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

TL;DR

This paper develops a theoretical framework for understanding edge states in tight-binding models based on the Hermiticity of Hamiltonian operators, providing a new way to analyze boundary phenomena in lattice systems.

Contribution

It introduces a Hermiticity-based approach to derive boundary conditions and edge state Hamiltonians, applicable to models like Hofstadter, graphene, and HOTIs, with a focus on nearest-neighbor interactions.

Findings

Edge states can be described by separate Hamiltonians at system boundaries.

Hermiticity conditions lead to natural boundary conditions for lattice models.

The theory applies to various models including Hofstadter, graphene, and HOTIs.

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).