Backbone exponents of the two-dimensional q-state Potts model: a Monte Carlo investigation.

We determine the backbone exponent X(b) of several critical and tricritical q-state Potts models in two dimensions. The critical systems include the bond percolation, the Ising, the q=2-sqrt[3], 3, and 4 state Potts, and the Baxter-Wu model, and the tricritical ones include the q=1 Potts model and the Blume-Capel model. For this purpose, we formulate several efficient Monte Carlo methods and sample the probability P2 of a pair of points connected via at least two independent paths. Finite-size-scaling analysis of P2 yields X(b) as 0.3566(2), 0.2696(3), 0.2105(3), and 0.127(4) for the critical q=2-sqrt[3], 1,2, 3, and 4 state Potts model, respectively. At tricriticality, we obtain X(b)=0.0520(3) and 0.0753(6) for the q=1 and 2 Potts model, respectively. For the critical q-->0 Potts model it is derived that X(b)=3/4. From a scaling argument, we find that, at tricriticality, X(b) reduces to the magnetic exponent, as confirmed by the numerical results.

[1]  P. Zinn-Justin,et al.  A transfer matrix for the backbone exponent of two-dimensional percolation , 2001, cond-mat/0111374.

[2]  Coniglio.  Fractal structure of Ising and Potts clusters: Exact results. , 1989, Physical review letters.

[3]  Wolff,et al.  Collective Monte Carlo updating for spin systems. , 1989, Physical review letters.

[4]  Antonio Coniglio,et al.  Clusters and droplets in the q-state Potts model , 1982 .

[5]  R. Baxter,et al.  Ising Model on a Triangular Lattice with Three-spin Interactions. I. The Eigenvalue Equation , 1974 .

[6]  P. Grassberger,et al.  Conductivity exponent and backbone dimension in 2-d percolation , 1998, cond-mat/9808095.

[7]  U. Wolff.  Comparison Between Cluster Monte Carlo Algorithms in the Ising Model , 1989 .

[8]  Blöte,et al.  Geometrical aspects of critical Ising configurations in two dimensions. , 1992, Physical review letters.

[9]  Beyond blobs in percolation cluster structure: the distribution of 3-blocks at the percolation threshold. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  R. Baxter,et al.  Exact Solution of an Ising Model with Three-Spin Interactions on a Triangular Lattice , 1973 .

[11]  P. Grassberger.  Spreading and backbone dimensions of 2D percolation , 1992 .

[12]  Monochromatic path crossing exponents and graph connectivity in two-dimensional percolation. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  H. Eugene Stanley,et al.  Building blocks of percolation clusters: volatile fractals , 1984 .

[14]  M. Blume.  THEORY OF THE FIRST-ORDER MAGNETIC PHASE CHANGE IN UO$sub 2$ , 1966 .

[15]  Li,et al.  Rigorous lower bound on the dynamic critical exponents of the Swendsen-Wang algorithm. , 1989, Physical review letters.

[16]  B. Duplantier,et al.  Statistical mechanics of polymer networks of any topology , 1989 .

[17]  H E Stanley,et al.  Backbone and elastic backbone of percolation clusters obtained by the new method of 'burning' , 1984 .

[18]  Henk W. J. Blöte,et al.  The simple-cubic lattice gas with nearest-neighbour exclusion: Ising universality , 1996 .

[19]  T. A. Larsson.  Possibly exact fractal dimensions from conformal invariance , 1987 .

[20]  B. Nienhuis,et al.  Analytical calculation of two leading exponents of the dilute Potts model , 1982 .

[21]  Henk W. J. Blöte,et al.  Geometric cluster Monte Carlo simulation , 1998 .

[22]  H. E. Stanley,et al.  The fractal dimension of the minimum path in two- and three-dimensional percolation , 1988 .

[23]  Ferdinando Gliozzi.  Simulation of Potts models with real q and no critical slowing down. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  R. B. Potts.  Some generalized order-disorder transformations , 1952, Mathematical Proceedings of the Cambridge Philosophical Society.

[25]  Fractal dimensions of Potts clusters , 1992 .

[26]  E. N. Miranda.  Geometrical properties of Swendsen-Wang clusters , 1991 .

[27]  POLYMERS AND PERCOLATION IN TWO DIMENSIONS AND TWISTED N=2 SUPERSYMMETRY , 1991, hep-th/9111007.

[28]  C. Fortuin,et al.  On the random-cluster model: I. Introduction and relation to other models , 1972 .

[29]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .

[30]  F. Y. Wu.  The Potts model , 1982 .