Numerical methods of characterizing symmetry protected topological states in one dimension

Zhuang, Ye

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https://hdl.handle.net/2142/102788 Copy

Description

Title

Numerical methods of characterizing symmetry protected topological states in one dimension

Author(s)

Zhuang, Ye

Issue Date

2018-11-05

Director of Research (if dissertation) or Advisor (if thesis)

Hughes, Taylor L.

Doctoral Committee Chair(s)

Gadway, Bryce

Committee Member(s)

• Mason, Nadya
• Clark, Bryan K.

Department of Study

Physics

Discipline

Physics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• topological phases
• numerical

Abstract

In this dissertation, we use numerical methods to study one dimensional symmetry protected topological (SPT) phases. We focus on the density matrix renormalization group (DMRG) methods and explore the machine learning methods. We investigated different SPT phases in the context of interactions and disorders. The application of machine learning methods reveals new insights into the topological phases. We begin by studying the Z3 parafermionic chain, the simplest generalization of the Kitaev p-wave wire. The quantum entanglement diagnostics we performed allow us to determine phase boundaries, and the nature of the phase transitions. An intervening incommensurate phase is found between the topological and trivial phases. We locate and characterize a putative tricritical point in the phase diagram where the three above mentioned phases meet at a single point. The phase diagram is predicted to contain a Lifshitz type transition which we con rm using entanglement measures. As another generalization of the Kitaev p-wave wire, we study the interacting inversion symmetric superconductor. We introduce interaction and inversion symmetry and preserve its original time-reversal, particle-hole and chiral symmetry. The symmetries indicates a Z2 classification. We study the quantum entanglement, teleportation and fractional Josephson effects of this system. The ground state of the topological phase is a condensation of four electrons instead of cooper-pairs. While there is a nonzero teleportation for cooper-pairs, the teleportation of one electron is suppressed. The inversion symmetry restricts the edge modes of the system to be cooper-pairs other than two uncorrelated electrons. It is also proved by the 2 pi periodicity in the fractional Josephson effects. At last we apply machine learning methods for classification of SPT phases when strong disorder is present. The entanglement spectrum is used as features to train the random forest model. We do the training using the data generated from a small fraction in the parameter space. The model can give high accuracy predictions to other regions in the phase space. It is even able to make correct predictions to system in a different symmetry class. A detailed analysis of the model indicates that it is able to capture the degeneracy in the entanglement spectrum.

Graduation Semester

2018-12

Type of Resource

text

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http://hdl.handle.net/2142/102788

Copyright and License Information

Copyright 2018 Ye Zhuang

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