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arXiv:cond-mat/0301492 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Classical infinite-range-interaction Heisenberg ferromagnetic model: Metastability and sensitivity to initial conditions

Fernando Dantas Nobre, Constantino Tsallis

Published Sep 19, 2003• 48 citations
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Abstract

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.

DOI
10.1103/physreve.68.036115

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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.
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.
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.
We have performed large-scale molecular dynamics simulations of a dynamical variant of the classical Heisenberg model in which the spins are replaced by interacting "linear molecules," each of which has two rotational degrees of freedom. The Hamiltonian consists of rotational kinetic-energy terms as well as nearest-neighbor Heisenberg-like interactions $\ensuremath{-}J{\mathrm{s}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{s}}_{j}$, where si{\mathrm{s}}_{i}si​ is a unit vector specifying the orientation of the molecule at lattice site iii. Systems of size up to 643{64}^{3}643 spins on the simple cubic lattice have been studied, and thermostatting is used to maintain strictly constant temperature. We determined the dynamic structure function from the time- and space-displaced correlation functions, and found good agreement with the predictions of the dynamic scaling theory with a dynamic critical exponent of z=1.5z=1.5z=1.5. Results are compared and contrasted with data from spin-dynamics calculations on classical Heisenberg ferromagnets and antiferromagnets.
The thermodynamic properties of a classical spin-SSS Heisenberg ferromagnetic chain with long-range interactions in the presence of an external magnetic field are investigated by means of the spectral density method within the framework of the two-time Green's functions in classical statistical mechanics. For exchange interactions decaying as ${r}^{\ensuremath{-}\ensuremath{\alpha}},$ it is shown that long-range order exists at finite temperature for $1<\ensuremath{\alpha}<2$ and that no transition to a ferromagnetic phase at any finite temperature occurs for $\ensuremath{\alpha}>~2,$ consistently with a recent extension of the Mermin-Wagner theorem to spin models with long-range interactions. Besides, by means of a suitable modification of the lowest-order moment equations for the spectral density, the critical temperature and the main critical exponents for $1<\ensuremath{\alpha}<2$ and the low-temperature behavior of the paramagnetic susceptibility for $\ensuremath{\alpha}>~2$ are also obtained. Our results appear in a reasonable agreement with the available exact ones and Monte Carlo calculations.
We analyze the equilibrium properties of a chain of ferromagnetically coupled rotators which interact through a force that decays as ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ where rrr is the interparticle distance and $\ensuremath{\alpha}\ensuremath{\ge}0$. By integrating the equations of motion we obtain the microcanonical time averages of both the magnetization and the kinetic energy. We detect three different regimes depending on whether $\ensuremath{\alpha}$ belongs to the intervals [0,1)[0,1)[0,1), (1,2)(1,2)(1,2), or $(2,\ensuremath{\infty})$. For $0<\ensuremath{\alpha}<1$, the microcanonical averages agree, after a scaling, with those obtained in the canonical ensemble for the mean-field case $(\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0)$. This correspondence offers a mathematically tractable way of dealing with systems governed by slowly decaying long-range interactions.
We study equilibrium states of quantum spin systems with nonadditive long-range interactions by adopting an appropriate scaling of the interaction strength, i.e., the so-called Kac prescription. In classical spin systems, it is known that the equilibrium free energy is obtained by minimizing the free-energy functional over the coarse-grained magnetization. Here we show that it is also true for quantum spin systems. From this observation, it is found that when the canonical ensemble and the microcanonical ensemble are not equivalent in some parameter region, it is not necessarily justified to replace the actual long-range interaction by the infinite-range interaction (Curie-Weiss-type interaction). On the other hand, in the parameter region where the two ensembles are equivalent, this replacement is always justified. We examine the Heisenberg XXZ model as an illustrative example and discuss the relation to experiments.