University of Illinois Urbana-Champaign

Geometry-charge responses in topological phases of matter

May-Mann, Julian

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MAY-MANN-DISSERTATION-2023.pdf

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https://hdl.handle.net/2142/121921

Description

Title
Geometry-charge responses in topological phases of matter
Author(s)
May-Mann, Julian
Issue Date
2023-07-26
Director of Research (if dissertation) or Advisor (if thesis)
Hughes, Taylor L
Doctoral Committee Chair(s)
Fradkin, Eduardo
Committee Member(s)
  • Madhavan, Vidya
  • Leigh, Robert
Department of Study
Physics
Discipline
Physics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
  • Theoretical physics
  • condensed matter
  • topology
  • geometry
Abstract
Topological phases of matter have been a corner stone of modern condensed matter physics over the last several decades. One reason that topological phases have attracted so much attention is that they can display novel quantized responses, known as topological responses. The paradigmatic topological response is the quantized Hall conductance displayed by two-dimensional quantum Hall insulators. Such responses are of practical importance, as they serve as smoking-gun signatures of topological physics in experimental settings. In theoretical contexts, topological responses have greatly informed the modern understanding of topological quantum field theory and quantum anomalies. This thesis concerns itself with analyzing the quantized geometry-charge responses that occur in topological phases of crystalline matter. Formally, geometry-charge responses are responses where charge fluctuations are driven by fluxes of spatial symmetries. We are specifically concerned with the responses that are driven by fluxes of the discrete rotation symmetry of a crystalline lattice. Physically, rotation symmetry fluxes are disclination defects of the lattice. A notable example of a topological geometry-charge response occurs in two- dimensions, where a quantized charge is bound to disclination defects. This response and the generalization we will consider in this thesis, provide new insights into how topology and geometry can become intertwined in condensed matter systems. It is also potentially possible to observe these responses in experimental studies of imperfect crystalline material, or designer metamaterials. The aforementioned two-dimensional disclination-charge response has been shown to occur in certain higher-order topological insulators (insulators with gapped bulks and boundaries, but anomalous corner modes). In Chapter 2, we show that this geometry-charge response is an essential ingredient for describing the universal low-energy physics of rotation-invariant higher-order topological insulators. We derive an effective topological field theory description of rotation-invariant higher-order topological insulators that provides a unified description of the corner and disclination charge responses. This theory consists of a physical geometry (which encodes the disclinations), and an effective geometry (which encodes the corners). In this formulation, the corner physics is related to the disclination response via a quantum anomaly at the surface of the higher-order topological insulator. We also extend the effective field theory to interacting systems, and ii consider topologically ordered counterparts to higher-order topological insulators. Based on the results of our analysis of two-dimensional insulators, we consider the geometry-charge responses of three-dimensional topological crystalline insulators in Chapters 3 and 4. In Chapter 3 we show that three-dimensional insulators with rotation symmetry can possess a novel response where disclination-lines of the lattice carry electric polarization. There is an accompanying dual response, where magnetic monopoles in the bulk of the insulator bind angular momentum (analogously to how magnetic monopoles bind electric charge in the Witten effect). These responses are described by a three-dimensional topological response term that couples the lattice curvature to the electromagnetic field. The addition of mirror or particle- hole symmetry quantizes the mixed geometry-charge response and defines a new class of rotation-invariant topological crystalline insulators. Notably, the charge bound to disclinations on gapped surfaces of these insulators is half the minimal amount that can occur in purely two-dimensional insulators with the same symmetries. We construct lattice models of these rotation-invariant topological phases and numerically verify that they exhibit the three-dimensional geometry-charge responses. In Chapter 4 we use geometry-charge responses to characterize the correlated insulating phases that form when the translation symmetry of a three-dimensional Dirac semimetal is spontaneously broken by a charge density wave. We show that disclination-lines of these Dirac-charge density wave insulators bind a quantized charge per charge density wave period when defined on periodic geometries. Provided the charge density wave preserves inversion symmetry, the Dirac-charge density wave insulators can also display a disclination filling anomaly: a quantized difference in the charge bound to disclination for systems with open and periodic boundaries. The disclination charge, and the disclination filling anomaly can respectively be thought of as crystalline analogs to the quantum anomalous Hall effect and quantized axion electrodynamics displayed by Weyl-charge density wave insulators (the insulators that form when a charge density wave breaks the translation symmetry of a Weyl semimetal). We present an effective topological response theory that captures the quantized geometry-charge responses of the Dirac-charge density wave insulator. In particular, we show that the response theories for insulators with and without a disclination line filling anomaly differ by the topological term we first presented in Chapter 3.
Graduation Semester
2023-12
Type of Resource
Thesis
Handle URL
https://hdl.handle.net/2142/121921
Copyright and License Information
Copyright 2023 Julian May-Mann

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Graduate Theses and Dissertations at Illinois