Percolation between vacancies in the two-dimensional Blume-Capel model

Abstract

Using suitable Monte Carlo methods and finite-size scaling, we investigate the Blume-Capel model on the square lattice. We construct percolation clusters by placing nearest-neighbor bonds between vacancies with a variable bond probability p(b) . At the tricritical point, we locate the percolation threshold of these vacancy clusters at p(bc) =0.706 33 (6) . At this point, we determine the fractal dimension of the vacancy clusters as Xf =0.1308 (5) approximately equal to 21/160, and the exponent governing the renormalization flow in the p(b) direction as y(p) =0.426 (2) approximately equal to 17/40 . For bond probability p(b) > p(bc) , the vacancy clusters maintain strong critical correlations; the fractal dimension is Xf =0.0750 (2) approximately equal to 3/40 and the leading correction exponent is y(p) =-0.45 (2) approximately equal to -19/40 . The above values fit well in the Kac table for the tricritical Ising model. These vacancy clusters have much analogy with those consisting of Ising spins of the same sign, although the associated quantities rho and magnetization m are energylike and magnetic quantities, respectively. However, along the critical line of the Blume-Capel model, the vacancies are more or less uniformly distributed over the whole lattice. In this case, no critical percolation correlations are observed in the vacancy clusters, at least in the physical region p(b) < or = 1 .