The similarity of attractive and repulsive forces on a lattice

TL;DR

This paper explores how on a lattice, the bounded kinetic energy blurs the line between attractive and repulsive forces, showing that their effects can be transformed into each other and analyzing bound states in such systems.

Contribution

It introduces a transformation linking spectra of systems with repulsive and attractive forces on a lattice, and demonstrates how to tune continuum parameters to mimic lattice energy bands.

Findings

A transformation between repulsive and attractive force spectra.

Bound states can exist under repulsive forces on a lattice.

Parameters can be tuned to isolate the first energy band in a continuum system.

Abstract

On a lattice, as the momentum space is compact, the kinetic energy is bounded not only from below but also from above. It is shown that this, somehow removes the distinction between repulsive and attractive forces. In particular, it is seen that a region with attractive force would appear forbidden for states with energies higher than a certain value, while repulsive forces could develop bound-states. An explicit transformation is introduced which transforms the spectrum of a system corresponding to a repulsive force, to that of a similar system corresponding to an attractive force. Explicit numerical examples are presented for discrete energies of bound-states of a particle experiencing repulsive force by a piecewise constant potential. Finally, the parameters of a specific one dimensional translationally invariant system on continuum are tuned so that the energy of the system resembles the kinetic energy of a system on a one dimensional lattice. In particular, it is shown that the parameters could be tuned so that while the width of the first energy band and its position are kept finite, the energy gap between the first energy band and the next energy band go to infinity, so that effectively only the first energy band is relevant.