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Journal of Physics A Mathematical and General

The triangular lattice Ising model with first and second neighbour interactions

J. Oitmaa

Published Feb 1, 1982• 7 citations
Publisher

Abstract

Reports studies of the critical behaviour of the Ising model with first and second neighbour interactions J 1 , J 2 on the triangular lattice. High-temperature series have been used to determine the locations of the ferromagnetic critical line and the critical line in the region (J 1 <0, J 2 >0). On the latter line the ordering susceptibility appears to diverge algebraically with the exponent gamma approximately 2.4. The existence of a sinusoidal phase and an associated Lifshitz point are conjectured.

DOI
10.1088/0305-4470/15/2/024

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International Journal of Modern Physics B•Dec 20, 2002•0 citations
Reports studies of the critical behaviour of the Ising model with first and second neighbour interactions J 1 , J 2 on the triangular lattice. High-temperature series have been used to determine the locations of the ferromagnetic critical line and the critical line in the region (J 1 <0, J 2 >0). On the latter line the ordering susceptibility appears to diverge algebraically with the exponent gamma approximately 2.4. The existence of a sinusoidal phase and an associated Lifshitz point are conjectured.
The critical behaviour of the square-lattice Ising model, with nearest and next-nearest neighbour interactions of either sign, has been investigated by means of high-temperature series. lhe location of the critical lines in the coupling constant plane has been accurately determined. Along the critical line which corresponds to transitions to the layered or superantiferromagnetic state a breakdown of universality is observed and explicit numerical estimates obtained for the exponent of the ordering susceptibility.
Positions of critical temperatures of antiferromagnetic Ising model with ferromagnetic next nearest neighbour interactions on the triangular lattice are investigated by making use of their properties of the Kosterlitz-Thouless type of phase transition. In particular, finite-size scaling method and Monte Carlo Renormalization Group methods (MCRG) with flow diagrams of the anomalous dimension η show the fixed line structure where the correlation length diverges. Another MCRG with flow diagram of the stiffness constant also shows a critical property clearly. The temperature dependence of the long range order is also discussed in order to locate the lower critical point.
The critical behaviour of the Ising model on the square lattice with ferromagnetic nearest-neighbour interactions (J) and anti-ferromagnetic next-nearest neighbour interactions (J') is discussed for small lambda =J/ mod J' mod . The singular part of the free energy is calculated to second order in lambda by perturbation theory. Near the critical temperature of the unperturbed system (J=0), this expansion is found to have a form which may be exponentiated yielding, for non-zero lambda , a non-universal critical line along which the exponents vary continuously with lambda . For small lambda , the specific heat exponent, alpha = alpha 1 lambda 2 +O( lambda 2 ) with alpha 1 approximately=1.5.
We have calculated the low-temperature series expansions of the spontaneous magnetization and the zero-field susceptibility of the square-lattice ferromagnetic Ising model with first-neighbour interaction J1 and second-neighbour interaction J2 to the 30th and 26th order respectively by computer. Our results extend the previous calculations by Lee and Lin to six more orders. We use the Padé approximants to estimate the critical exponents and the critical temperature for different ratios of R = J2/J1. The estimated critical temperature as a function of R agrees with the estimation by Oitmaa from high-temperature series expansions.
We have calculated the high-temperature series expansion of the zero-field susceptibility of the square-lattice Ising model with first and second neighbour interactions to the 20th order by computer. Our results extend the previous calculation by Hsiao and Lin to two more orders. We use the Padé approximants to estimate the critical exponent γ and the critical temperature. Our result 1.747 < γ < 1.753 supports the universality conjecture that all two-dimensional Ising models have the same critical exponent γ = 1.75.