Emergence of a collective crystal in a classical system with long-range interactions

TL;DR

This paper introduces a one-dimensional long-range classical rotator model exhibiting a phase transition to a collective crystal phase at low energies, with analytical and molecular dynamics analyses revealing a transient crystalline state near zero energy.

Contribution

It presents a new long-range rotator model demonstrating a collective crystal phase and analyzes its phase transition and dynamical properties.

Findings

A second order phase transition at critical energy $ ext{ε}_c=0.75$.

Emergence of a self-organized crystal phase at low energies.

Crystal phase persists as a long-lasting transient near zero energy.

Abstract

A one-dimensional long-range model of classical rotators with an extended degree of complexity, as compared to paradigmatic long-range systems, is introduced and studied. Working at constant density, in the thermodynamic limit one can prove the statistical equivalence with the Hamiltonian Mean Field model (HMF) and $\alpha$-HMF: a second order phase transition is indeed observed at the critical energy threshold $\varepsilon_c=0.75$. Conversely, when the thermodynamic limit is performed at infinite density (while keeping the length of the hosting interval $L$ constant), the critical energy $\varepsilon_c$ is modulated as a function of $L$. At low energy, a self-organized collective crystal phase is reported to emerge, which converges to a perfect crystal in the limit $\epsilon \rightarrow 0$. To analyze the phenomenon, the equilibrium one particle density function is analytically computed by maximizing the entropy. The transition and the associated critical energy between the gaseous and the crystal phase is computed. Molecular dynamics show that the crystal phase is apparently split into two distinct regimes, depending on the the energy per particle $\varepsilon$. For small $\varepsilon$, particles are exactly located on the lattice sites; above an energy threshold $\varepsilon{*}$, particles can travel from one site to another. However, $\varepsilon{*}$ does not signal a phase transition but reflects the finite time of observation: the perfect crystal observed for $\varepsilon >0$ corresponds to a long lasting dynamical transient, whose life time increases when the $\varepsilon >0$ approaches zero.