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Physical Review B

Phase transitions in the classical n -vector model on the fcc lattice

J. Ricardo de Sousa, J. A. Plascak

Published Jan 18, 2008• 6 citations
Publisher

Abstract

The classical $n$-vector model on the face-centered cubic lattice with ferromagnetic and antiferromagnetic interactions is studied by using the framework of an effective field theory approach in cluster with two spins. For $n=1$ (Ising), $n=2$ ($XY$-planar rotator), $n=3$ (Heisenberg), and $n\ensuremath{\rightarrow}\ensuremath{\infty}$ (spherical) models the ferromagnetic system undergoes a second-order phase transition while the antiferromagnetic system presents a first-order behavior. The first-order character for $n=3$, which is inconclusive in the literature, seems actually to be the case for the isotropic Heisenberg antiferromagnet, and the same is expected to hold for the less studied $XY$ and spherical models. In addition, the transition temperatures are quantitatively comparable to the exact one (when available) and to those from Monte Carlo simulations and series expansion.

DOI
10.1103/physrevb.77.024419

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The classical nnn-vector model on the face-centered cubic lattice with ferromagnetic and antiferromagnetic interactions is studied by using the framework of an effective field theory approach in cluster with two spins. For n=1n=1n=1 (Ising), n=2n=2n=2 (XYXYXY-planar rotator), n=3n=3n=3 (Heisenberg), and n\ensuremath→\ensuremath∞n\ensuremath{\rightarrow}\ensuremath{\infty}n\ensuremath→\ensuremath∞ (spherical) models the ferromagnetic system undergoes a second-order phase transition while the antiferromagnetic system presents a first-order behavior. The first-order character for n=3n=3n=3, which is inconclusive in the literature, seems actually to be the case for the isotropic Heisenberg antiferromagnet, and the same is expected to hold for the less studied XYXYXY and spherical models. In addition, the transition temperatures are quantitatively comparable to the exact one (when available) and to those from Monte Carlo simulations and series expansion.
The Ising antiferromagnet on a face-centered cubic (fcc) lattice with nearest-neighbor interaction only is well known to exhibit a macroscopic (exponential in the system size LLL) ground-state degeneracy. With increasing temperature, this degeneracy is expected to be lifted and the model undergoes a first-order phase transition. For a model with an exponential degeneracy in the whole low-temperature phase, it was recently found that the finite-size scaling behavior is governed by leading correction terms $\ensuremath{\sim}{L}^{\ensuremath{-}2}$ instead of $\ensuremath{\sim}{L}^{\ensuremath{-}3}$ as usual. To test the conjecture that such a transmuted behavior may effectively persist also for the fcc antiferromagnet up to some crossover system size, we have performed parallel multicanonical Monte Carlo simulations for lattices of linear size $L\ensuremath{\le}18$ with periodic boundary conditions and determined various inverse pseudo phase transition temperatures, as well as the extremal values of the specific heat and the energetic Binder parameter. We indeed find that, for the simulated lattice sizes, the conjectured transmuted finite-size scaling ansatz fits the data better than the standard ansatz. On this basis, we extrapolate for the transition temperature an estimate of T0=1.735047(46){T}_{0}=1.735047(46)T0​=1.735047(46).
The present paper addresses a simple lattice gas model, consisting of particles that carry a three-component classical spin (i.e., a three-component unit vector), and associated with a simple-cubic lattice; each site can host one particle at most, thus implicitly allowing for hard-core repulsion; the pair interaction, restricted to nearest neighbors, is ferromagnetic and isotropic in spin space. We have studied thermodynamic properties and phase diagrams of the model by means of mean-field and two-site cluster treatments, and identified the tricritical points. Note that the fluctuating occupation numbers give rise to additional fluidlike observables in comparison with the usual saturated-lattice situation. The case of zero chemical potential has been investigated by grand-canonical Monte Carlo simulations, so as to obtain a quantitative test of the named approximations, as well as a more detailed physical characterization. The mean-field treatment was found to yield only qualitative agreement with simulation results, and the two-site cluster treatment recognizably improved upon it. Extensive simulations point to a second-order transition, whose universality class was found to be the same as its saturated-lattice counterpart.
Using Monte Carlo methods we investigate the phase transition of an Ising system on the fcc lattice, with antiferromagnetic nearest-neighbor interaction JJJ and ferromagnetic next-nearest-neighbor interaction $\ensuremath{\alpha}J$ for \ensuremath{\alpha} values up to 2. The scaling of the microcanonical entropy sL{s}_{L}sL​ with system size LLL agrees with the assumption of a first-order transition. Since the asymptotic behavior has not yet been reached, even for the largest systems with L=40,L=40,L=40, our calculations cannot determine the order of the transition unambiguously.
The phase diagram of the Heisenberg ferromagnetic model in the presence of a magnetic random field (we have used bimodal distribution) of spin S=1/2 (quantum case) and $S=\infty $ (classical case) on a simple cubic lattice is studied within the framework of the effective-field theory in finite cluster (we have chosen N=2 spins). Integrating out the part of order parameter (equation of state), we obtained an effective Landau expansion for the free energy written in terms of the order parameter Ψ(m)Ψ(m)Ψ(m). Using Maxwell construction we have obtained the phase diagram in the T−HT-HT−H plane for all interval of field. The first-order transition temperature is calculated by the discontinuity of the magnetization at Tc∗(H)T_{c}^{\ast}(H)Tc∗​(H), on the other hand in the continuous transition the magnetization is null at T=Tc(H)T=T_{c}(H)T=Tc​(H). At null temperature (T=0) we have found the \textbf{coexistence} field Hc=3.23JH_{c}=3.23JHc​=3.23J that is independent of spin value. The transition temperature Tc(H)T_{c}(H)Tc​(H) for the classical case ($S=\infty $), in the T−HT-HT−H plane, is larger than the quantum case (S=1/2).
Abstract—Phase transitions in the antiferromagnetic Heisenberg model on the body-centered cubic lattice were studied using the replica algorithm of the Monte Carlo method, taking into account the interactions between the first and second neighbors. The investigations were performed for the ratio of the value of the exchange interactions between the nearest and second neighbors in the range 0.0 ≤ r ≤ 1.0. The phase diagram of critical temperature versus the value of interaction between the second neighbors was constructed. The region in the diagram where the transition from antiferromagnetic to paramagnetic phase is of the first order was discovered for the first time in this study.
We have considered a classical spin system, consisting of 3-component unit vectors, associated with a two-dimensional lattice {uk, k ∈ Z2}, and interacting via a translationally and rotationally invariant pair potential, of the long-range ferromagnetic form [Formula: see text] Here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), and xj denotes dimensionless coordinates of the lattice sites. This potential model is known rigorously to possess an ordering transition at finite temperature, and has been characterized quantitatively by Monte Carlo simulation, whose results suggest a continuous transition taking place at [Formula: see text]; comparison is also reported with other theoretical treatments, such as Spherical Model, Molecular Field and Two Site Cluster approximations.