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arXiv:cond-mat/0608547 · Journal of Physics A Mathematical and Theoretical

Geometric properties of two-dimensional O(n) loop configurations

Chengxiang Ding, Youjin Deng, Wenan Guo, Xiaofeng Qian, Henk W. J. Blöte

Published Mar 14, 2007• 13 citations

Abstract

We study the fractal geometry of O( n ) loop configurations in two dimensions by means of scaling and a Monte Carlo method, and compare the results with predictions based on the Coulomb gas technique. The Monte Carlo algorithm is applicable to models with noninteger n and uses local updates. Although these updates typically lead to nonlocal modifications of loop connectivities, the number of operations required per update is only of order 1. The Monte Carlo algorithm is applied to the honeycomb O( n ) model for several values of n , including noninteger ones. We thus determine scaling exponents that describe the fractal nature of O( n ) loops at criticality. The results of the numerical analysis agree with the theoretical predictions.

DOI: 10.1088/1751-8113/40/13/001

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We study the fractal geometry of O( n ) loop configurations in two dimensions by means of scaling and a Monte Carlo method, and compare the results with predictions based on the Coulomb gas technique. The Monte Carlo algorithm is applicable to models with noninteger n and uses local updates. Although these updates typically lead to nonlocal modifications of loop connectivities, the number of operations required per update is only of order 1. The Monte Carlo algorithm is applied to the honeycomb O( n ) model for several values of n , including noninteger ones. We thus determine scaling exponents that describe the fractal nature of O( n ) loops at criticality. The results of the numerical analysis agree with the theoretical predictions.

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The fractal structure of the Ising model at the critical point Tc is studied in the present paper.The fractal dimensionality D of the total magnetization at Tc was estimated numerically as D = 1.86 ±O.Ol for the two-dimensional square lattice and D=2.46±O.Ol for the three-dimensional simple cubic lattice by Monte Carlo simulations.These values agree very well with the value D=1.875 obtained from the exact critical exponents and D=2.48 obtained from the known critical exponents, re-~pectively, through the relation D=d-fJ/v=(d+r/v)/2.This fractalness yields the hyperscaling relation dv=2fJ+r.It was also observed how the fractal nature of the relevant system disappears as the system deviates from the critical point.The dimensionality d of the relevant lattice is observed at temperatures lower than Tc and the random percolation value d/2 at higher temperatures.

We investigate the Eulerian bond-cubic model on the square lattice by means of Monte Carlo simulations, using an efficient cluster algorithm and a finite-size scaling analysis. The critical points and four critical exponents of the model are determined for several values of n. Two of the exponents are fractal dimensions, which are obtained numerically for the first time. Our results are consistent with the Coulomb gas predictions for the critical O( n ) branch for n < 2 and the results obtained by previous transfer matrix calculations. For n = 2, we find that the thermal exponent, the magnetic exponent and the fractal dimension of the largest critical Eulerian bond component are different from those of the critical O(2) loop model. These results confirm that the cubic anisotropy is marginal at n = 2 but irrelevant for n < 2.