Dynamical threshold in the relaxation of long-range systems

The dynamics of systems with mean-field interaction, when all the particles interact with equal strength, is characterized by very long equilibration times: Indeed, the system gets trapped in quasi-stationary states, where macroscopic quantities (temperature, magnetization, etc) evolve very slowly toward their equilibrium value. The lifetime of these states generally increases with the system size and, as a consequence, the time for the system to reach equilibrium diverges in the thermodynamic limit.

We here investigate the equilibration of lattice systems with long-range pair interactions, decaying like $1/r^\alpha$ with the distance $r$. The long-range regime corresponds to $\alpha<d$, with $d$ the dimension of the system. We characterize the relaxation times and show that long-range lattice systems also exhibit quasi-stationarity, as well as relaxation times that diverge with the size of the system.

However, upon varying the interaction range $\alpha$, we find evidence for the existence of a threshold at $\alpha=d/2$, at which the relaxation behaviour changes qualitatively and the corresponding scaling exponents switch to a different regime. Since our observation is based on the behaviour of both a quantum and a classical system, investigated analytically and numerically, for ferro- and anti-ferromagnetic interactions, we conjecture this threshold and some of its characteristic properties to be universal.