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Boundary Physics and Bulk-Boundary Correspondence in Topological Phases of Matter (Paperback, 1st ed. 2019)
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Boundary Physics and Bulk-Boundary Correspondence in Topological Phases of Matter (Paperback, 1st ed. 2019)
Series: Springer Theses
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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.
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