Computation of Order Parameters in an Ising Lattice by the Monte Carlo Method

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Abstract
The long‐range and short‐range order parameters are computed for the Ising lattice using a Monte Carlo sampling scheme. The square lattice, the simple cubic lattice, and the body‐centered cubic lattice are considered. In the three‐dimensional calculations both the antiferromagnetic and ferromagnetic cases are considered as well as the coupling to an external magnetic field of various strengths. Good agreement is found where the results can be compared with the exact two‐dimensional results, and in the three‐dimensional case the results agree well with those obtained from series approximations in the regions where the series approximations are valid. The present method appears to give good results for the short‐range order even very close to the critical temperature, but in this neighborhood the long‐range order estimate is crude. The computations were performed on the high‐speed computer ILLIAC, located at the University of Illinois.