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Topological Phases of Matter: Exactly Solvable Models and Classification


Wang, Zitao (2019) Topological Phases of Matter: Exactly Solvable Models and Classification. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BXJR-1M62.


In this thesis, we study gapped topological phases of matter in systems with strong inter-particle interaction. They are challenging to analyze theoretically, because interaction not only gives rise to a plethora of phases that are otherwise absent, but also renders methods used to analyze non-interacting systems inadequate. By now, people have had a relatively systematic understanding of topological orders in two spatial dimensions. However, less is known about the higher dimensional cases. In Chapter 2, we will explore three dimensional long-range entangled topological orders in the framework of Walker-Wang models, which are a class of exactly solvable models for three-dimensional topological phases that are not known previously to be able to capture these phases. We find that they can represent a class of twisted discrete gauge theories, which were discovered using a different formalism. Meanwhile, a systematic theory of bosonic symmetry protected topological (SPT) phases in all spatial dimensions have been developed based on group cohomology. A generalization of the theory to group supercohomology has been proposed to classify and characterize fermionic SPT phases in all dimensions. However, it can only handle cases where the symmetry group of the system is a product of discrete unitary symmetries. Furthermore, the classification is known to be incomplete for certain symmetries. In Chapter 3, we will construct an exactly solvable model for the two-dimensional time-reversal-invariant topological superconductors, which could be valuable as a first attempt to a systematic understanding of strongly interacting fermionic SPT phases with anti-unitary symmetries in terms of exactly solvable models. In Chapter 4, we will propose an alternative classification of fermionic SPT phases using the spin cobordism theory, which hopefully can capture all the phases missing in the supercohomology classification. We test this proposal in the case of fermionic SPT phases with Z2 symmetry, where Z2 is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known fermionic SPT phases in space dimensions less than or equal to 3.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Condensed Matter Theory; Topological Phases; Exactly Solvable Models; Cobordism Theory
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Physics
Minor Option:Computer Science
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Chen, Xie
Group:Institute for Quantum Information and Matter, Walter Burke Institute for Theoretical Physics
Thesis Committee:
  • Kapustin, Anton N. (chair)
  • Alicea, Jason F.
  • Endres, Manuel A.
  • Chen, Xie
Defense Date:5 April 2019
Non-Caltech Author Email:zitaowang1 (AT)
Record Number:CaltechTHESIS:04242019-205929726
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for Chapter 2. adapted for Chapter 3. adapted for Chapter 4.
Wang, Zitao0000-0002-2326-2674
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:11488
Deposited By: Zitao Wang
Deposited On:30 Apr 2019 18:25
Last Modified:28 Feb 2023 19:08

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