scholarly journals Engineering spectral properties of non-interacting lattice Hamiltonians

2021 ◽  
Vol 11 (6) ◽  
Author(s):  
Ali Moghaddam ◽  
Dmitry Chernyavsky ◽  
Corentin Morice ◽  
Jasper van Wezel ◽  
Jeroen van den Brink
Keyword(s):  
Spectral Properties ◽  
Tight Binding ◽  
Integral Form ◽  
Higher Dimensions ◽  
Exact Diagonalization ◽  
Wannier Functions ◽  
One Dimensional ◽  
Binding Models ◽  
Bounded Functions ◽  
Position Dependence

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.

scholarly journals Persistent Currents in Continuous One-Dimensional Disordered Rings within the Hartree–Fock Approximation

1997 ◽  
Vol 11 (15) ◽  
pp. 1845-1863 ◽  
Author(s):  
A. Cohen ◽  
R. Berkovits ◽  
A. Heinrich
Keyword(s):  
Energy Levels ◽  
Tight Binding ◽  
Zero Temperature ◽  
Persistent Currents ◽  
One Dimensional ◽  
Hartree Fock ◽  
Self Consistent ◽  
Binding Models ◽  
Interacting Electrons ◽  
Disorder Potential

We present numerical results for the zero temperature persistent currents carried by interacting spinless electrons in disordered one-dimensional continuous rings. The disorder potential is described by a collection of δ-functions at random locations and strengths. The calculations are performed by a self-consistent Hartree–Fock (HF) approximation. Because the HF approximation retains the concept of single-electron levels, we compare the statistics of energy levels of noninteracting electrons with those of interacting electrons as well as of the level persistent currents. We find that the e–e interactions alter the levels and samples persistent currents and introduces a preffered diamagnetic current direction. In contrast to the analogous calculations that recently appeared in the literature for interacting spinless electrons in the presence of moderate disorder in tight-binding models we find no suppression of the persistent currents due to the e–e interactions.

Diffraction on a Barrier

2018 ◽  
Author(s):  
Klaus Morawetz
Keyword(s):  
Organic Molecules ◽  
Tight Binding ◽  
Higher Dimensions ◽  
Initial Condition ◽  
Tight Binding Model ◽  
Binding Model ◽  
One Dimensional ◽  
Set Up ◽  
Planar Heterojunction ◽  
Function Matrix

The transport through a one-dimensional barrier is calculated within the tight-binding model. The surface Green’s functions are introduced as a method to invert the Green’s function matrix and to set-up convenient boundary conditions for simulations. The formalism is applied to calculate the transport properties of parallel stacked organic molecules. The extension to higher dimensions and multiband crystals is discussed. In this section we apply the GKB formalism to diffraction of electrons on a barrier. The system we study is a planar heterojunction of two ideal semi-infinite crystals or a surface of a crystal. As an initial condition we take a stream of electrons with a sharp momentum.

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