We study three-dimensional
Z2 topological phases enriched by
Z2T time-reversal symmetry with bosonic bulk excitations. Some of these phases can be constructed by simply coupling the three-dimensional symmetry protected topological phases with ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{T}$ symmetry to a deconfined
Z2 gauge field. Besides these simple phases, we also construct two special root phases, whose boundary can have an extra
Z2 topological order in addition to their bulk topological order, and the boundary anyon excitations can have fractional time-reversal transformation
T with ${\mathcal{T}}^{4}=\ensuremath{-}1$. In particular, the boundary
e and
m anyons of one of the two root phases are interchanged under time-reversal transformation, and they must be degenerate and orthogonal with each other. Eventually we obtain ${({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}$ classification (8 different phases) for
Z2T-enriched
Z2 topological phases with bosonic bulk excitations in three dimensions.
In topological phases in
2+1 dimensions, anyons fall into representations of quantum group symmetries. As proposed in our work [Hung and Wan, Int. J. Mod. Phys. B 28, 1450172 (2014)], the physics of a symmetry enriched phase can be extracted by the mathematics of (hidden) quantum group symmetry breaking of a ``parent phase.'' This offers a unified framework and classification of the symmetry enriched (topological) phases, including symmetry protected trivial phases as well. In this paper, we extend our investigation to the case where the ``parent'' phases are non-Abelian topological phases. We show explicitly how one can obtain the topological data and symmetry transformations of the symmetry enriched phases from that of the ``parent'' non-Abelian phase. Two examples are computed: (1) the $\text{Ising}\ifmmode\times\else\texttimes\fi{}\overline{\text{Ising}}$ phase breaks into the
Z2 toric code with
Z2 global symmetry; (2) the
SU(2)8 phase breaks into the chiral Fibonacci $\ifmmode\times\else\texttimes\fi{}$ Fibonacci phase with a
Z2 symmetry, a first non-Abelian example of symmetry enriched topological phase beyond the gauge-theory construction.
This thesis contributes to the understanding of symmetry-enriched topological phases focusing on their descriptions in terms of tensor network states. The Projected Entangled Pair State (PEPS) formalism allows us to locally encode the main properties of the models (like topological order, symmetries and their interplay) in the tensors. We have used that encoding to classify, construct and detect some classes of symmetry-enriched topological phases in 2D PEPS. For that purpose, we have studied what the allowed freedom in the tensors generating the same tensor network state is. These results are the so-called 'fundamental theorems' and they give the necessary knowledge to properly study symmetries (actions that leave the states invariant). We focus our study on the family of PEPS describing quantum double models of
G, the so-called
G-injective PEPS, and on global on-site symmetries coming from a finite group
Q: as a result of the classification we have a finite number of phases closely related to the theory of group extensions. We also provide the maps that appear in group extension theory with physical meaning and we characterize their actions on the ground subspace and on the anyons. We construct a representative of each class of our classification. Moreover, we connect our construction with an interesting physical phenomenon, the so-called anyon condensation, which describes topological phase transitions. We have also proposed a family of gauge invariant quantities and their corresponding order parameters (in the form of expectation values of a local operator in 2D) to detect the corresponding quantum phases, in particular their symmetry fractionalization patterns. We conclude this thesis by touching on some mathematical open problems in PEPS.