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Journal of Physics A Mathematical and General

A finite-size scaling study of the 4D Ising model

Eduardo Sánchez-Velasco

Published Oct 1, 1987• 9 citations
Publisher

Abstract

The author studies the finite-size scaling behaviour of the susceptibility and of the second derivative of the susceptibility with respect to the magnetic field at the critical point for the four-dimensional Ising model with zero magnetic field using Monte Carlo techniques. The finite-size scaling exponents are found to be in good agreement with the form predicted by mean-field theory critical exponents.

DOI
10.1088/0305-4470/20/14/041

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The author studies the finite-size scaling behaviour of the susceptibility and of the second derivative of the susceptibility with respect to the magnetic field at the critical point for the four-dimensional Ising model with zero magnetic field using Monte Carlo techniques. The finite-size scaling exponents are found to be in good agreement with the form predicted by mean-field theory critical exponents.
The four-dimensional Ising model is studied to probe the possibility of observing in Monte Carlo simulations the logarithmic corrections to the mean-field theory near criticality. The finite-size-scaling behavior for the correlation length is proposed. The scaling forms of the finite-size renormalized coupling, susceptibility, and fourth field derivative at the renormalized tree-level approximation are derived. These results are used to analyze simulation data of the simple hypercubic lattices of sizes 4\ensuremath{\le}L\ensuremath{\le}14, near criticality. Our simulation results are in agreement with the presence of logarithmic corrections and recent field-theoretical calculations of the specific heat. The approach to the nonscattering theory is observed and is consistent with the predicted logarithmic finite-size dependence.
In this work we have considered the Taylor series expansion of the dynamic scaling relation of the magnetization with respect to small initial magnetization values in order to study the dynamic scaling behaviour of 2- and 3-dimensional Ising models. We have used the literature values of the critical exponents and of the new dynamic exponent x0 to observe the dynamic finite-size scaling behaviour of the time evolution of the magnetization during early stages of the Monte Carlo simulation. For 3-dimensional Ising Model we have also presented that this method opens the possibility of calculating z and x0 separately. Our results show good agreement with the literature values. Measurements done on lattices with different sizes seem to give very good scaling.
The five-dimensional Ising model with free boundary conditions has recently received a renewed interest in a debate concerning the finite-size scaling of the susceptibility near the critical temperature. We provide evidence in favour of the conventional scaling picture, where the susceptibility scales as L2 inside a critical scaling window of width 1/L2. Our results are based on Monte Carlo data gathered on system sizes up to L=79 (ca. three billion spins) for a wide range of temperatures near the critical point. We analyse the magnetisation distribution, the susceptibility and also the scaling and distribution of the size of the Fortuin–Kasteleyn cluster containing the origin. The probability of this cluster reaching the boundary determines the correlation length, and its behaviour agrees with the mean field critical exponent δ=3, that the scaling window has width 1/L2.
In this work we have considered the Taylor series expansion of the dynamic scaling relation of the magnetization with respect to small initial magnetization values in order to study the dynamic scaling behavior of two- and three-dimensional Ising models. We have used the literature values of the critical exponents and of the new dynamic exponent x0 to observe the dynamic finite-size scaling behavior of the time evolution of the magnetization during early stages of the Monte Carlo simulation. For the three-dimensional Ising model we have also presented that this method opens the possibility of calculating z and x0 separately. Our results show good agreement with the literature values. Measurements done on lattices with different sizes seem to give very good scaling.