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International Journal of Modern Physics B

HIGH-TEMPERATURE SUSCEPTIBILITY SERIES FOR SQUARE-LATTICE ISING MODEL WITH FIRST AND SECOND NEIGHBOUR INTERACTIONS

Keh Ying Lin, MALL CHEN

Published Dec 20, 2002• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979202015819

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Series expansions for the zero-temperature transverse Ising model

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Series expansions are presented for the magnetisation, susceptibility, magnetic field derivatives of the susceptibility and 'specific heat' of the zero-temperature transverse Ising model. Coefficients in these series have been calculated to tenth order in lambda 2 (where lambda is the transverse field) for the linear chain, to eighth order for the square lattice and to seventh order for the triangular lattice. These series yield estimates of the low-temperature critical exponents alpha ', beta ', gamma ' and Delta ' of the two- and three-dimensional Ising models. They provide good evidence for the symmetry of exponents above and below the critical point, e.g. gamma '= gamma .

Journal of Physics A Mathematical and General•Aug 1, 1981•50 citations
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.
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.
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.
We investigate the complex-temperature singularities of the susceptibility of the 2D Ising model on a square lattice. From an analysis of low-temperature series expansions, we find evidence that, as one approaches the point u=u s =-1 (where u=e '4K ) from within the complex extensions of the FM or AFM phases, the susceptibility has a divergent singularity of the form X approximately A s '(1+u)(- gamma s ') with exponent gamma s '=3/2. The critical amplitude A s ' is calculated. Other critical exponents are found to be alpha s '= sigma s =0 and beta s = 1/4 , so that the scaling relation alpha s '+2 beta s + gamma s ' is satisfied. However, using exact results for beta s on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: beta s is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length and hence susceptibility are finite as one approaches the point u=-1 from within the symmetric phase. This is confirmed by an explicit study of high-temperature series expansions.
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.
Series expansions are presented for the magnetisation, susceptibility, magnetic field derivatives of the susceptibility and 'specific heat' of the zero-temperature transverse Ising model. Coefficients in these series have been calculated to tenth order in lambda 2 (where lambda is the transverse field) for the linear chain, to eighth order for the square lattice and to seventh order for the triangular lattice. These series yield estimates of the low-temperature critical exponents alpha ', beta ', gamma ' and Delta ' of the two- and three-dimensional Ising models. They provide good evidence for the symmetry of exponents above and below the critical point, e.g. gamma '= gamma .