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Super Pivotal Categories, Fermion Condensation, and Fermionic Topological Phases


Aasen, David (2018) Super Pivotal Categories, Fermion Condensation, and Fermionic Topological Phases. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/P9A4-MH26.


We describe a systematic way of producing fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases which contain an emergent 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 fermionic theories, which we call “m-type” and “q-type” particles. The endomorphism algebras of q-type particles are complex Clifford algebras, and they have no analogues in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations in fermionic topological phases. We then prove a series of results relating data in condensed theories to data in their parent theories; for example, if C is a modular tensor category containing a fermion, then the tube category of 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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Topological order
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied Physics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Alicea, Jason F.
Group:Institute for Quantum Information and Matter
Thesis Committee:
  • Alicea, Jason F. (chair)
  • Motrunich, Olexei I.
  • Chen, Xie
  • Nadj-Perge, Stevan
Defense Date:18 May 2018
Non-Caltech Author Email:david (AT)
Record Number:CaltechTHESIS:05312018-132922155
Persistent URL:
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Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10982
Deposited By: David Aasen
Deposited On:01 Jun 2018 22:54
Last Modified:28 Feb 2023 19:17

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