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). We also obtain estimates of the free energy, internal energy, and entropy of both the ordered and disordered states at
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
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
kBS(0) at the critical fields
Jhc=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
L) 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).
Using Monte Carlo methods we investigate the phase transition of an Ising system on the fcc lattice, with antiferromagnetic nearest-neighbor interaction
J and ferromagnetic next-nearest-neighbor interaction $\ensuremath{\alpha}J$ for \ensuremath{\alpha} values up to 2. The scaling of the microcanonical entropy
sL with system size
L 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, our calculations cannot determine the order of the transition unambiguously.
The mean-field approximation of the critical behavior of the antiferromagnetic Ising model in a magnetic field shows that the critical field increases first at low temperature. However, the exact solution of a decorated square lattice proves that the critical field rapidly decreases with increasing temperature for the two-dimensional lattice. Three-dimensional lattices of this model have been studied as well using the high temperature expansion, low temperature expansion, and Monte Carlo simulation. The results are different. In this article we have investigated the critical field of the body centered cubic lattice of this model using the linked-cluster expansion technique. An eighth-order perturbation series for the free energy has been calculated. The magnetization and staggered magnetic susceptibility series have been calculated from the free energy. These series have been analyzed using Padé approximants to obtain the phase diagram in the field-temperature plane. A comparison is made with Monte Carlo simulation calculations.
The phase diagrams of Ising antiferromagnets in a magnetic field
H are investigated for various values of the ratio
R between nearest- and next-nearest-neighbor interaction. While meanfield approximations and the existing real-space renormalization-group treatments yield phase diagrams which are sometimes even qualitatively incorrect, accurate results are obtained from Monte Carlo calculations. For
R<0 only an antiferromagnetically ordered phase exists. Its transition to the disordered phase is first order for temperatures below the tricritical point (
Tt,
Ht). For $R\ensuremath{\rightarrow}0$ also ${T}_{t}\ensuremath{\rightarrow}0$. For
R=0 we find very good agreement with the results of M\"uller-Hartmann and Zittartz. For
R>0 and
H1<H<H2 a new phase with anomalous high ground-state degeneracy is found (two sublattices have only one-dimensional order). These sublattices undergo order-disorder transitions at
T=0, such that for
T>0 one is left with a "superantiferromagnetic" phase. At low temperatures in this phase a pronounced tendency is observed to form a simpler (2 \ifmmode\times\else\texttimes\fi{} 2) superstructure but with many antiphase domain boundaries. For $R\ensuremath{\rightarrow}\frac{1}{2}$ and
H<H1 the regime of the antiferromagnetic phases goes to zero temperature, while for
R>21 the superantiferromagnetic phase exists also for
H<H1. The order-disorder transition associated with this phase seems to have non-Ising critical exponents which vary as a function of
R and
H. Estimates for the exponents lead us to suggest that Suzuki's "weak universality" is valid. The behavior of the model at
T=0 is related to known results on hard-core lattice gases. It is shown that it is useful to interpret the transitions at
T=0 as generalized percolation transitions. Since the model may have applications to adsorbate phases in registered structures at (100) surfaces of cubic crystals, the transcription of our results to temperature-coverage phase diagrams and adsorption isotherms is discussed in detail, and possible experimental applications are mentioned.