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

References in corpus (10)

• 99%-fidelity ballistic quantum-state transfer through long uniform channels
• A generalization of Bloch's theorem for arbitrary boundary conditions: Theory
• Generalized Aubry-André-Harper model with p-wave superconducting pairing
• One-dimensional topological insulators with noncentered inversion symmetry axis
• Externally controlled local magnetic field in a conducting mesoscopic ring coupled to a quantum wire
• The New Phase due to Symmetry Protected Piecewise Berry Phases; Enhanced Pumping and Non-reciprocity in Trimer Lattices
• Realizing the Harper model with Ultracold Atoms in a Ring Lattice
• Local Quench, Majorana Zero Modes, and Disturbance Propagation in the Ising chain
• NOON States via Quantum Walk of Bound Particles
• Edge States of a Periodic Chain with Four-Band Energy Spectrum