Some statistical equilibrium mechanics and stability properties of a class of two-dimensional Hamiltonian mean-field models

• Cold atoms may become a playground to prepare long-range systems in the laboratory.
• In this purpose, a class of 2D long-range mean-field models is introduced.
• The long-range interaction may be attractive, repulsive or a mixture of both cases.
• Microcanonical Monte Carlo computations allow us to determine equilibrium.
• A Bravais-like lattice is introduced to classify the low-energy real 2D-space structure.

A two-dimensional class of mean-field models that may serve as a minimal model to study the properties of long-range systems in two space dimensions is considered. The statistical equilibrium mechanics is derived in the microcanonical ensemble using Monte Carlo simulations for different combinations of the coupling constants in the potential leading to fully repulsive, fully attractive and mixed attractive–repulsive potential along the Cartesian axis and diagonals. Then, having in mind potential realizations of long-range systems using cold atoms, the linear theory of this two-dimensional mean-field Hamiltonian models is derived in the low temperature limit.