http://arxiv.org/abs/1212.5121
Modular transformation and bosonic/fermionic topological orders in Abelian fractional quantum Hall states
Xiao-Gang Wen
(Submitted on 20 Dec 2012)
The non-Abelian geometric phases of the degenerate ground states was proposed as a physically measurable defining properties of topological order in 1990. In this paper we discuss in detail such a quantitative characterization of topological order, using generic Abelian fractional quantum Hall states as examples. We show that the non-Abelian geometric phases not only contain information about the quasi-particle statistics, they also contain information about the Hall viscosity and the chiral central charge of the edge states. The chiral central charge appears as the universal 1/A correction to the Hall viscosity (where A is the area of the space). Thus, the non-Abelian geometric phases (both the Abelian part and the non-Abelian part) may provide a way to completely characterize 2D topological order. Also the non-Abelian part of the geometric phases gives rise to a projective representation of the modular group (or SL(2,Z)).
http://arxiv.org/abs/1212.4863
Boundary Degeneracy of Topological Order
Juven Wang, Xiao-Gang Wen
(Submitted on 19 Dec 2012)
We introduce the notion of boundary degeneracy of topologically ordered states on a compact orientable spatial manifold with boundaries, and emphasize that it provides richer information than the bulk degeneracy. Beyond the bulk-edge correspondence, we find the ground state degeneracy of fully gapped edge states depends on boundary gapping conditions. We develop a quantitative description of different types of boundary gapping conditions by viewing them as different ways of non-fractionalized particle condensation on the boundary. This allows us to derive the ground state degeneracy formula in terms of boundary gapping conditions, which reveals the fusion algebra of fractionalized quasiparticles. We apply our results to Toric code and Levin-Wen string-net models. By measuring the boundary degeneracy on a cylinder, we predict Z_k gauge theory and U(1)_k x U(1)_k non-chiral fractional quantum hall state at even integer k can be experimentally distinguished. Our works refine definitions of symmetry protected topological order and intrinsic topological order.
http://arxiv.org/abs/1212.2121
2D Lattice Model Construction of Symmetry-Protected Topological Phases
Peng Ye, Xiao-Gang Wen
(Submitted on 10 Dec 2012)
We propose a general approach to construct symmetry protected topological (SPT) states (ie the short-range entangled states with symmetry) in 2D spin/boson systems on lattice. In our approach, we fractionalize spins/bosons into different fermions, which occupy nontrivial Chern bands. After the Gutzwiller projection of the free fermion state obtained by filling the Chern bands, we can obtain SPT states on lattice. In particular, we constructed a U(1) SPT state, a SO(3) SPT state, and a SU(2) SPT state on lattice.
http://arxiv.org/abs/1212.1827
Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
Ling-Yan Hung, Xiao-Gang Wen
(Submitted on 8 Dec 2012)
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 $\mathcal{H}^d(G,\R/\Z)$. When $d=3$ and/or when $G_s$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (ie gapped long-range entangled phases with symmetry). Thus those SET phases are classified by $\mathcal{H}^d(G,\R/\Z)$, 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 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).
To be honest, the deep architectures are genuinely