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Progress of Theoretical Physics

Specific Heat of Three-Dimensional Ising Model

Hiroshi Katsumori

Published Nov 1, 1974• 1 citation
Publisher

Abstract

The eigenvalue problem of three-dimensional Ising model is solved in successive screw approximation on a computer. Suitable extrapolation of the results suggests that the maximum value of the specific heat of the simple cubic lattice tends to infinity, in proportion to the logarithm of the number of spins making up each layer. It does not seem unreasonable to consider that the temperature at which the specific heat has a maximum gradually approaches to the critical temperature predicted from the Padé approximant method as the number of spins in each layer increases.

DOI
10.1143/ptp.52.1473

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The eigenvalue problem of three-dimensional Ising model is solved in successive screw approximation on a computer. Suitable extrapolation of the results suggests that the maximum value of the specific heat of the simple cubic lattice tends to infinity, in proportion to the logarithm of the number of spins making up each layer. It does not seem unreasonable to consider that the temperature at which the specific heat has a maximum gradually approaches to the critical temperature predicted from the Padé approximant method as the number of spins in each layer increases.
The partition function of a three-dimensional Ising model was calculated by a high-speed computer for a cubic array of twenty-seven points. The periodicity condition was imposed. The specific heat of the 3×3×3 lattice was calculated as a function of temperature. The critical temperature estimated from the specific heat curve seems slightly lower that obtained from the Padé approximation. The zeros of the partition function of the complex temperature plane were also obtained. The critical temperature estimated from the zeros near the positive real axis seems slightly higher than both of the above values. Similar calculations were carried out for the case of the two-dimensional Ising model. Several results are compared with those obtained from Onsager's exact solution.
For pt. I see abstr. A37614 of 1967. High temperature series expansions for the specific heat of the Ising model of a ferromagnet are given for the face-centred cubic, body-centred cubic, and simple cubic lattices. From a numerical study it is concluded that the critical index ( alpha ) is lattice independent and that in three dimensions alpha approximately=1/8. A numerical representation of the specific heat in the range T C <or=T<or= infinity is given in each case, together with estimates of the critical energy, entropy, and free energy. Asymptotic forms for the specific heat and energy are given.
We use the Pad\'e approximant method to investigate the nature of the singularity in the specific heat for some three-dimensional Ising model lattices and to investigate the nature and location of the singularity in the magnetic susceptibility for some three-dimensional Heisenberg model lattices. We find that the three-dimensional Ising model specific heat becomes singular like $log|T\ensuremath{-}{T}_{c}|$ with a different coefficient above and below the singular point. For the Heisenberg model we find that the magnetic susceptibility does not behave like the Curie-Weiss law, but tends to infinity more rapidly than $\frac{1}{(T\ensuremath{-}{T}_{c})}$.
We study the energy fluctuations in 3D Ising model near the phase transition point. Specific heat is a relevant quantity which is directly related to the mean squared amplitude of the energy fluctuations in the system. We have made extensive Monte Carlo simulations in 3D Ising model to clarify the character of the singularity of the specific heat Cv based on the finite-size scaling of its maximal values Cvmax depending on the linear size of the lattice L. An original iterative method has been used which automatically finds the pseudocritical temperature corresponding to the maximum of Cv. The simulations made up to L ≤ 128 with application of the Wolff's cluster algorithm allowed us to verify the possible power-like as well as logarithmic singularity of the specific heat predicted by different theoretical treatments. The most challenging and interesting result we have obtained is that the finite-size scaling of Cvmax in 3D Ising model is well described by a logarithmic rather than power-like ansatz, just like in 2D case. Another modification of our iterative method has been considered to estimate the critical coupling of 3D Ising model from the Binder cumulant data within L ε [96; 384]. Furthermore, the critical exponent β has been evaluated from the simulated magnetization data within the range of reduced temperatures t ≥ 0.000086 and system sizes L ≤ 410.
We have developed a method to change the transfer matrix V1 of 3D Ising model into a much smaller transfer matrix. V1 can be written as the linear combination of basic vectors. We discover that these basic vectors can be divided into several groups. The point product of one basic vector and V1 can be written as the linear combination of basic vectors from the same group. Based on that, we establish a new transfer matrix V2 to describe the relationship between basic vectors of the same group. The number of members from the same group is the order of V2. V2 is much smaller than V1 and contains the largest eigenvalue of V1. We use this method to calculate the specific heat and magnetic momentum per atom. The results show that there’s a pair of temperature and magnetic field strength that can make the specific heat get to its maximum value. When N1N2 increases, the maximum value of specific heat increases too.