The surprising convergence of the Monte Carlo renormalization group for the d = 3 Ising Model

The Monte Carlo renormalization group (MCRG) method is a systematic procedure for computing critical properties of lattice spin models[1, 2]. It has been shown to be both flexible and effective in the calculation of critical exponents, critical temperatures, and renormalized couplings constants[3–9]. A particularly interesting application of MCRG is the three-dimensional Ising model[10, 11]. This model has proven to be one of the most difficult to obtain accurate estimates for, because the approach to the fixed point is so slow. Attempts have been made to bring the fixed point closer to the nearest-neighbor model[12], but these have been controversial[13], and have not resulted in improved results. The most encouraging result has been that of Blöte et al.[14] who used a three-parameter approximation to the fixed point, along with a modified majority rule for the RG transformation. We have discovered a particularly simple modification of this calculation, which simulates the nearest-neighbor critical point and optimizes the RG transformation for the even and odd exponents separately. In the following sections, we recall the MCRG method, illustrate the slow convergence with the majority rule, and introduce a tunable RG transformation in section IV[14]. The improved convergence of the tuned RG transformation is then demonstrated in sections V through IX. Finally, we present our conclusions and discuss future work.