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Topological Phases of Matter: Classification, Stacking Law, and Relation to Topological Quantum Field Theory

Citation

You, Minyoung (2020) Topological Phases of Matter: Classification, Stacking Law, and Relation to Topological Quantum Field Theory. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4xq7-gh61. https://resolver.caltech.edu/CaltechTHESIS:08272020-235955490

Abstract

We study aspects of gapped phases of matter, focusing on their classification, including the group law under stacking, and their relation to topological quantum field theories (TQFT). In one spatial dimension, it is well-known that Matrix Product States (MPS) efficiently approximate ground states of gapped systems; by showing that these states arise naturally in 1 + 1-dimensional lattice TQFT, which in turn are closely related to continuum TQFT, we provide a concrete connection between ground states of lattice systems and TQFT in 1 + 1 dimensions. We generalize this to systems with symmetries and fermions, and obtain a classification and group law for the stacking of 1 + 1-dimensional symmetry-protected topological phases. Further, we study the effect of turning on/off interactions for the classification: the phase classification of a given symmetry class of Hamiltonians can be different depending on whether we allow interactions or not, and in low dimensions we provide some concrete formulas relating the phases under the non-interacting classification and those under the interacting classification. Lastly, we study the phases of the 2 + 1-dimensional topological superconductor, and show that for all 16 phases braiding statistics of vortices, which determine the underlying TQFT, can be obtained by stacking layers of the basic p + ip superconductor.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Topological phases, condensed matter physics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Physics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Kapustin, Anton N.
Thesis Committee:
  • Chen, Xie (chair)
  • Marcolli, Matilde
  • Motrunich, Olexei I.
  • Kapustin, Anton N.
Defense Date:12 June 2020
Record Number:CaltechTHESIS:08272020-235955490
Persistent URL:https://resolver.caltech.edu/CaltechTHESIS:08272020-235955490
DOI:10.7907/4xq7-gh61
Related URLs:
URLURL TypeDescription
https://doi.org/10.1103/PhysRevB.96.075125DOIArticle adapted for Chapter 4, Sections 1 through 4.
https://doi.org/10.1103/PhysRevB.98.125101DOIArticle adapted for Chapter 4, Sections 5 through 8.
https://doi.org/10.1103/PhysRevB.99.035103DOIArticle adapted for Chapter 4, Sections 8 through 10.
https://doi.org/10.1103/PhysRevB.100.195128DOIArticle adapted for Chapter 5.
https://arxiv.org/abs/2008.00154arXivArticle adapted for Chapter 6.
ORCID:
AuthorORCID
You, Minyoung0000-0002-8251-953X
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:13859
Collection:CaltechTHESIS
Deposited By: Minyoung You
Deposited On:10 Sep 2020 19:09
Last Modified:10 Dec 2020 00:04

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