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Journal of the Physical Society of Japan

Monte Carlo Study of the Spontaneous Magnetizationof the Three-Dimensional Ising Model

Nobuyasu Ito, Masuo Suzuki

Published Jun 15, 1991• 34 citations
Publisher

Abstract

The spontaneous magnetization of the three-dimensional Ising model is studied numerically by the Monte Carlo method. The values of squared magnetizations for finite simple-cubic lattices are extrapolated to those of the infinite system to estimate the values of spontaneous magnetization. The extrapolation parameter is the inverse of the system size. By assuming that the spontaneous magnetization is a simple pole function near the critical point, the values of the critical point K c and the exponent β of the spontaneous magnetization are estimated to be 0.221657(3) and 0.3205(6), respectively. If we assume that it is the form of simple pole function plus regular function, they are estimated to be 0.221651(4) and 0.3265(9), respectively. Therefore they are estimated to be 0.221654(6) and 0.324(4), respectively, from the mid values of these estimations.

DOI
10.1143/jpsj.60.1978

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The spontaneous magnetization of the three-dimensional Ising model is studied numerically by the Monte Carlo method. The values of squared magnetizations for finite simple-cubic lattices are extrapolated to those of the infinite system to estimate the values of spontaneous magnetization. The extrapolation parameter is the inverse of the system size. By assuming that the spontaneous magnetization is a simple pole function near the critical point, the values of the critical point K c and the exponent β of the spontaneous magnetization are estimated to be 0.221657(3) and 0.3205(6), respectively. If we assume that it is the form of simple pole function plus regular function, they are estimated to be 0.221651(4) and 0.3265(9), respectively. Therefore they are estimated to be 0.221654(6) and 0.324(4), respectively, from the mid values of these estimations.
The values of the spontaneous magnetization, mS, of cubic‐lattice Ising model which have been estimated using the Monte Carlo simulation and the extrapolation method are analyzed. The critical indices are estimated from its behavior using the least square fitting and Akaike’s information criterion (AIC). The AIC analysis indicates that the ms is a simple single‐pole function. Based on this type of function, the values of Kc and β are estimated to be 0.2216566(17) and 0.31977(38), respectively.
While the three-dimensional Ising model has defied analytic solution, various numerical methods like Monte Carlo, Monte Carlo renormalization group, and series expansion have provided precise information about the phase transition. Using Monte Carlo simulation that employs the Wolff cluster flipping algorithm with both 32-bit and 53-bit random number generators and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising Model, with lattice sizes ranging from 16^{3} to 1024^{3}. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, e.g., logarithmic derivatives of magnetization and derivatives of magnetization cumulants, we have obtained the critical inverse temperature K_{c}=0.221654626(5) and the critical exponent of the correlation length ν=0.629912(86) with precision that exceeds all previous Monte Carlo estimates.
The Monte Carlo renormalization group is applied to the three-dimensional Ising model on simple cubic lattices with 83{8}^{3}83, 163{16}^{3}163, 323{32}^{3}323, and 643{64}^{3}643 sites. The comparison of block-spin correlation functions from the largest lattices yields the nearest-neighbor critical coupling K1c=0.221654(6){K}_{1}^{c}=0.221654(6)K1c​=0.221654(6). After allowing for (i) interpolation to this best estimate for K1c{K}_{1}^{c}K1c​, (ii) an apparent finite-size effect in the renormalization-group transformation due to the measurement of correlation functions of too few (seven) operators, and (iii) the extrapolation for the effect of a slow transient towards the fixed point, the values $\ensuremath{\nu}=0.629(4)$ and $\ensuremath{\eta}=0.031(5)$ are obtained for the thermal and magnetic exponents. The correction-to-scaling exponent $\ensuremath{\omega}$ is estimated to be around 1; to obtain an accuracy competitive with other methods requires measurements with more than seven operators. We briefly review the problem of redundant operators and indicate the future prospects for this kind of calculation.
We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).
The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.
The fractal structure of the Ising model at the critical point Tc is studied in the present paper.The fractal dimensionality D of the total magnetization at Tc was estimated numerically as D = 1.86 ±O.Ol for the two-dimensional square lattice and D=2.46±O.Ol for the three-dimensional simple cubic lattice by Monte Carlo simulations.These values agree very well with the value D=1.875 obtained from the exact critical exponents and D=2.48 obtained from the known critical exponents, re-~pectively, through the relation D=d-fJ/v=(d+r/v)/2.This fractalness yields the hyperscaling relation dv=2fJ+r.It was also observed how the fractal nature of the relevant system disappears as the system deviates from the critical point.The dimensionality d of the relevant lattice is observed at temperatures lower than Tc and the random percolation value d/2 at higher temperatures.