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Physical review. B, Solid state

Kadanoff block transformation by the Monte Carlo technique

Zvi Friedman, J. Felsteiner

Published Jun 1, 1977• 42 citations
Publisher

Abstract

A program combining the Monte Carlo technique and the Kadanoff block transformation is carried out for the Ising model in two and three dimensions, and the critical point and the critical indices $\ensuremath{\nu}$ and $\ensuremath{\delta}$ are obtained. A sequence of approximations seems to converge to the exact Onsager result in two dimensions, although the convergence of $\ensuremath{\nu}$ is rather slow. In three dimensions, the numbers obtained seem to tend towards the results from exact series expansions.

DOI
10.1103/physrevb.15.5317

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A program combining the Monte Carlo technique and the Kadanoff block transformation is carried out for the Ising model in two and three dimensions, and the critical point and the critical indices \ensuremathν\ensuremath{\nu}\ensuremathν and \ensuremathδ\ensuremath{\delta}\ensuremathδ are obtained. A sequence of approximations seems to converge to the exact Onsager result in two dimensions, although the convergence of \ensuremathν\ensuremath{\nu}\ensuremathν is rather slow. In three dimensions, the numbers obtained seem to tend towards the results from exact series expansions.
We review the assumptions on which the Monte Carlo renormalization technique is based, in particular, the analyticity of the block-spin transformations. On this basis, we select an optimized Kadanoff blocking rule in combination with the simulation of a d=3d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3d=3 Ising model with reduced corrections to scaling. This is achieved by including interactions with second and third neighbors. As a consequence of the improved transformation, this Monte Carlo renormalization method yields a fast convergence and a high accuracy. The results for the critical exponents are yH=2.481(1){y}_{H}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2.481(1)yH​=2.481(1) and yT=1.585(3){y}_{T}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.585(3)yT​=1.585(3).
The majority-rule, block-spin, renormalization-group transformation for the d=3d=3d=3 Ising model is studied, with the use of a recently developed technique for calculating renormalized coupling parameters from a single Monte Carlo simulation. Renormalization-group trajectories and the approximate fixed point are calculated. Improvements in the convergence of the Monte Carlo renormalization-group calculations of critical exponents are discussed.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.