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Journal of Physics C Solid State Physics

Theoretical analysis of the Ising ferromagnetic transition for cubic lattices

B. Frank, C Y Cheung, Ole G. Mouritsen

Published Feb 28, 1982• 12 citations
Publisher

Abstract

New relations, the 'i- delta relations', are presented for the evaluation of higher-order critical correlation functions for the three-dimensional spin-1/2 Ising model. The critical temperatures Tc for the cubic lattices are then calculated and are found to be within 1/2% of the results of series analysis. Monte Carlo calculations are performed on the simple cubic lattice, and are found to bear out the theoretical predictions for the correlation functions to within 1.3%, for the basic Tc-determining equations to within 0.5%, and for the i- delta relations to within 2.5%. An earlier theory, which uses the critical correlation function approximation, is critically examined both theoretically and by comparison with the Monte Carlo results.

DOI
10.1088/0022-3719/15/6/018

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Physical review. B, Condensed matter•Mar 1, 1991•129 citations
New relations, the 'i- delta relations', are presented for the evaluation of higher-order critical correlation functions for the three-dimensional spin-1/2 Ising model. The critical temperatures Tc for the cubic lattices are then calculated and are found to be within 1/2% of the results of series analysis. Monte Carlo calculations are performed on the simple cubic lattice, and are found to bear out the theoretical predictions for the correlation functions to within 1.3%, for the basic Tc-determining equations to within 0.5%, and for the i- delta relations to within 2.5%. An earlier theory, which uses the critical correlation function approximation, is critically examined both theoretically and by comparison with the Monte Carlo results.
A theory is presented for the determination of the transition temperature T c for spin- 1 / 2 Ising models on cubic lattices with pair (J 2 ) and quartet (J 4 ) interactions. A linear relationship between T c and J 4 /J 2 is found to have a base wider in scope than that of previous theories. The calculation of the linear coefficient is performed through the derivation of a new equation for special four-spin correlation functions which is solved in a novel way by the use of a ratio approximation. The results are compared with those derived from previous theories, series analysis and Monte Carlo calculations which are also presented here. The linear coefficient is shown to depend on a parameter which turns out to have approximately the same value for all cubic lattices, thus suggesting a set of scaled variables in terms of which the phase boundary for all cubic lattices may be fitted by a common curve. From the authors' work a coherent picture emerges for the phase behaviour of cubic Ising lattices with pair and quartet interactions, involving regions of continuous and first-order transitions separated by a tricritical point which occurs at the same value of the scaled variables for all cubic lattices.
The rational values beta = 5 / 16 , delta =5, alpha = 1 / 8 , nu = 5 / 8 , and eta =0 of the critical exponents for the three-dimensional spin 1 / 2 Ising ferromagnet are shown to be consistent with a recently-derived linearised equation for the Ising two-spin correlation function. This equation arises from an exact integral representation which previously, on application of a straightforward approximation for higher-order cumulant averages, was shown to yield transition temperatures T c to within 2% of their correct values for the cubic lattices. In the search for beta , the requirement is imposed that the correction terms for the magnetisation and for M 2 , a quantity closely related to the system internal energy, be analytic in some power of T c -T (which turns out to be 1 / 8 ) when the external magnetic field h is zero.
A pair-correlation function ansatz, previously used in the derivation of the Ising model critical exponent η for the square lattice from a criticality equation within the i-δ approximation, is investigated further. Various methods are used to calculate the multispin correlation functions entering the criticality equation. The calculations are extended to the three-dimensional cubic lattices. It is found that the values obtained for η are relatively insensitive to the specific method used. In two dimensions, η takes the values 0.2506 (i-δ method), 0.2471, and 0.2490 (renormalized Hamiltonian methods), thereby remaining within 1.2% of the exact value 1/4. In three dimensions, η ranges from 0.030 to 0.075, being of the same order of magnitude as the series value 0.041 ± 0.01.
The critical properties of the spin- 1 / 2 Ising ferromagnet on a diamond lattice are calculated using the Monte Carlo technique. The heat capacity, the susceptibility, the magnetisation, and various pair and multispin correlation functions are calculated as functions of the temperature. The critical temperature, and the critical parameters pertaining to the magnetisation, are determined. The results are consistent with series analyses. Using the critical values of the correlation functions, it is shown that the basic assumption in the theory of Frank and Mitran (1978) is not exact for the diamond lattice.
We report the results of a Monte Carlo investigation of the (equilibrium) time-displaced correlation functions for the magnetization and energy of a simple cubic Ising model as a function of time, temperature, and lattice size. The simulations were carried out on a CDC CYBER 205 supercomputer employing a high-speed, vectorized multispin coding program and using a total of 5\ifmmode\times\else\texttimes\fi{}1012{10}^{12}1012 Monte Carlo spin-flip trials. We used L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices with periodic boundary conditions and L as large as 96. The short-time and long-time behaviors of the correlation functions are analyzed by fits to a sum of exponential decays, and the critical exponent z for the largest relaxation time is extracted using a finite-size-scaling analysis. Our estimate z=2.04\ifmmode\pm\else\textpm\fi{}0.03 resolves an intriguing contradiction in the literature; it satisfies the theoretical lower bound and is in agreement with the prediction obtained by \ensuremath{\epsilon} expansion. We also consider various small systematic errors that typically occur in the analysis of relaxation functions and show how they can lead to spurious results if sufficient care is not exercised.