# Two-dimensional Wigner crystals of classical Lennard-Jones particles

## TL;DR

This paper investigates the ground-state configurations of 2D classical particles interacting via Lennard-Jones potentials, extending previous work to complex lattice structures with multiple particles per unit cell and analyzing phase transitions and energy limits.

## Contribution

It extends Bétren's analysis to multi-particle unit cells, proposing new lattice structures and systematically studying their energies and phase transitions.

## Key findings

- Second-order phase transitions are of mean-field type.
- Ground-state energy approaches that of phase-separated hexagonal lattice as particles per cell increase.
- Analytical results for low-density and hard-core limits of Lennard-Jones interactions.

## Abstract

The ground-state of two-dimensional (2D) systems of classical particles interacting pairwisely by the generalized Lennard-Jones potential is studied. Taking the surface area per particle $A$ as a free parameter and restricting oneself to periodic Bravais lattices with one particle per unit cell, B\'etermin L [2018 Nonlinearity {\bf 31} 3973] proved that the hexagonal, rhombic, square and rectangular structures minimize successively the interaction energy per particle as $A$ increases. We show here that the second-order transitions between the rhombic/square and square/rectangular phases are of mean-field type. The aim of this paper is to extend B\'etermin's analysis to periodic 2D lattices with more than one particle per elementary cell. Being motivated by previous works dealing with other kinds of models, we propose as the ground-state the extensions of the 2D rectangular (1-chain) lattice, namely the "zig-zag" (2-chain), 3-chain, 4-chain, etc. structures possessing 2, 3, 4, etc. particles per unit cell, respectively. By using a recent technique of lattice summation we find for the standard Lennard-Jones potential that their ground-state energy per particle approaches systematically as the number of particles per unit cell increases to the one of a phase separated state (the optimal hexagonal lattice). We analyze analytically the low-density limit $A\to\infty$ and the limiting hard-core case of the generalized Lennard-Jones potential.