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 the 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.
Highlights
The density of states (DOS) is a key physical quantity in condensed matter physics of noninteracting and weakly interacting systems – a plethora of electronic properties of solids depend on it, such as conductivities, thermoelectric coefficients, and screening effects [1]
The form of the integrand in the DOS relation change for these cases, we show that an integral form for the DOS always exists and can be evaluated at least numerically
By considering the hopping between distant sites and tuning their relative strengths ξm, we can engineer the DOS. This will serve as a kind of correspondence between the position-dependent and position-independent lattice models with short- and long-range hoppings, respectively, meaning that their DOS becomes identical in the limit of very large lattice sizes
Summary
The density of states (DOS) is a key physical quantity in condensed matter physics of noninteracting and weakly interacting systems – a plethora of electronic properties of solids depend on it, such as conductivities, thermoelectric coefficients, and screening effects [1]. We will focus on tight-binding models, our findings can be applied to all other position-dependent lattice models as long as they can be mapped to some non-interacting Hamiltonian This includes photonic or acoustic metamaterials, as well as spin models which effectively map to a non-interacting problem, and even a mean-field superconducting Hamiltonian with smooth spatial variations in pairing potential. Such position-dependent lattice models have been discussed in the context of modeling curved spacetimes [17] to provide gravitational analogies in quantum condensed matter systems [18].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
