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 Physics , 2014, DOI: 10.1103/PhysRevB.91.121103 Abstract: Topologically ordered phases of matter, in particular so-called symmetry enriched topological (SET) phases, can exhibit quantum number fractionalization in the presence of global symmetry. In Z_2 topologically ordered states in two dimensions, fundamental translations T_x and T_y acting on anyons can either commute or anticommute. This property, crystal momentum fractionalization, can be seen in a periodicity of the excited-state spectrum in the Brillouin zone. We present a numerical method to detect the presence of this form of symmetry enrichment given a projected entangled pair state (PEPS); we study the minima of spectrum of correlation lengths of the transfer matrix for a cylinder. As a benchmark, we demonstrate our method using a modified toric code model with perturbation. An enhanced periodicity in momentum clearly reveals the anticommutation relation {T_x,T_y}=0$for the corresponding quasiparticles in the system.  Physics , 2013, DOI: 10.1103/PhysRevB.89.045127 Abstract: Recently, there is a considerable study on gapped symmetric phases of bosons that do not break any symmetry. Even without symmetry breaking, the bosons can still be in many exotic new states of matter, such as symmetry-protected trivial (SPT) phases which are short-range entangled and symmetry-enriched topological (SET) phases which are long-range entangled. It is well-known that non-interacting fermionic topological insulators are SPT states protected by time-reversal symmetry and U(1) fermion number conservation symmetry. In this paper, we construct three-dimensional exotic phases of bosons with time-reversal symmetry and boson number conservation U(1) symmetry by means of fermionic projective construction. We first construct an algebraic bosonic insulator which is a symmetric bosonic state with an emergent U(1) gapless gauge field. We then obtain many gapped bosonic states that do not break the time-reversal symmetry and boson number conservation via proper dyon condensations. We identify the constructed states by calculating the allowed electric and magnetic charges of their excitations, as well as the statistics and the symmetric transformation properties of those excitations. This allows us to show that our constructed states can be trivial SPT states (i.e. trivial Mott insulators of bosons with symmetry), non-trivial SPT states (i.e. bosonic topological insulators) and SET states (i.e. fractional bosonic topological insulators). In non-trivial SPT states, the elementary monopole (carrying zero electric charge but unit magnetic charge) and elementary dyon (carrying both unit electric charge and unit magnetic charge) are fermionic and bosonic, respectively. In SET states, intrinsic excitations may carry fractional charge.  Physics , 2014, DOI: 10.1103/PhysRevB.90.245125 Abstract: In topological phases in$2+1$dimensions, anyons fall into representations of quantum group symmetries. As proposed in our work (arXiv:1308.4673), 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$\text{Ising}\times\overline{\text{Ising}}$phase breaks into the$\mathbb{Z}_2$toric code with$\mathbb{Z}_2$global symmetry; (2) the$SU(2)_8$phase breaks into the chiral Fibonacci$\times$Fibonacci phase with a$\mathbb{Z}_2$symmetry, a first non-Abelian example of symmetry enriched topological phase beyond the gauge theory construction.  Physics , 2012, DOI: 10.1103/PhysRevB.87.165107 Abstract: We study the quantized topological terms in a weak-coupling gauge theory with gauge group$G_g$and a global symmetry$G_s$in$d$space-time dimensions. We show that the quantized topological terms are classified by a pair$(G,\nu_d)$, where$G$is an extension of$G_s$by$G_g$and$\nu_d$an element in group cohomology$\cH^d(G,\R/\Z)$. When$d=3$and/or when$G_g$is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e. gapped long-range entangled phases with symmetry). Thus, those SET phases are classified by$\cH^d(G,\R/\Z)$, where$G/G_g=G_s$. We also apply our theory to a simple case$G_s=G_g=Z_2$, which leads to 12 different SET phases in 2+1D, where quasiparticles have different patterns of fractional$G_s=Z_2$quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry$G_s$, which may lead to different fractionalizations of$G_s\$ quantum numbers and different fractional statistics (if in 2+1D).