Engineering spectral properties of non-interacting lattice Hamiltonians

TL;DR

This paper derives an exact integral form for the density of states in one-dimensional lattices with position-dependent parameters and solves the inverse problem of designing lattice models with desired spectral properties, extending to multi-orbital and higher-dimensional systems.

Contribution

It provides an exact integral expression for the DOS and an inverse construction method for designing lattice Hamiltonians with specific spectral features, including generalizations to complex models.

Findings

Exact integral form for DOS in 1D lattices with position-dependent parameters

Analytic solutions for inverse design of lattice hopping and potentials

Extension of DOS formulas to multi-orbital and higher-dimensional models

Abstract

We investigate the spectral properties of one-dimensional lattices with position-dependent hopping amplitudes and on-site potentials that are smooth bounded functions of position. We find an exact integral form for the density of states (DOS) in the limit of an infinite number of sites, which we derive using a mixed Bloch-Wannier basis consisting of piecewise Wannier functions. Next, we provide an exact solution for the inverse problem of constructing the position-dependence of hopping in a lattice model yielding a given DOS. We confirm analytic results by comparing them to numerics obtained by exact diagonalization for various incarnations of position-dependent hoppings and on-site potentials. Finally, we generalize the DOS integral form to multi-orbital tight-binding models with longer-range hoppings and in higher dimensions.