Energy bands in a three dimension simple cubic lattice of contact potential

TL;DR

This paper analyzes energy bands in a 3D simple cubic lattice with contact potential, revealing how lattice potential influences negative energy states, degeneracy, and the existence of dark states through group theory and explicit wave function construction.

Contribution

It introduces a detailed analysis of energy bands in a cubic lattice with contact potential, including the effects on negative states, degeneracy, and the identification of dark states using group theory.

Findings

Negative energy states exist for any weak periodic potential.

Degeneracy explained via group theory and wave function symmetry.

Existence of dark states unaffected by lattice potential.

Abstract

In this work, we investigate energy bands in a three dimensional simple cubic lattice of contact potential. The energy bands in the first Brillouin Zone are obtained with Ewald's summation method. In comparison with single point potential, the presence of lattice potential changes the existence condition of negative energy states near zero energy. It is found that the system always has negative energy states for an arbitrarily weak periodic potential. In addition, we prove that if an irreducible unitary representation is not a trivial representation of group of wave vector, the corresponding wave functions at lattice sites would be zero. With this theorem, the degeneracy of energy bands is explained with group theory. Furthermore, we find that there exists some energy bands which are not affected by the lattice potential. We call their corresponding eigenstates as dark states. The physical mechanism of the dark states is explained by explicitly constructing the standing wave-type Bloch wave functions.