Boundary Physics and Bulk-Boundary Correspondence in Topological Phases of Matter

1. Front Matter

   Pages i-xvii

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2. Introduction

   • Abhijeet Alase

   Pages 1-12

3. Generalization of Bloch’s Theorem to Systems with Boundary

   • Abhijeet Alase

   Pages 13-63

4. Investigation of Topological Boundary States via Generalized Bloch Theorem

   • Abhijeet Alase

   Pages 65-122

5. Matrix Factorization Approach to Bulk-Boundary Correspondence

   • Abhijeet Alase

   Pages 123-157

6. Mathematical Foundations to the Generalized Bloch Theorem

   • Abhijeet Alase

   Pages 159-190

7. Summary and Outlook

   • Abhijeet Alase

   Pages 191-198

8. Back Matter

   Pages 199-200

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This thesis extends our understanding of systems of independent electrons by developing a generalization of Bloch’s Theorem which is applicable whenever translational symmetry is broken solely due to arbitrary boundary conditions. The thesis begins with a historical overview of topological condensed matter physics, placing the work in context, before introducing the generalized form of Bloch's Theorem. A cornerstone of electronic band structure and transport theory in crystalline matter, Bloch's Theorem is generalized via a reformulation of the diagonalization problem in terms of corner-modified block-Toeplitz matrices and, physically, by allowing the crystal momentum to take complex values. This formulation provides exact expressions for all the energy eigenvalues and eigenstates of the single-particle Hamiltonian. By precisely capturing the interplay between bulk and boundary properties, this affords an exact analysis of several prototypical models relevant to symmetry-protected topological phases of matter, including a characterization of zero-energy localized boundary excitations in both topological insulators and superconductors. Notably, in combination with suitable matrix factorization techniques, the generalized Bloch Hamiltonian is also shown to provide a natural starting point for a unified derivation of bulk-boundary correspondence for all symmetry classes in one dimension.

• Bloch's theorem
• Bulk-boundary correspondence
• Altland-Zirnbauer symmetry class
• topological boundary states
• gapless quasiparticle excitation
• stability of zero modes
• bulk fermionic wavefunction

• Institute for Quantum Science and Technology, University of Calgary, Calgary, Canada

  Abhijeet Alase

Abhijeet Alase is a postdoctoral researcher at the Institute for Quantum Science and Technology of the University of Calgary. He received his PhD from Dartmouth College in 2019.

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