Crystalline-electromagnetic responses of topological semimetals and insulators

Hirsbrunner, Mark R.

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https://hdl.handle.net/2142/125513 Copy

Description

Title

Crystalline-electromagnetic responses of topological semimetals and insulators

Author(s)

Hirsbrunner, Mark R.

Issue Date

2024-06-03

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

Hughes, Taylor L

Doctoral Committee Chair(s)

Bradlyn, Barry

Committee Member(s)

• Madhavan, Vidya
• Covey, Jacob

Department of Study

Physics

Discipline

Physics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• crystalline symmetry
• topological insulators
• topological semimetals
• lattice defects

Abstract

Topological phases of quantum matter have been at the forefront of physics research for the past forty years, ever since the observation of the integer quantum Hall effect (IQHE) and, soon after, the discovery that its precise quantization arises from the topology of the groundstate wavefunction. The realization that the topology of the wavefunction can have meaningful physical consequences has been incredibly impactful, and topology is now a core aspect of nearly every field of physics. In the context of condensed matter physics, understanding the topology of the groundstate provides a means to further resolve distinct phases of matter beyond the Landau-Ginzburg paradigm of spontaneous symmetry breaking. Rather than being characterized by a local order parameter, topological quantum matter possesses robust global properties that are of fundamental theoretical and practical importance. These global properties are often well-described by topological effective actions, also known as topological response theories, that capture how the system responds to probe gauge fields. The paradigmatic example of this is the electromagnetic response of the IQHE, which is captured by the Abelian Chern-Simons action. These topological responses are an invaluable tool, as they both provide insight into experimental signatures of topological phases of matter and have significantly furthered our understand of quantum field theory. In condensed matter physics, the geometry of the lattice plays a central role. The crystal lattice greatly reduces the symmetry of the system from the full Poincare group to that of a space group, enabling the emergence of exotic phenomena that are otherwise prohibited. In topological phases of matter, these reduced symmetries allow the further resolution of topological phases into those that cannot be adiabatically deformed into one another while preserving certain symmetries. These symmetry protected topological phases (SPTs) are distinct from phases exhibiting topological order, which require no symmetry protection. Our understanding of SPTs protected by crystalline symmetries is currently quite robust, with the recent advent of topological quantum chemistry and the development of symmetry indicators for topological phases. In parallel to these developments, there has been a recent resurgence of interest in gauging crystalline symmetries. Crystalline gauge fields are a tool that allow us to formally treat lattice defects as fluxes of crystalline symmetries, placing them on the same footing as electromagnetic gauge fields that describe magnetic fluxes (fluxes of U(1) symmetry). As such, crystalline gauge fields enable the construction of topological effective actions that capture the coupling of electrons to the lattice geometry. These effective actions describe mixed crystalline-electromagnetic response theories in which fluxes of crystalline symmetries induce electric charge fluctuations, and, conversely, electromagnetic fluxes induces fluctuations of charges associated with crystalline symmetries (namely crystal momentum and orbital angular momentum). In this thesis we utilize the tool of crystalline gauge fields to study the crystalline-electromagnetic response of a range of topological semimetals and insulators. In Chapter 2 we begin by developing a unifying framework based on translation gauge fields that allows us to study the translation-electromagnetic responses of topological semimetals in up to D=3 dimensions. This framework not only illuminates relations between previously-known topological semimetals, but also allows us to identify a new class of quadrupolar nodal line semimetals. Within this framework, we exhaustively construct all topological effective actions that can be built by combining electromagnetic and translation gauge fields. In Chapter 3 we identify tight-binding lattice models that host each such action and confirm the predicted responses through microscopic and numerical calculations. We find that the coefficients of these responses are universally proportional to weighted momentum-energy multipole moments of the nodal points (or lines) of the semimetal. In Chapter 4 we expand the study of translation-electromagnetic responses to higher order topological semimetals. To do so we construct a model Hamiltonian for a time-reversal symmetric Weyl semimetal with a quadrupolar arrangement of higher-order Weyl nodes. It is known that similar first order Weyl semimetals exhibit a translation-electromagnetic response in which charge is bound to screw dislocations and momentum is bound to magnetic flux, both in an amount proportional to the chirality-weighted quadrupole moment of the Weyl nodes in the momentum space. We show that the higher-order nature of the Weyl nodes in our model leads to the emergence of an additional translation-electromagnetic surface response with a coefficient proportional to the dipole moment of the surface Dirac nodes that emerge with open boundary conditions. We also find that the quadrupole moment of the crystal momentum, a previously unstudied quantity, provides a link between the bulk and surface translation-electromagnetic response coefficients. Finally, we study in Chapter 5 the rotation-electromagnetic response of the insulator that emerges from a Dirac semimetal coupled to charge density wave order. Using analytic and numeric methods we show the following. First, when the CDW is lattice-commensurate, disclination-line defects of the lattice have a quantized charge per length. Second, when the CDW is inversion-symmetric, disclinations of the lattice have a quantized electric polarization. Third, when the CDW is lattice-commensurate and inversion-symmetric, disclinations are characterized by a ``disclination filling anomaly'' -- a quantized difference in the total charge bound to disclination-lines of Dirac-CDW with open and periodic boundaries. We construct an effective response theory that captures the topological responses of the Dirac-CDW insulators in terms of a total derivative term, denoted the R^F term. The R^F term describes the crystalline analog of the axion electrodynamics that are found in Weyl semimetal-CDW insulators. We also use the rotation-electromagnetic response theory to classify the strongly correlated topological phases of three-dimensional charge-ordered Dirac-semimetals.

Graduation Semester

2024-08

Type of Resource

Thesis

Handle URL

https://hdl.handle.net/2142/125513

Copyright and License Information

Copyright 2024 Mark Hirsbrunner

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