Analytical solution of open crystalline linear 1D tight-binding models

Abstract

A method for finding the exact analytical solutions for the bulk and edge energy levels and corresponding eigenstates for all commensurate Aubry-André/Harper single-particle models under open boundary conditions is presented here, both for integer and non-integer number of unit cells. The solutions are ultimately found to be dependent on the behavior of phase factors whose compact formulas, provided here, make this method simple to implement computationally. The derivation employs the properties of the Hamiltonians of these models, all of which can be written as Hermitian block-tridiagonal Toeplitz matrices. The concept of energy spectrum is generalized to incorporate both bulk and edge bands, where the latter are a function of a complex momentum. The method is then extended to solve the case where one of these chains is coupled at one end to an arbitrary cluster/impurity. Future developments based on these results are discussed.

18 pages, 10 figures, 2 tables

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• Anomalous topological edge modes in a periodically-driven trimer lattice
• Topological Phases of Tight-Binding Trimer Lattice in the BDI Symmetry Class
• Generalized Lieb's theorem for noninteracting non-Hermitian $n$-partite tight-binding lattices
• Impurity flat band states in the diamond chain
• High-root topological edge-state bands
• Exact and mean-field analysis of the role of Hubbard interactions on flux driven circular current in a quantum ring
• Electronic Structure of Multilayer Graphene with Arbitrary Stackings