43,720 research outputs found

    Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions

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    It has been shown that the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, there exist bosonic SPT phases beyond the current Chern-Simons theory framework. We also apply the theory to fermionic SPT phases with Zm\mathbb{Z}_m symmetry and find the classification of SPT phases depends on the parity of mm: for even mm there are 2m2m classes, mm out of which is intrinsically fermionic SPT phases and can not be realized in any bosonic system. Finally we propose a classification scheme of fermionic SPT phases for any finite, Abelian symmetry.Comment: published versio

    Non-Divergence of Unipotent Flows on Quotients of Rank One Semisimple Groups

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    Let GG be a semisimple Lie group of rank 11 and Γ\Gamma be a torsion free discrete subgroup of GG. We show that in G/ΓG/\Gamma, given ϵ>0\epsilon>0, any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than δ \delta for 1ϵ1-\epsilon proportion of the time for some δ>0\delta>0. The result also holds for any finitely generated discrete subgroup Γ\Gamma and this generalizes Dani's quantitative nondivergence theorem \cite{D} for lattices of rank one semisimple groups. Furthermore, for a fixed ϵ>0\epsilon>0 there exists an injectivity radius δ\delta such that for any unipotent trajectory {utx}t[0,T]\{u_tx\}_{t\in [0,T]}, either it spends at least 1ϵ1-\epsilon proportion of the time in the set with injectivity radius larger than δ\delta for all large T>0T>0 or there exists a {ut}tR\{u_t\}_{t\in\mathbb{R}}-normalized abelian subgroup LL of GG which intersects gΓg1g\Gamma g^{-1} in a small covolume lattice. We also extend these results when GG is the product of rank-11 semisimple groups and Γ\Gamma a discrete subgroup of GG whose projection onto each nontrivial factor is torsion free.Comment: 23 page