topological Phases, Entanglement and Boson Condensation

Abstract

This dissertation investigates the boson condensation of topological phases and the

entanglement 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 classies 16 dierent chiral superconductors

in (2+1)D; (2) we prove that in any layers of topological theories SO(3)k with 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 deconned particles braid trivially with the

condensed bosons while the conned 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 rst 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.