Controlling the Range of Interactions in the Classical Inertial Ferromagnetic Heisenberg Model: Analysis of Metastable States

Abstract

A numerical analysis of a one-dimensional Hamiltonian system, composed by $N$ classical localized Heisenberg rotators on a ring, is presented. A distance $r_{ij}$ between rotators at sites $i$ and $j$ is introduced, such that the corresponding two-body interaction decays with $r_{ij}$ as a power-law, $1/r_{ij}^{\alpha}$ ($\alpha \ge 0$). The index $\alpha$ controls the range of the interactions, in such a way that one recovers both the fully-coupled (i.e., mean-field limit) and nearest-neighbour-interaction models in the particular limits $\alpha=0$ and $\alpha\to\infty$, respectively. The dynamics of the model is investigated for energies $U$ below its critical value ($U<U_{c}$), with initial conditions corresponding to zero magnetization. The presence of quasi-stationary states (QSSs), whose durations $t_{\rm QSS}$ increase for increasing values of $N$, is verified for values of $\alpha$ in the range $0 \leq \alpha <1$, like the ones found for the similar model of XY rotators. Moreover, for a given energy $U$, our numerical analysis indicates that $t_{\rm QSS} \sim N^{\gamma}$, where the exponent $\gamma$ decreases for increasing $\alpha$ in the range $0 \leq \alpha <1$, and particularly, our results suggest that $\gamma \to 0$ as $\alpha \to 1$. The growth of $t_{\rm QSS}$ with $N$ could be interpreted as a breakdown of ergodicity, which is shown herein to occur for any value of $\alpha$ in this interval.