Fractal Structure of High-Temperature Graphs of O($N$) Models in Two Dimensions

Details Fractal Structure of High-Temperature Graphs of O($N$) Models in Two Dimensions Fractal Structure of High-Temperature Graphs of O($N$) Models in Two Dimensions

2005-02-02

arxiv.org/abs/cond-mat/0502062v2

Phys. Rev. Lett. 95, 135702 (2005)

Physics

Condensed Matter

Statistical Mechanics

4 pages, no figures; 2nd version: Introduction rewritten, comparison of prediction with recent high-precision Monte Carlo data

Scientific paper

10.1103/PhysRevLett.95.135702

The fractal structure and critical properties of the high-temperature graphs of the two-dimensional O($N)$ model close to criticality are investigated. Based on Monte Carlo simulations, De Gennes' results for polymer chains, corresponding to the limit $N \to 0$, are generalized to random loops for arbitrary $-2 \leq N \leq 2$. The loops are also studied close to their tricritical point, known as the $\Theta$ point in the context of polymers, where they collapse. The corresponding fractal dimensions are argued to be in one-to-one correspondence with those at the critical point, leading to an analytic prediction for the magnetic scaling dimension at the O($N)$ tricritical point.

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