The square Ising lattice with unequal first- and second-neighbor interactions has been considered. Among the exact results discussed, we show that the lowest energy state can be ferromagnetic, antiferromagnetic, or superantiferromagnetic, and the transition temperature should vanish in some cases. It is also shown that this problem is a special case of a more general problem arising in the statistical consideration of the hydrogen-bonded crystals. A well-defined approximation procedure is then introduced to solve the latter problem and to derive the (approximate) critical condition and expressions for the thermodynamic functions. The critical temperature thus determined is exact for the regular Ising lattice and for the lattices with ${T}_{c}=0$ while for the equivalent neighbor model the error is less than 2%. The specific heat possesses the usual logarithmic singularity in all cases.
The square Ising lattice with unequal first- and second-neighbor interactions has been considered. Among the exact results discussed, we show that the lowest energy state can be ferromagnetic, antiferromagnetic, or superantiferromagnetic, and the transition temperature should vanish in some cases. It is also shown that this problem is a special case of a more general problem arising in the statistical consideration of the hydrogen-bonded crystals. A well-defined approximation procedure is then introduced to solve the latter problem and to derive the (approximate) critical condition and expressions for the thermodynamic functions. The critical temperature thus determined is exact for the regular Ising lattice and for the lattices with Tc=0 while for the equivalent neighbor model the error is less than 2%. The specific heat possesses the usual logarithmic singularity in all cases.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858, comparable to the exponent 4/7≈0.5714 found from renormalization arguments.
The phase diagram of the Ising square lattice with first- and second-nearest-neighbor antiferromagnetic interactions (J1 and J2) is calculated within the framework of the cluster-variation method. The Hamiltonian is solved within the nine-point-cluster approximation. In contrast to previous studies we find that the phase transition from the superantiferromagnetic to the paramagnetic state is of first order in the range 0.5R=J1/${\mathit{J}}_{2}$1.144. The non-Ising critical behavior observed in that range of interactions might be due to that phenomenon.
The method of developing exact power-series expansions for the partition function ZN and related thermodynamic functions for the Ising model valid below the critical point is generalized to include exchange interactions between first-, second-, and third-neighbor pairs. Expansions of the spontaneous magnetization M0(T) and zero field susceptibility χ0(T) are derived through to sixth order of perturbation for the s.q. lattice, and through to fifth order of perturbation for the Δ′r, b.c.c., s.c., and f.c.c. lattices, when interactions J1SizSjz and J2SkzSlz are present between first- and second-neighbor spins, respectively (second-neighbor model). These expansions have also been obtained for the case where interactions of equal magnitude (J1 = J2 = J3) are present between first-, second-, and third-neighbor pairs (third-equivalent-neighbor model); here expansions through to fifth order of perturbation are obtained for the s.q., Δ′r, b.c.c., and s.c. lattices and through to fourth order for the f.c.c. lattice. The Padé approximant·procedure is employed to discuss the effects of an extended but finite range of interaction on the behavior of M0(T) and χ0(T) for T→Tc− as characterized by the critical exponents β and γ′, respectively. For the second-equivalent-neighbor model lattices, it is found that 0.122 ≤ β ≤ 0.134 in two dimensions, and that 0.308 ≤ β ≤ 0.328 in three dimensions; from which it is concluded that β remains unchanged from its value in the nearest-neighbor model. The corresponding limits for γ′ in three dimensions are 1.18 ≤ γ′ ≤ 1.28; from this and the results for the b.c.c. lattice in particular, it is concluded that γ′ is probably 54 and hence the transition in χ0 is symmetrical about Tc (γ′ = γ). A repetition of this analysis for the third-equivalent-neighbor model three-dimensional lattices shows a marked shift in the estimated range of β and γ′; the results are 0.345 ≤ β ≤ 0.365, and 1.01 ≤ γ′ ≤ 1.14. In each of the above cases, the corresponding high-temperature (T > Tc) expansions of χ0(T) obtained previously have been analyzed to yield estimates of the critical exponent γ. The over-all results and in particular the estimates of γ for the s.q. and b.c.c. lattices suggest that this index is unaffected by extending the range of interaction, and that if γ is a rational fraction then it is the same fraction for the n.n. model and second- and third-equivalent neighbor models. Finally the high-temperature expansions of ZN in zero field, and of χ0(T) for the second-neighbor model are used to examine the dependence of the critical temperature Tc, the critical energy (E∞ − Ec)/kTc, and the critical entropy (S∞ − Sc)/k on the relative strengths of J1 and J2 for values of J2/J1 in the range 0 to 1. It is found that the variation of the critical point is well represented by Tc(α)=Tc(0)[1+m1α],where α = J2/J1 and lies in the range 0 ≤ α ≤ 1; and Tc(0) is the critical temperature of the nearest-neighbor model. The values of m1 are 0.61, 2.47, 0.84, 1.45, and 1.35 for the f.c.c., s.c., b.c.c., s.q., and Δ′r lattices, respectively. All these calculations are compared with the corresponding results for the Heisenberg model.
The critical surface of the quenched bond-mixed square-lattice spin-21 first-neighbor-interaction ferromagnetic Ising model (with exchange interactions J1 and J2) has been investigated. Through renormalization group and heuristical procedures, a very accurate [error inferior to 3 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$ in the variables ${t}_{i}\ensuremath{\equiv}tanh(\frac{{J}_{i}}{{k}_{B}T})$] approximate numerical proposal for all points of this surface is presented. This proposal simultaneously satisfies all the available exact results concerning the surface, namely pc=21, ${t}_{c}=\sqrt{2}\ensuremath{-}1$, both limiting slopes in these points, and ${t}_{2}=\frac{(1\ensuremath{-}{t}_{1})}{(1+{t}_{1})}$ for p=21. Furthermore an analytic approximation [namely, $(1\ensuremath{-}p)\mathrm{ln}(1+{t}_{1})+p\mathrm{ln}(1+{t}_{2})=\frac{1}{2}\mathrm{ln}2$] is also proposed. In what concerns the available exact results, it only fails in reproducing one of the two limiting slopes, where there is an error of 1% in the derivative: These facts result in an estimated error less than ${10}^{\ensuremath{-}3}$ (in the t variables) for any point in the surface.
Statistical mechanics of the Bethe lattice was developed. Properties of the Bethe lattice, i. e., (of the Bethe approximation) were summarized including low temperature susceptibilities of ferro- and antiferromagnets. The equivalence of the exact solution of the Bethe lattice to the Bethe approximation of the Ising model in the case of the antiferromagnetism was shown. The partition function of the Bethe lattice with first and second neighbor interactions was obtained and the critical temperature was calculated in the case where the first neighbor interaction is ferromagnetic.
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.