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Fermion condensation and super pivotal categories

Aasen, David and Lake, Ethan and Walker, Kevin (2019) Fermion condensation and super pivotal categories. Journal of Mathematical Physics, 60 (12). Art. No. 121901. ISSN 0022-2488. doi:10.1063/1.5045669.

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We study fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases that contain a fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions and condensing pairs of physical and emergent fermions. There are two distinct types of objects in the resulting fermionic fusion categories, which we call “m-type” and “q-type” objects. The endomorphism algebras of q-type objects are complex Clifford algebras, and they have no analogs in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations arising from the condensed theories. We prove a series of results relating data in fermionic theories to data in their parent bosonic theories; for example, if C is a modular tensor category containing a fermion, then the tube category constructed from the condensed theory satisfies Tube(C/ψ)≅C×(C/ψ). We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted to three detailed examples of performing fermion condensation to produce fermionic topological phases: we condense fermions in the Ising theory, the SO(3)₆ theory, and the ½E₆ theory and compute the quasiparticle excitation spectrum in each of the condensed theories.

Item Type:Article
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URLURL TypeDescription Paper
Aasen, David0000-0002-6552-488X
Additional Information:Published under license by AIP Publishing. Submitted: 22 June 2018 • Accepted: 19 August 2019 • Published Online: 24 December 2019 Ethan Lake and Dave Aasen are grateful to Nick Bultinck, Nicolas Tarantino, Ryan Thorngren, Brayden Ware, and Dominic Williamson for helpful discussions. Dave Aasen is thankful to Parsa Bonderson for explaining his unpublished work at early stages of this project. Dave Aasen gratefully acknowledges support from the KITP Graduate Fellows Program, the National Science Foundation through Grant No. DMR-1723367, and the Caltech Institute for Quantum Information and Matter, an NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation through Grant No. GBMF1250. Ethan Lake was supported by the Fannie and John Hertz Foundation. Dave Aasen and Ethan Lake acknowledge support from the 2016 Boulder Summer School for Condensed Matter and Materials Physics through NSF Grant No. DMR-13001648. Kevin Walker is thankful to Zhenghan Wang and Scott Morrison for helpful conversations and to the Aspen Center for Physics and the Mathematisches Forschungsinstitut Oberwolfach for providing stimulating research environments.
Group:Institute for Quantum Information and Matter
Funding AgencyGrant Number
Kavli Institute for Theoretical PhysicsUNSPECIFIED
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
Gordon and Betty Moore FoundationGBMF1250
Fannie and John Hertz FoundationUNSPECIFIED
Issue or Number:12
Record Number:CaltechAUTHORS:20180129-085443018
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:84555
Deposited By: Tony Diaz
Deposited On:31 Jan 2018 19:07
Last Modified:15 Nov 2021 20:20

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