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Physical review. B, Condensed matter

Computer simulation study of hysteresis and free energy in the fcc Ising antiferromagnet

Hagai Meirovitch

Published Sep 1, 1984• 48 citations
Publisher

Abstract

A method for estimating the entropy $S$ 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 $h$. The estimated accuracy for $S$ 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 $\frac{S(0)}{{k}_{B}}$ at the critical fields $\frac{{h}_{c}}{J}=12$ 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.

DOI
10.1103/physrevb.30.2866

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Journal of Experimental and Theoretical Physics•Sep 1, 1999•0 citations
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, \ensuremathσu\ensuremath′=0.2398{\ensuremath{\sigma}}_{u}^{\ensuremath{'}}=0.2398\ensuremathσu\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.
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.
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.
Inspired by the work by Kämmerer, Dünweg, Binder, and d'Onorio de Meo we investigated the spin-[Formula: see text] Ising antiferromagnet on a fcc lattice in a homogeneous external magnetic field H with uniform nearest-neighbor interaction J by Monte Carlo simulations. In order to answer the open question whether this system exhibits a triple point or a multicritical point for positive H, we mainly studied lattices with 4 · 323 and 4 · 723 spins. We found a multicritical point at (Hc, kBTc) = (3.49 ± 0.02, 1.02 ± 0.01) J in the thermodynamic limit.
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).
We employ the Monte Carlo method to provide a computer description of the thermodynamic properties of the disordered antiferromagnetic Ising model for a simple cubic lattice with allowance for the nearest neighbors interaction and the next-nearest neighbors interaction. We construct the phase diagram for the model in an external magnetic field for systems with spin concentrations p =1.0, 0.95, and 0.8. We also establish the tricritical behavior of the model. Finally, we study the critical behavior of the model in weak fields.