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Physical Review B

Thermodynamic properties of the fcc Ising antiferromagnet obtained from precision density of states calculations

A. D. Beath, Dominic H Ryan

Published May 17, 2006• 14 citations
Publisher

Abstract

We calculate the density of states for the face-centered-cubic (fcc) Ising model with nearest-neighbor interactions using a Wang-Landau algorithm. This allows us to calculate thermodynamic quantities at all temperatures for both the ferromagnetic (FM) and antiferromagnetic (AF) models from the same data set, while avoiding the hysteresis usually occurring in models undergoing a first-order phase transition. For the FM model, our results are in agreement with high-temperature (HT) series expansion results, and are of the same precision. For the AF model which has a first-order transition, and where precise estimates of the critical behavior are lacking, we obtain ${T}_{N}=1.7217(8)$. We also obtain estimates of the free energy, internal energy, and entropy of both the ordered and disordered states at ${T}_{N}$ with a precision comparable to that obtained in the HT series for the FM model. Details of the finite-size scaling for the AF model are discussed, and a different convergence criterion for the Wang-Landau method is introduced.

DOI
10.1103/physrevb.73.174416

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Physical Review B•Jan 18, 2008•6 citations
We calculate the density of states for the face-centered-cubic (fcc) Ising model with nearest-neighbor interactions using a Wang-Landau algorithm. This allows us to calculate thermodynamic quantities at all temperatures for both the ferromagnetic (FM) and antiferromagnetic (AF) models from the same data set, while avoiding the hysteresis usually occurring in models undergoing a first-order phase transition. For the FM model, our results are in agreement with high-temperature (HT) series expansion results, and are of the same precision. For the AF model which has a first-order transition, and where precise estimates of the critical behavior are lacking, we obtain TN=1.7217(8){T}_{N}=1.7217(8)TN​=1.7217(8). We also obtain estimates of the free energy, internal energy, and entropy of both the ordered and disordered states at TN{T}_{N}TN​ with a precision comparable to that obtained in the HT series for the FM model. Details of the finite-size scaling for the AF model are discussed, and a different convergence criterion for the Wang-Landau method is introduced.
A method for estimating the entropy SSS from a Monte Carlo simulation, suggested previously by the author, is applied to the fcc Ising antiferromagnet with nearest-neighbor interactions and external magnetic field hhh. The estimated accuracy for SSS is 0.01% to 0.5%, which is, at least, 1 order of magnitude higher than has been obtained with thermodynamic integration by Binder. Comparison of the free-energy results for the coexisting ordered and disordered phases enables us to determine with high accuracy the first-order transition points and the discontinuities in the thermodynamic functions. For the residual ground-state entropies per spin S(0)kB\frac{S(0)}{{k}_{B}}kB​S(0)​ at the critical fields hcJ=12\frac{{h}_{c}}{J}=12Jhc​​=12 and 4, we obtain 0.249 89(2) [=0.369 51(3)ln2] and 0.239(1) [=0.345(1)ln2], respectively. The first result is significantly larger than the theoretical upper bound, ${\ensuremath{\sigma}}_{u}^{\ensuremath{'}}=0.2398$ (=0.3460ln2), conjectured by Hadjukovi\ifmmode \acute{c}\else \'{c}\fi{} and Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{}. Our results are also significantly larger than Binder's result, ~⅓ ln2, obtained for both critical fields.
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).
Monte Carlo methods are employed to measure the internal energy, sublattice magnetization, and spin-spin correlation function of the fcc antiferromagnetic Ising model as a function of temperature. The internal energy of both the ordered and disordered phases is fitted by appropriate series expansions, and the free energy is obtained analytically from the series. The ordering transition is seen to be of first order with a transition temperature of 1.736\ifmmode\pm\else\textpm\fi{}0.001 in units of the nearest-neighbor coupling JJJ. These results are compared with earlier approximations of the model, in particular, the low-temperature series expansion and the Kikuchi tetrahedron approximation, and other Monte Carlo results. The spin-spin correlation function was measured in the disordered phase up to eight lattice spacings in the [100] direction. The correlation length at the transition is found to be $\ensuremath{\sim}2.5a$. The behavior of the correlation length is approximately mean-field-like.
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 original collective variables method for calculating the thermodynamic characteristics of a three-dimensional Ising-like system is developed in the approximation of the sextic distribution for modes of spin-density oscillations $({\ensuremath{\rho}}^{6}$ model), taking into account the first confluent correction. The calculations within the framework of this higher non-Gaussian approximation are performed in the temperature regions above and below the critical value of Tc{T}_{c}Tc​ and are illustrated by an example of a simple cubic lattice and an exponentially decreasing interaction potential. The collective variables method makes it possible to obtain universal and nonuniversal quantities by using a unified approach. The dependence of the critical region size and phase-transition temperature on the microscopic parameters is investigated. A microscopic analog of the Landau free energy is calculated. Explicit expressions for the thermodynamic characteristics are obtained as functions of temperature and microscopic parameters of the system. It is shown that each of the leading critical amplitudes and correction-to-scaling amplitudes can be represented as a product of a universal factor not depending on the microscopic parameters of the system and a nonuniversal factor depending on these parameters. The thermodynamic characteristics near Tc{T}_{c}Tc​ and their amplitudes are given for various values of the effective radius of the exponentially decreasing interaction potential (including the values corresponding to the nearest-neighbor interaction and the interactions between the nearest and next-nearest neighbors and between the nearest, next-nearest, and third neighbors). The results of calculations and their comparison with other authors' data show that the ${\ensuremath{\rho}}^{6}$ model provides a better quantitative description of the critical behavior of a three-dimensional Ising ferromagnet than the ${\ensuremath{\rho}}^{4}$ model.
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 $n\ensuremath{\rightarrow}\ensuremath{\infty}$ (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=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.