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Journal of the Physical Society of Japan

Approximate Treatment of the Two-Dimensional Ising Ferromagnet with Second-Neighbor Interaction

Sei-ichi Takase

Published Mar 1, 1974• 8 citations
Publisher

Abstract

The two-dimensional square Ising ferromagnet with the nearest-neighbor interaction - J and the second-neighbor interaction - J ' is investigated starting from the identity < s i >=<tanh (ΣK i j s j )>. The Hartree approximation is used for treating the many-spin correlation functions. The critical point thus determined has two values for J '/ J <0, and is corrected by the study of the spontaneous magnetization and the free energy to give a resonable function T c ( J '/ J ). Our approximation may be valid at least qualitatively as a whole.

DOI
10.1143/jpsj.36.636

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The two-dimensional square Ising ferromagnet with the nearest-neighbor interaction - J and the second-neighbor interaction - J ' is investigated starting from the identity < s i >=<tanh (ΣK i j s j )>. The Hartree approximation is used for treating the many-spin correlation functions. The critical point thus determined has two values for J '/ J <0, and is corrected by the study of the spontaneous magnetization and the free energy to give a resonable function T c ( J '/ J ). Our approximation may be valid at least qualitatively as a whole.
The static properties are investigated by computer simulation for the square Ising ferromagnet in which the nearest-neighbor interactions are ferromagnetic and the second interactions antiferromagnetic. Energy and magnetization at various temperatures are obtained by Monte Carlo calculations performed on the lattice with 20×20 spins. The Curie point determined as the inflection point of the energy-vs-temperature curve agrees with the result by Fan and Wu rather than other analytical approximations. The behavior of the spontaneous magnetization M 0 ( T ) for T → T c -0 as characterized by the critical exponent β is shown by comparing the data with an analytical equation for the magnetization. The exponent β remains probably unchanged from the original value 1/8.
The square Ising ferromagnet is investigated by computer simulation in the region of competing interactions (nearest-neighbor interaction K >0 and second-neighbor one K '<0). Spontaneous magnetization M 0 and energy E are calculated from advanced Monte Carlo calculations. The estimates of M 0 and of the Curie point T c ( K '/ K ) determined by analyzing the data on E are compared with the results from analysis of low-temperature series based on the assumption β=1/8. The values of T c ( K '/ K ) are obtained from padé-approximant method for low-temperature series. The value of β does not vary from the Onsager's value 1/8.
The Ising of spin 1/2 with the nearest-neighbor interaction, in which nonmagnetic impurity atoms are dissolved, is investigated. Based on the identity < s i >=<tanh ( K Σ s j )>, decoupling approximation is introduced for treating the many-spin correlation functions. The dependences of the critical temperature and of the spontaneous magnetization on the impurity concentration are calculated for two- and three-dimensional lattices. The results for the critical temperature and the critical concentration are better than the results of the results of the extended molecular-field approximations of Mamada and Takano, or of Oguchi and Obokata. Especially for the three-dimensional lattices the critical obtained agrees with that of Sykes and Essam, estimated from the series expansion.
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 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.