Disentangling the Generalized Double Semion Model
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Communications in
Mathematical Physics
Disentangling the Generalized Double Semion Model Lukasz Fidkowski1 , Jeongwan Haah2 , Matthew B. Hastings2,3 , Nathanan Tantivasadakarn4 1 Department of Physics, University of Washington, Seattle, WA 98195, USA.
E-mail: [email protected]
2 Quantum Architectures and Computation, Microsoft Research, Redmond, WA 98052, USA. 3 Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA. 4 Department of Physics, Harvard University, Cambridge, MA 02138, USA.
Received: 24 June 2019 / Accepted: 14 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: We analyze the class of Generalized Double Semion (GDS) models in arbitrary dimensions from the point of view of lattice Hamiltonians. We show that on a d-dimensional spatial manifold M the dual of the GDS is equivalent, up to constant depth local quantum circuits, to a group cohomology theory tensored with lower dimensional cohomology models that depend on the manifold M. We comment on the space-time topological quantum field theory (TQFT) interpretation of this result. We also investigate the GDS in the presence of time reversal symmetry, showing that it forms a non-trivial symmetry enriched toric code phase in odd spatial dimensions. 1. Introduction Gapped quantum phases of matter are, roughly speaking, equivalence classes of gapped lattice Hamiltonians under smooth deformation of parameters, in the thermodynamic limit of large system size. Some (but not all [1]) gapped quantum phases also possess effective topological quantum field theory (TQFT) descriptions [2–4], which capture their universal low energy features, such as topological ground state degeneracy and quasiparticle braiding statistics. TQFTs can be used to distinguish such gapped phases, but it is not known whether they provide a complete set of invariants. One class of TQFTs that arise from gapped lattice Hamiltonians and can be defined in arbitrary dimension are Dijkgraaf-Witten gauge theories [5]. The generalized double semion model (GDS) [6] is another gapped lattice Hamiltonian that can be defined in arbitrary dimension. Given the relative scarcity of TQFTs in spatial dimensions greater than 2, it is natural to ask what, if any, is the relation between the two. It was shown [6] that for odd space dimension d the GDS model is equivalent to a toric code up to a local quantum circuit, but in even dimensions d ≥ 4 its ground state degeneracy does not match that of any Z2 Dijkgraaf-Witten theory [5]. It was shown [7] that one could define a TQFT such that on any closed manifold M, the state spaces of the TQFT match the space of low-energy states of the GDS model, and this isomorphism is compatible with
L. Fidkowski, J. Haah, M. B. Hastings, N. Tantivasadakarn
the actions of the mapping class group of M on both spaces. This TQFT is a so-called “gauge-gravity TQFT.” This term is used because the action in the TQFT combines the Z2 gauge field with Stiefel-Whitney classes of the underlying manifold. Roughly speaking, the action includes
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