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arXiv:cond-mat/9604079 · Journal of Physics A Mathematical and General

Extrapolation procedure for low-temperature series for the square lattice spin-1 Ising model

Iwan Jensen, Anthony John Guttmann

Published Jul 21, 1996• 7 citations
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Abstract

The finite-lattice method of series expansions has been combined with a new extrapolation procedure to extend the low-temperature series for the specific heat, spontaneous magnetization, and susceptibility of the spin-1 Ising model on the square lattice. The extended series were derived by directly calculating the series to order 99 (in the variable and using the new extrapolation procedure to calculate an additional 13 - 14 terms.

DOI
10.1088/0305-4470/29/14/010

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The finite-lattice method of series expansions has been combined with a new extrapolation procedure to extend the low-temperature series for the specific heat, spontaneous magnetization, and susceptibility of the spin-1 Ising model on the square lattice. The extended series were derived by directly calculating the series to order 99 (in the variable and using the new extrapolation procedure to calculate an additional 13 - 14 terms.
The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the spin-1 Ising model on the square lattice. A new formalism is described which uses two distinct transfer-matrix approaches in order to significantly reduce computer memory requirements and which permits the derivation of the series to 79th order. Subsequent analysis of the series clearly confirms that the spin-1 model has the same dominant critical exponents as the spin- 1/2 Ising model. Accurate estimates for both the critical temperature and non-physical singularities are obtained. In addition, evidence for a non-analytic confluent correction with exponent Delta 1 =1.1+or-0.1 is found.
We derive low-temperature series (in the variable ) for the spontaneous magnetization, susceptibility, and specific heat of the spin-S Ising model on the square lattice for , 2, , and 3. We determine the location of the physical critical point and non-physical singularities. The number of non-physical singularities closer to the origin than the physical critical point grows quite rapidly with S. The critical exponents at the singularities which are closest to the origin and for which we have reasonably accurate estimates are independent of S. Due to the many non-physical singularities, the estimates for the physical critical point and exponents are poor for higher values of S, though consistent with universality.
Extended high-temperature series expansions are presented for the spin-s Ising model on the face-centred cubic lattice. Series coefficients for the zero-field internal energy (U 0 ) and the second moment ( mu 2 ) of the correlation function are derived, for 1 / 2 <or=s<or= 9 / 2 , to order 14 in the high-temperature expansion variable K(=J/kT). For s= 1 / 2 , the fourth field derivative of the free energy is calculated to order K 14 and the magnetisation series is given to order 13 and 27 in the temperature and magnetic field variables respectively. For s=1 to 9 / 2 , the zero-field susceptibility series are derived to order K 14 .
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.
For pt. III, see abstr. A21890 of 1970. A new low-temperature expansion variable for the Ising model is given. Expansions in the new variable, x=1-tanh(J/kT), then converge right up to the critical point. Power series in the new variable are derived, and then analysed by the Pade method and the method of N-point fits. For the spontaneous magnetization of the Ising model in three dimensions. beta =0.312+or-0.002 for the simple cubic lattice, beta =0.3125+or-0.0015 for the body-centred cubic lattice, and beta =0.312+or-0.004 for the face-centred cubic lattice. For the low-temperature susceptibility gamma =1.30+or-0.02, gamma =1.28+or-0.04, and gamma '=1.29+or-0.04 for the simple cubic, body-centred cubic and face-centred cubic lattices respectively. For the low-temperature specific heat the authors estimate 1 / 16 <or= alpha =0.2 for all three lattices.