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.
Motived by the $H\text{\ensuremath{-}}T$ phase diagram in the bcc Ising antiferromagnetic with nearest-neighbor interactions obtained by Monte Carlo simulation [Landau, Phys. Rev. B 16, 4164 (1977)] that shows a reentrant behavior at low temperature, with two critical temperatures in magnetic field about 2% greater than the critical value
Hc=8J, we apply the effective field renormalization group (EFRG) approach in this model on three-dimensional lattices (simple cubic-sc and body centered cubic-bcc). We find that the critical curve
TN(H) exhibits a maximum point around of $H\ensuremath{\simeq}{H}_{c}$ only in the bcc lattice case. We also discuss the critical behavior by the effective field theory in clusters with one (EFT-1) and two (EFT-2) spins, and a reentrant behavior is observed for the sc and bcc lattices. We have compared our results of EFRG in the bcc lattice with Monte Carlo and series expansion, and we observe a good accordance between the methods.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
The critical exponents of the non-interacting hard-square lattice gas model are determined by means of series analysis and finite-size scaling. The series analysis leads to results for the exponents that are consistent with the expected Ising universal values. A scaling analysis is performed on different types of numerical finite-size data obtained by means of the transfer-matrix method. The results support Ising universality. A procedure based on conformal invariance reproduces the known Ising temperature and magnetic exponents within 10 -6 . Furthermore we estimate the critical density of the hard-square model as rho c =0.367 743 000 (5).
We investigate the Ising model with nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic interactions on the square and body-centered-cubic lattices. For
0<J2J1<23 on the bcc lattice the model has two nontrivially degenerate ground states and has been conjectured to have
XY (
n=2) critical exponents. For
J2J1=23 there are three degenerate ground states and Heisenberg critical exponents (
n=3) are expected. We have obtained nine terms in the high-temperature expansion of the staggered susceptibility. For
0<J2J1< 1 we find $\ensuremath{\gamma}=1.25$ (
n=1). At
J2J1=23 the results are not conclusive, but $\ensuremath{\gamma}=1.33$ (
n=2) is not ruled out by the series analysis whereas $\ensuremath{\gamma}=1.4$ (
n=3) is not probable. Results for the square lattice are inconclusive.
Field-induced phase transitions in the bcc Ising antiferromagnet with nearest-neighbor interactions have been studied using a Monte Carlo method. The properties of $N\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ lattices with periodic boundary conditions have been determined for $N\ensuremath{\le}16$ and the size effects analyzed using finite-size scaling theory. We find no field dependence of the critical exponents and only a slow variation of critical amplitudes in small fields in agreement with the "smoothness postulate." At low temperatures the critical-field curve shows a weak, but clear maximum which is about 2% greater than the zero-temperature value
Hc. Crossover to noncritical exponential behavior due to thermally induced spin flips is observed as we move away from the low-temperature phase boundary.