Boehm and Bak [11] have analysed, within the mean field approximation, a model which leads to a phase diagram including multiple phase transitions between commensurate phases. It happens that an exactly soluble two-dimensional Ising model exhibiting similar mean field features, is available. This model is built on a brickwork lattice, and its exact partition function is obtained within the following assumptions : i) In the vertical direction, nearest neighbours are coupled by ferromagnetic interactions 2 J' (J' > 0). ii) In the horizontal direction, nearest neighbours are coupled by ferromagnetic interactions J (J > 0) while nextnearest neighbours are coupled by arbitrary interactions J2. Results are discussed with respect to the value of the ratio J2/J. It is found in particular that : 1) For J2/J > — 1/2, the system orders ferromagnetically at a critical temperature Tc = 1/kB βc defined by tanh 2 βc J'. sinh 2 βc J. exp 4 βc J 2 = 1. 2) For J2/J ≤ - 1/2, no ordering occurs even at zero temperature. Ground states and their degeneracy are examined. These results are compared with those obtained for the same system in the mean field approximation, which predicts the existence of a devil's staircase behaviour for the periodicity versus temperature curve, and with those given for the two-dimensional ANNNI model by a recent Monte-Carlo study [13].
Boehm and Bak [11] have analysed, within the mean field approximation, a model which leads to a phase diagram including multiple phase transitions between commensurate phases. It happens that an exactly soluble two-dimensional Ising model exhibiting similar mean field features, is available. This model is built on a brickwork lattice, and its exact partition function is obtained within the following assumptions : i) In the vertical direction, nearest neighbours are coupled by ferromagnetic interactions 2 J' (J' > 0). ii) In the horizontal direction, nearest neighbours are coupled by ferromagnetic interactions J (J > 0) while nextnearest neighbours are coupled by arbitrary interactions J2. Results are discussed with respect to the value of the ratio J2/J. It is found in particular that : 1) For J2/J > — 1/2, the system orders ferromagnetically at a critical temperature Tc = 1/kB βc defined by tanh 2 βc J'. sinh 2 βc J. exp 4 βc J 2 = 1. 2) For J2/J ≤ - 1/2, no ordering occurs even at zero temperature. Ground states and their degeneracy are examined. These results are compared with those obtained for the same system in the mean field approximation, which predicts the existence of a devil's staircase behaviour for the periodicity versus temperature curve, and with those given for the two-dimensional ANNNI model by a recent Monte-Carlo study [13].
We study the phase diagram of the four dimensional Ising model with first and second neighbour couplings, specially in the antiferromagnetic region, by using Mean Field and Monte Carlo methods. From the later, all the transition lines seem to be first order except that between ferromagnetic and disordered phases in a region including the first-neighbour Ising transition point. 1 Antiferromagnetism has been considered in a great variety of models in order to find properties not present in the purely ferromagnetic (FM) system. Mostly they have been models in two and three dimensions. For instance, in several works on High Temperature Superconductivity the transition from paramagnetic (PM) to non-pure FM ordered phases has been studied [1, 2, 3, 4]. On the other hand, in relation to the finite temperature phase transition in pure gauge SU(3), the d = 3 three state Potts model with negative second neighbour couplings was unsuccessfully considered to find a new critical behaviour [5, 6]. In four dimensions, in diluted systems recently new critical exponents have been
Abstract. We use improved Monte-Carlo algorithms to study the antiferromagnetic 2D-Ising model with competing interactions J1 on nearest neighbour and J2 on next-nearest neighbour bonds. The finite-temperature phase diagram is divided by a critical point at J2 = J1/2 where the groundstate is highly degenerate. To analyse the phase boundaries we look at the specific heat and the energy distribution for various ratios of J2/J1. We find a first order transition for small J2> J1/2 and the transition temperature suppressed to TC = 0 at the critical point. 1.
The phase diagrams of the ferromagnetic two-dimensional mixed spin-1 and spin-1/2 Ising model with four-spin interaction J4 and next-nearest couplings J’ are investigated by the use of the finite cluster approximation based on single-site cluster theory. The state equations are derived for the two-dimensional square lattice. The phase diagram is qualitatively and quantitatively different from that obtained when the interaction J’ is ignored. In fact, the system exhibits a variety of interesting features resulting from the introduction of the next-nearest neighbor interaction (NNN). It undergoes two kinds of behavior according to the negative and the positive value of J’. In particular, for J4/J2=-4, the system keeps the coexistence of the two ground state up to J’-dependent finite temperature.
Making use of the Stratonovich linearization procedure for bilinear exponents and certain eigenvalue properties of the generalised Jacobi matrices, an exact expression has been obtained, in a closed form, for the partition function of the d (=3) dimensional, nearest neighbour coupled, Ising model for finite continuous spins(=D) in the presence of an external field. A Gaussian measure has been chosen for integration over the spin space. The method admits complete generalisation with respect to the dimensionality, the range of interaction and the sign of the interaction constant J (ferro- or antiferromagnetic). All thermodynamical quantities of interest can be obtained by taking suitable partial derivatives of the partition function. The critical indices can be ascertained for the cases of d=3 for which the theory predicts a second-order phase transition. For the ferromagnetic case(J>0), the spontaneous magnetisation vanishes identically the temperature T>T (the transition temperarture) and the magnetic susceptibility follows essentially the Curie Weiss Law.
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.
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.