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arXiv:0911.5670 · International Symposium on Lattice Field Theory (LATTICE)

Cluster Algorithm Renormalization Group Method

Guillermo Palma

Published Jun 30, 2009• 0 citations
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Abstract

We present a self consistent method based on cluster algorithms and Renormalization Group on the lattice to study critical systems numerically. We illustrate it by means of the 2D Ising model. We compute the critical exponents $ν$ and $η$ and the renormalization group flow of the probability density function of the magnetization. The results, compared to the standard Monte Carlo Renormalization Group proposed by Swendsen [1], are very accurate and the method works faster by a factor which grows monotonically with the lattice size. This allows to simulate larger lattices in reachable computational times.

DOI
10.22323/1.066.0043

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We present a self consistent method based on cluster algorithms and Renormalization Group on the lattice to study critical systems numerically. We illustrate it by means of the 2D Ising model. We compute the critical exponents ννν and ηηη and the renormalization group flow of the probability density function of the magnetization. The results, compared to the standard Monte Carlo Renormalization Group proposed by Swendsen [1], are very accurate and the method works faster by a factor which grows monotonically with the lattice size. This allows to simulate larger lattices in reachable computational times.
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The critical relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.
We introduce a new numerical method for the solution of self-consistent equations in the cluster mean-field theory. The method uses the density matrix renormalization group method to solve the associated cluster problem. We obtain an accurate critical value of the supersolid-superfluid transitions in the hard-core bosonic Hubbard model on a triangular lattice, which is comparable with the recent quantum Monte Carlo results. This algorithm is applicable to more general classes of models with a larger number of degrees of freedom.
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We present a histogram Monte Carlo method to calculate the existence probability Ep{\mathit{E}}_{\mathit{p}}Ep​, the percolation probability P, and the mean-cluster size S for percolation problems as continuous functions of the bond or site probability p. We then use Ep{\mathit{E}}_{\mathit{p}}Ep​ and P in a percolation renormalization-group method to calculate the critical point and exponents. Our method gives quite accurate results for percolation problems.