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arXiv:1409.5825 · Journal of Statistical Mechanics Theory and Experiment

Controlling the range of interactions in the classical inertial ferromagnetic Heisenberg model: analysis of metastable states

Leonardo J L Cirto, Leonardo S Lima, Fernando D Nobre

Published Apr 29, 2015• 27 citations
arXivHTMLPDF

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 i j α ( α ⩾ 0). The index α 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 α = 0 and α → ∞, 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 QSS increase for increasing values of N , is verified for values of α in the range 0 ⩽ α < 1, like the ones found for the similar model of XY rotators. Moreover, for a given energy U , our numerical analysis indicates that t QSS ∼ N γ , where the exponent γ decreases for increasing α in the range 0 ⩽ α < 1 and particularly, our results suggest that γ → 0 as α → 1. The growth of t QSS with N could be interpreted as a breakdown of ergodicity, which is shown herein to occur for any value of α in this interval.

DOI
10.1088/1742-5468/2015/04/p04012

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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 i j α ( α ⩾ 0). The index α 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 α = 0 and α → ∞, 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 QSS increase for increasing values of N , is verified for values of α in the range 0 ⩽ α < 1, like the ones found for the similar model of XY rotators. Moreover, for a given energy U , our numerical analysis indicates that t QSS ∼ N γ , where the exponent γ decreases for increasing α in the range 0 ⩽ α < 1 and particularly, our results suggest that γ → 0 as α → 1. The growth of t QSS with N could be interpreted as a breakdown of ergodicity, which is shown herein to occur for any value of α in this interval.
An N-sized inertial classical Heisenberg ferromagnet, which consists of a modification of the well-known standard model, where the spins are replaced by classical rotators, is studied in the limit of infinite-range interactions. The usual canonical-ensemble mean-field solution of the inertial classical n-vector ferromagnet (for which n=3 recovers the particular Heisenberg model considered herein) is briefly reviewed, showing the well-known second-order phase transition. This Heisenberg model is studied numerically within the microcanonical ensemble through molecular dynamics. In what concerns the caloric curve, it is shown that, far from criticality, the kinetic temperature obtained at the long-time-limit microcanonical-ensemble simulation recovers well the equilibrium canonical-ensemble estimate, whereas, close to criticality, a discrepancy (presumably due to finite-size effects) is found. The time evolution of the kinetic temperature indicates that a basin of attraction exists for the initial conditions for which the system evolves into a metastable state, whose duration diverges as N--> infinity, before attaining the terminal thermal equilibrium. Such a metastable state is observed for a whole range of energies, which starts right below criticality and extends up to very high energies (in fact, the gap between the kinetic temperatures associated with the metastable and the terminal-equilibrium states is expected to disappear only as one approaches infinite energy). To the best our knowledge, this has never before been observed on similar Hamiltonian models, in a noticeable way, for such a large range of energies. For example, for the XY (n=2) version of the present model, such a behavior was observed only near criticality. It is shown also that the (metastable state) maximum Lyapunov exponent decreases with N like lambda(max) approximately N-kappa, where for the initial conditions employed herein (maximal magnetization), kappa=0.225+/-0.030, both above and below the critical point.
Abstract We study, by means of first-principle molecular-dynamics, the temperature T QSS of the quasi-stationary state (QSS) of a d -dimensional classical inertial α -Heisenberg ferromagnetic model, consisting of a set of rotators with interactions decaying with distance as r − α ( α ⩾ 0 ) . We find, through detailed finite-size calculations, the scaling relation T QSS − T ∞ ∝ N − β , where T ∞ ≡ U − 1 / 2 , with energy per particle U < U c ( U c = 5 / 6 ), and total number of particles N , which holds for the very long-range interactions regime 0 ⩽ α / d ⩽ 1 , in which the QSS exists. We show that the scaling exponent β does not depend on U but only in the quotient α / d , being β ( α / d ) ≃ 1 for 0 ⩽ α / d ⩽ 0.5 , and decaying linearly for α / d > 0.5 , so <
The duration of the quasistationary states (QSSs) emerging in the ddd-dimensional classical inertial $\ensuremath{\alpha}$-Heisenberg model, i.e., NNN three-dimensional rotators whose interactions decay with distance rij{r}_{ij}rij​ as $1/{r}_{ij}^{\ensuremath{\alpha}}$ ($\ensuremath{\alpha}\ensuremath{\ge}0$), is studied through first-principle molecular dynamics. These QSSs appear for the very-long-range interaction regime ($0\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}1$), for an average energy per rotator U<UcU<{U}_{c}U<Uc​ (Uc=5/6{U}_{c}=5/6Uc​=5/6), and they do not exist for U>UcU>{U}_{c}U>Uc​. They are characterized by a kinetic temperature TQSS{T}_{\text{QSS}}TQSS​, before a crossover to a second plateau occurring at the Boltzmann-Gibbs temperature TBG>TQSS{T}_{\text{BG}}>{T}_{\text{QSS}}TBG​>TQSS​. We investigate here the behavior of their duration tQSS{t}_{\text{QSS}}tQSS​ when UUU approaches Uc{U}_{c}Uc​ from below, for large values of NNN. The QSS gradually disappears as $U\ensuremath{\rightarrow}{U}_{c}$, while its duration undergoes a critical phenomenon, namely ${t}_{\text{QSS}}\ensuremath{\propto}{({U}_{c}\ensuremath{-}U)}^{\ensuremath{-}\ensuremath{\xi}}$. Universality is found for the critical exponent $\ensuremath{\xi}=1\text{.}67\ifmmode\pm\else\textpm\fi{}0\text{.}02$ throughout the very-long-range interaction regime.
Long-lived quasi-stationary states (QSSs) are a signature characteristic of long-range interacting systems both in the classical and in the quantum realms. Often, they emerge after a sudden quench of the Hamiltonian internal parameters and present a macroscopic lifetime, which increases with the system size. Despite their ubiquity, the fundamental mechanism at their root remains unknown. Here, we show that the spectrum of systems with power-law decaying couplings remains discrete up to the thermodynamic limit. As a consequence, several traditional results on the chaotic nature of the spectrum in many-body quantum systems are not satisfied in the presence of long-range interactions. In particular, the existence of QSSs may be traced back to the finiteness of Poincaré recurrence times. This picture justifies and extends known results on the anomalous magnetization dynamics in the quantum Ising model with power-law decaying couplings. The comparison between the discrete spectrum of long-range systems and more conventional examples of pure point spectra in the disordered case is also discussed.
Systems with long-range interactions, while relaxing towards equilibrium, sometimes get trapped in long-lived non-Boltzmann quasistationary states (QSS) which have lifetimes that grow algebraically with the system size. Such states have been observed in models of globally coupled particles that move under Hamiltonian dynamics either on a unit circle or on a unit spherical surface. Here, we address the ubiquity of QSS in long-range systems by considering a different dynamical setting. Thus, we consider an anisotropic Heisenberg model consisting of classical Heisenberg spins with mean-field interactions and evolving under classical spin dynamics. Our analysis of the corresponding Vlasov equation for time evolution of the phase space distribution shows that in a certain energy interval, relaxation of a class of initial states occurs over a timescale which grows algebraically with the system size. We support these findings by extensive numerical simulations. This work further supports the generality of occurrence of QSS in long-range systems evolving under Hamiltonian dynamics.