The Ising model is the simplest model for ferromagnetism that predicts phase transitions and critical phenomena. The spins are discrete and have only two possible states. This model, introduced by Lenz in 1920 [Lenz:20a], was solved in one dimension by Ising in 1925 [Ising:25a], and in two dimensions by Onsager in 1944 [Onsager:44a]. However, it has not been solved analytically in three dimensions, so Monte Carlo computer simulation methods have been one of the methods used to obtain numerical solutions. One of the best available techniques for this is the Monte Carlo Renormalization Group (MCRG) method [Wilson:80a], [Swendsen:79a]. The Ising model exhibits a second-order phase transition in d=3 dimensions at a critical temperature . As T approaches , the correlation length diverges as a power law with critical exponent :
and the pair correlation function at falls off to zero with distance r as a power law defining the critical exponent :
, and determine the critical behavior of the 3-D Ising model and it is their values we wish to determine using MCRG.
In 1984, this was done by Pawley, Swendsen, Wallace and Wilson [Pawley:84a] in Edinburgh on the ICL DAP computer with high statistics. They ran on four lattice sizes-, , and -measuring seven even and six odd spin operators. We are essentially repeating their calculation on the new AMT DAP computer. Why should we do this? First, to investigate finite size effects-we have run on the biggest lattice used by Edinburgh, , and on a bigger one, . Second, to investigate truncation effects-qualitatively the more operators we measure for MCRG, the better, so we have included 53 even and 46 odd operators. Third, we are making use of the new cluster-updating algorithm due to Swendsen and Wang [Swendsen:87a], implemented according to Wolff [Wolff:89b]. Fourth, we would like to try to measure another critical exponent more accurately-the correction-to-scaling exponent , which plays an important role in the analysis.
The idea behind MCRG is that the correlation length diverges at the critical point, so that certain quantities should be invariant under ``renormalization'', which here means a transformation of the length scale. On the lattice, we can double the lattice size by, for example, ``blocking'' the spin values on a square plaquette into a single spin value on a lattice with 1/4 the number of sites. For the Ising model, the blocked spin value is given the value taken by the majority of the 4 plaquette spins, with a random tie-breaker for the case where there are 2 spins in either state. Since quantities are only invariant under this MCRG procedure at the critical point, this provides a method for finding the critical point.
In order to calculate the quantities of interest using MCRG, one must evaluate the spin operators . In [Pawley:84a], the calculation was restricted to seven even spin operators and six odd; we evaluated 53 and 46, respectively [Baillie:91d]. Specifically, we decided to evaluate the most important operators in a cube [Baillie:88h]. To determine the critical coupling (or inverse temperature), , one performs independent Monte Carlo simulations on a large lattice L of size and on smaller lattices S of size , , and compares the operators measured on the large lattice blocked m times more than the smaller lattices. when they are the same. Since the effective lattice sizes are the same, unknown finite size effects should cancel. The critical exponents, , are obtained directly from the eigenvalues, , of the stability matrix, , which measures changes between different blocking levels, according to . In particular, the leading eigenvalue of for the even gives from , and, similarly, from the odd eigenvalue of .
The Distributed Array Processor (DAP) is a SIMD computer consisting of bit-serial processing elements (PEs) configured as a cyclic two-dimensional grid with nearest-neighbor connectivity. The Ising model computer simulation is well suited to such a machine since the spins can be represented as single-bit (logical) variables. In three-dimensions, the system of spins is configured as an simple cubic lattice, which is ``crinkle mapped'' onto the DAP by storing pieces of each of M planes in each PE: , with . Our Monte Carlo simulation uses a hybrid algorithm in which each sweep consists of 10 standard Metropolis [Metropolis:53a] spin updates followed by one cluster update using Wolff's single-cluster variant of the Swendsen and Wang algorithm. On the lattice, the autocorrelation time of the magnetization reduces from sets of 100 sweeps for Metropolis alone to sets of 10 Metropolis plus one cluster update for the hybrid algorithm. In order to measure the spin operators, , the DAP code simply histograms the spin configurations so that an analysis program can later pick out each particular spin operator using a look-up table. Currently, the code requires the same time to do one histogram measurement, one Wolff single-cluster update or 100 Metropolis updates. Therefore, our hybrid of 10 Metropolis plus one cluster update takes about the same time as a measurement. On a DAP 510, this hybrid update takes on average 127 secs (13.5 secs) for the () lattices. We have performed simulations on and lattices at two values of the coupling: (Edinburgh's best estimate of the critical coupling) and . We accumulated measurements for each of the simulations and for the so that the total time used for this calculation is roughly 11,000 hours. For error analysis, this is divided into bins of measurements.
In analyzing our results, the first thing we have to decide is the order in which to arrange our 53 even and 46 odd spin operators. Naively, they can be arranged in order of increasing total distance between the spins [Baillie:88h] (as was done in [Pawley:84a]). However, the ranking of a spin operator is determined physically by how much it contributes to the energy of the system. Thus, we did our analysis initially with the operators in the naive order to calculate their energies, then subsequently we used the ``physical'' order dictated by these energies. This physical order of the first 20 even operators is shown in Figure 4.12 with 6 of Edinburgh's operators indicated; the 7th Edinburgh operator (E-6) is our 21st. This order is important in assessing the systematic effects of truncation, as we are going to analyze our data as a function of the number of operators included. Specifically, we successively diagonalize the , , , ( for even, for odd) stability matrix to obtain its eigenvalues and, thus, the critical exponents.
Figure 4.12: Our Order for Even Spin Operators
We present our results in terms of the eigenvalues of the even and odd parts of . The leading even eigenvalue on the first four blocking levels starting from the lattice is plotted against the number of operators included in the analysis in Figure 4.13, and on the first five blocking levels starting from the lattice in Figure 4.14. Similarly, the leading odd eigenvalues for and lattices are shown in Figures 4.15 and 4.16, respectively. First of all, note that there are significant truncation effects-the value of the eigenvalues do not settle down until at least 30 and perhaps 40 operators are included. We note also that our value agrees with Edinburgh's when around 7 operators are included-this is a significant verification that the two calculations are consistent. With most or all of the operators included, our values on the two different lattice sizes agree, and the agreement improves with increasing blocking levels. Thus, we feel that we have overcome the finite size effects so that a lattice is just large enough. However, the advantage in going to is obvious in Figures 4.14 and 4.16: There, we can perform one more blocking , which reveals that the results on the fourth and fifth blocking levels are consistent. This means that we have eliminated most of the transient effects near the fixed point in the MCRG procedure. We also see that the main limitation of our calculation is statistics-the error bars are still rather large for the highest blocking level.
Now in order to obtain values for and , we must extrapolate our results from a finite number of blocking levels to an infinite number. This is done by fitting the corresponding eigenvalues and according to
where is the extrapolated value and is the correction-to-scaling exponent. Therefore, we first need to calculate , which comes directly from the second leading even eigenvalue: . Our best estimate is in the interval -0.85, and we use the value 0.85 for the purpose of extrapolation, since it gives the best fits. The final results are , , where the first errors are statistical and the second errors are estimates of the systematic error coming from the uncertainty in .
Figure 4.13: Leading Even Eigenvalue on Lattice
Figure 4.14: Leading Even Eigenvalue on Lattice
Figure 4.15: Leading Odd Eigenvalue on Lattice
Figure 4.16: Leading Odd Eigenvalue on Lattice
Finally, perhaps the most important number, because it can be determined the most accurately, is . By comparing the fifth blocking level on the lattice to the fourth on the lattice for both coupling values and taking a weighted mean, we obtain , where again the first error is statistical and the second is systematic.
Thus, MCRG calculations give us very accurate values for the three critical parameters , , and , and give a reasonable estimate for . Each parameter is obtained independently and directly from the data. We have shown that truncation and finite-size errors at all but the highest blocking level have been reduced to below the statistical errors. Future high statistics simulations on lattices will significantly reduce the remaining errors and allow us to determine the exponents very accurately.