University of Illinois Urbana-Champaign

Probing bulk crystalline topology with broken translational symmetry: position-space markers & other techniques

Velury, Saavanth

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VELURY-DISSERTATION-2024.pdf

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https://hdl.handle.net/2142/127137

Description

Title
Probing bulk crystalline topology with broken translational symmetry: position-space markers & other techniques
Author(s)
Velury, Saavanth
Issue Date
2024-08-12
Director of Research (if dissertation) or Advisor (if thesis)
Hughes, Taylor L
Doctoral Committee Chair(s)
Bradlyn, Barry
Committee Member(s)
  • Covey, Jacob
  • Vishveshwara, Smitha
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 crystalline phases
  • defects
  • disorder
  • translational symmetry breaking
Abstract
This dissertation is comprised of works focused on theoretical investigations related to topological crystalline phases of matter. Within this field of study, tremendous progress has been achieved in the past several years in classifying non-interacting fermionic symmetry-protected topological phases, which are characterized by bulk topological invariants. Symmetries of the lattice can be exploited to reduce the complexity of characterizing the bulk nature of topological phases, namely translational symmetry, which is the assumption of a crystal with a perfect, periodic structure, along with internal symmetries (such as time-reversal, particle-hole, and chiral symmetries) and crystalline symmetries (such as rotations, reflections, etc.). The assumption of translational symmetry is quite strong, since it allows for single-particle eigenstates to be expressed as functions of the crystalline momentum, thereby allowing for the computation of topological invariants through a material's electronic band structure. However, real materials in nature do not exhibit this perfect, periodic structure because this can easily be destroyed through local impurities and defects, or through more extensive means such as disorder. The entirety of this thesis explores various aspects of breaking translational symmetry in the bulk of topological crystalline insulators. In the first chapter of this thesis, we provide a comprehensive overview of the field of topological insulators and superconductors and the extensive classification schemes developed to classify these phases based on symmetries. We also review some fundamental properties of topological crystalline insulators and topological semimetals, and review previous works examining the effects of disorder in topological phases and the topological marker approaches developed to characterize topology in bulk disordered phases. In the second chapter of this thesis, we investigate how the relationship between strong topological invariants, bulk response, and topological crystalline invariants holds under translational-symmetry breaking disorder that preserves the crystalline symmetry. We address this by studying the paradigmatic Su-Schrieffer-Heeger chain with inversion and chiral symmetries under disorder that preserves both inversion and chiral symmetries exactly. We investigate the relationship between the chiral winding number, quantized bulk polarization response, and inversion symmetry indicator invariant under strong inversion and chiral-symmetric disorder, and determine that when bulk disorder is present, the relationship between these three quantities no longer holds despite preserving the inversion symmetry, thereby illustrating that the presence of crystalline symmetry by itself is insufficient for maintaining this correspondence between strong topological invariants, bulk response, and topological crystalline invariants. Following this, in the third chapter of this thesis, we demonstrate how breaking translational symmetry through the introduction of defects in the bulk of a (topologically) trivial insulator can cause it to become topologically non-trivial, because of the topologically protected states bound to the defect despite being embedded in a trivial insulating background. To demonstrate this, we focus on D-dimensional trivial insulators composed of coupled multi-layers of d-dimensional topological semimetals (with d < D). Through defects of co-dimension (D-d), one can induce a transition from a D-dimensional trivial bulk insulator to a D-dimensional bulk topological semimetal, whereby this bulk property of the higher-dimensional system is inherited from the topological semimetal localized on the layer affected by the defect. Focusing on defects such as stacking faults and partial dislocations, we denote these defect-bound topological semimetals as embedded topological semimetals and detail various approaches of characterizing them and probing their topological signatures. Finally, in the fourth and final chapter of this thesis, we extend the momentum-space based classification of two-dimensional Chern insulators and atomic insulators protected by C_n (rotational) symmetries to position space, using basis-independent topological crystalline markers. These markers only use the ground state projector and the symmetry operators of the space group associated with the crystalline lattice. While previous approaches to classifying two-dimensional Chern insulators and atomic insulators re-expressed bulk quantities such as the Chern number, bulk polarization, and sector charge in terms of the number of occupied Bloch states carrying C_n rotation eigenvalues at each high symmetry momentum of the BZ, these methods are not generalizable to all C_n-symmetric lattices of finite dimension with different numbers of unit cells along each spatial dimension, with or without translational symmetry. Using the basis-independent markers, we extend this bulk topological classification of C_n-symmetric Chern insulators and atomic insulators to all C_n-symmetric finite-size lattices, in the Altland-Zirnbauer classes A, AI, AII, & D, encompassing atomic insulators, Chern insulators, and topological superconductors. Our reformulation of the Chern number, bulk polarization, and sector charge in terms of topological crystalline markers can be expanded in momentum-space or position-space, and we demonstrate how these markers can be used as local probes to measure bulk topology in the bulk of insulators with broken translation symmetry, but with regions where the crystalline symmetry is weakly preserved. The majority of the work discussed in this dissertation aims to expand the current methods of characterizing bulk crystalline topology without having to rely on the band structure, and provides a new framework for understanding the bulk nature of topological crystalline phases without with out translation symmetry. It is the hope of the author that the results presented in this portion of the dissertation lays the foundation for developing new methodologies that can be applied to studying crystalline systems with strong interactions and/or finite temperature, or exotic phases such as amorphous systems and quasicrystals.
Graduation Semester
2024-12
Type of Resource
Thesis
Handle URL
https://hdl.handle.net/2142/127137
Copyright and License Information
Copyright 2024 Saavanth Velury

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