Harmonic Stability Analysis of the 2D Square and Hexagonal Bravais Lattices for a Finite--Ranged Repulsive Pair Potential. Consequence for a 2D System of Ultracold Composite Bosons

TL;DR

This study analyzes the vibrational stability of 2D square and hexagonal lattices with finite-range repulsive interactions, concluding that hexagonal lattices are stable and likely represent the crystalline phase of ultracold composite bosons.

Contribution

It provides an analytical and numerical stability analysis of 2D lattice structures with finite-range interactions, highlighting the stability of hexagonal lattices for ultracold composite bosons.

Findings

Square lattice is unstable under transverse vibrations.

Hexagonal lattice can be stable depending on interaction range.

Crystalline phase of ultracold composite bosons likely forms a hexagonal lattice.

Abstract

We consider a classical, two-dimensional system of identical particles which interact via a finite-ranged, repulsive pair potential. We assume that the system is in a crystalline phase. We calculate the normal vibrational modes of a two-dimensional square Bravais lattice, first analytically within the nearest-neighbour approximation, and then numerically, relaxing the preceding hypothesis. We show that, in the harmonic approximation, the excitation of a transverse vibrational mode leads to the breakdown of the square lattice. We next study the case of the hexagonal Bravais lattice and we show that it can be stable with respect to lattice vibrations. We give a criterion determining whether or not it is stable in the nearest-neighbour approximation. Finally, we apply our results to a two-dimensional system of composite bosons and infer that the crystalline phase of such a system, if it exists, corresponds to a hexagonal lattice.