Theory of edge states based on the hermiticity of tight-binding Hamiltonian operators

Abstract

We develop a theory of edge states based on the Hermiticity of Hamiltonian operators for tight-binding models defined on lattices with boundaries. We describe Hamiltonians using shift operators which serve as differential operators in continuum theories. It turns out that such Hamiltonian operators are not necessarily Hermitian on lattices with boundaries, which is due to the boundary terms associated with the summation by parts. The Hermiticity of Hamiltonian operators leads to natural boundary conditions, and for models with nearest-neighbor (NN) hoppings only, there are reference states that satisfy the Hermiticity and boundary conditions simultaneously. Based on such reference states, we develop a Bloch-type theory for edge states of NN models on a half-plane. This enables us to extract Hamiltonians describing edge-states at one end, which are separated from the bulk contributions. It follows that we can describe edge states at the left and right ends separately by distinct Hamiltonians for systems of cylindrical geometry. We show various examples of such edge state Hamiltonians (ESHs), including Hofstadter model, graphene model, and higher-order topological insulators (HOTIs).

arXiv comment24.1 pages, 24 figures, v2: references added, Sec. VII B 3 added, v3: references added, typos corrected

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• Zeros of Green Functions in Topological Insulators
• Multi-Channel Luttinger Liquids at the Edge of Quantum Hall Systems
• Chern numbers associated with the periodic Toda lattice
• Hard wall edge confinement in 2D topological insulators and the energy of the Dirac Point
• Multiple topological corner states in the continuum of extended kagome lattice
• Loop unitary and phase band topological invariant in generic multi-band Chern insulators
• Boundary Condition Analysis of First and Second Order Topological Insulators