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.
Abstract We study the critical properties of the antiferromagnetic spin-1/2 Ising model in an external field on the square lattice. Using tomographic entropic sampling, a flat-histogram simulation method, we estimate the number of configurations, , and related microcanonical averages in the energy-magnetization space, for system sizes L = 10–30. The critical line and exponents are calculated using finite-size scaling analysis in the temperature-external field plane. With these estimates in hand, we perform detailed studies of critical behavior using Metropolis sampling of larger systems ( ). These results are compared to several approximate theoretical methods. Our estimates of critical exponents and Binder’s reduced fourth cumulant along the critical line are in very good agreement with their respective literature values for the two-dimensional Ising universality class. We verify as well that the specific heat scales along the critical line, as expected for an Ising-like critical point.
We apply the phenomenological scaling approach of Nightingale to the triangular Ising model with nearest-neighbor interactions in the presence of a magnetic field. In order to do so, we calculate the largest eigenvalues of the transfer matrix of semi-infinite strips with widths 3, 6, 9,..., etc. The technique is first applied to two exactly soluble cases, the ferromagnetic and the hard hexagon model. Employing maximum strip widths of 9 and 15, respectively, in these two cases, we obtain results for the thermal exponent
yT within 0.3% of exact values. Applied to the antiferromagnet with maximum strip width of 9, we obtain values of the exponent which vary with the applied field. This variation is only appreciable, however, near zero field where crossover to different critical behavior is anticipated. We formulate a criterion to determine the best values of the thermal exponent and obtain
yT=1.2002, in excellent agreement with the value 1.2 expected on the basis of symmetry.
We apply our Monte Carlo twist method to the antiferromagnetic Ising model on square lattices in several uniform fields imposing fixed boundary conditions. We have estimated the critical temperature T c and critical exponent ν at each field. The former well agree with the analytic results of Müller-Hartmann and Zittartz, and the latter are all close to ν=1. We have calculated the finite-size critical-point amplitude of the interfacial free energy Ξ 0 which is believed to be universal. Our estimate of Ξ 0 at zero field is in good agreement with the analytic result of Abraham and Švrakić. We also report for the first time estimates of Ξ 0 at finite fields. Further, comparison is made with results of Mon and Jasnow.
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.