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The relevant part of the sum as $\sum_{k^*,s_1,s_2}\delta_{k^*,s_1s_2}d_{s_1}d_{s_2}B^k_P$ Let me assume that the fusion category has no multiplicities, so $N_{ab}^c=0,1$, which I think Levin and Wen also assumed. We can write the sum as $\sum_{k^*,s_1}d_{s_1}B^k_P\sum_{s_2\in s_1\times \bar{k}}d_{s_2}$ This is because if $\delta_{k^*,s_1s_2}=1$, it ...


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Here is a partial answer that depends on a particular choice of local gauge constraint. In a U(1) gauge theory, the usual gauge constraint is just Gauss' Law, $$ \nabla \cdot \mathbf{E} = \rho. $$ This in turn implies Coulomb's Law $\mathbf{E} \sim 1/r$ for the electric field surrounding a deconfined point charge. Such a long-range interaction ought to be ...


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Is the existence of deconfined gauge charges a sufficient condition to ensure gaplessness? I think the answer is NO, such as the $Z_2$ gauge theory in 2+1D and 3+1D. I believe that the existence of deconfined gauge charges of a continuous gauge group is a sufficient condition to ensure gaplessness? Hastings and I have a paper ...


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Perhaps a way to understand all this in a slightly different way: when $\mu=0$ and $\Delta=t$, a half-second look at your Hamiltonian shows that $\gamma_{j}^{A}\gamma_{j+1}^{B}$ is an eigenmode of the problem. Similar with $\gamma_{j+1}^{A}\gamma_{j}^{B}$ when $\mu=0$ and $\Delta=-t$. In both cases two Majorana are left unpaired, because the sum runs only to ...


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It is not true that if $\mathbf{k}\neq -\mathbf{k}$ the matrix element $\langle u_i(\mathbf{k}|T|u_j(\mathbf{k})\rangle$ vanishes. Remember that $u(\mathbf{k})$ are Bloch wavefunctions, which are eigenvectors of the momentum space Hamiltonian $H(\mathbf{k})$ (e.g. $H(\mathbf{k})$ in Kane-Mele model is just a $4\times 4$ matrix at a given $\mathbf{k}$). ...


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My paper with Liang Kong, arXiv:1405.5858, presents the following conjecture: Bosonic topological orders in $n$-space-time dimensions (after quotient out the invertible topological orders) are described/classified by modular unitary $n$-categories with one object. The reason that modular unitary $n$-categories fail to classify topological orders is because ...



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