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Journal of Physics C Solid State Physics

Exact results and critical properties of the Ising model with competing interactions

Hidetoshi Nishimori

Published Jul 30, 1980• 116 citations
Publisher

Abstract

The random Ising model with competing interactions is investigated on the basis of the gauge-invariant formulation of the problem. Exact results for the internal energy, specific heat and gauge-invariant correlation function are derived. The critical exponent alpha is shown to be negative at the phase boundary of the paramagnetic and ferromagnetic phases if the latter exists at fairly low concentration of antiferromagnetic bonds.

DOI
10.1088/0022-3719/13/21/012

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The random Ising model with competing interactions is investigated on the basis of the gauge-invariant formulation of the problem. Exact results for the internal energy, specific heat and gauge-invariant correlation function are derived. The critical exponent alpha is shown to be negative at the phase boundary of the paramagnetic and ferromagnetic phases if the latter exists at fairly low concentration of antiferromagnetic bonds.
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{T}_{c}=0Tc​=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 ferromagnetic-phase stability limit and magnetization of a quenched bond-mixed spin-1/2 transverse Ising model in an anisotropic simple square lattice are studied within the framework of an effective-field theory considering both competing and noncompeting interactions. In the bond-disordered case, where bonds J1{J}_{1}J1​ and J2{J}_{2}J2​ occur with probabilities 1-p and p, respectively, critical curves are obtained for various \ensuremath{\alpha}'s (\ensuremath{\alpha}\ensuremath{\equiv}J1{J}_{1}J1​/J2{J}_{2}J2​), and the critical bond concentrations pc{p}_{c}pc​ at which the ferromagnetic phase breaks down are determined. As \ensuremath{\alpha}\ensuremath{\rightarrow}0, the mixed model reduces to the diluted Ising model and the results obtained are compared to those available in the literature. The effect of the transverse field (for different \ensuremath{\alpha}'s) in the thermal behavior of the magnetization as a function of the concentration is also discussed.
The two-dimensional random bond Ising model with two distinct ferromagnetic nearest-neighbour couplings, J and J ', distributed randomly over the lattice with concentration 1/2 is studied for various ratios J '/ J at criticality. The critical temperature is exactly known from self-duality. The size-dependent specific heat shows a crossover from the perfect Ising to the randomness-dominated critical behaviour of doubly logarithmic form. In the latter region, our estimates for the critical exponents of the size-dependent magnetization and susceptibility are in very good agreement with analytic results suggesting β/ν = 1/8 and γ/ν = 7/4.
We have performed Monte Carlo calculations of the thermodynamic properties of the bond-dilute Ising model on the square lattice. The results are obtained for the magnetization, internal energy, specific heat, and susceptibility for several concentrations of missing bonds. The dependence of the Curie temperature on dilution is in good agreement with the results obtained by other methods. The phase transition is found to remain sharp with the same value of the susceptibility exponent as for the pure Ising model.