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

Abstract

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 $-2\le N\le 2$. 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.

arXiv comment4 pages, no figures; 2nd version: Introduction rewritten, comparison of prediction with recent high-precision Monte Carlo data with figure included, references added

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Cited by in corpus (8)

• Geometric Properties of the Three-Dimensional Ising and XY Models
• Methods to determine the Hausdorff dimension of vortex loops in the three-dimensional XY model
• Exact characterization of O(n) tricriticality in two dimensions
• Critical Loop Gases and the Worm Algorithm
• Disordered O(n) Loop Model and Coupled Conformal Field Theories
• Invaded cluster algorithm for a tricritical point in a diluted Potts model
• Tricritical O(n) models in two dimensions
• Phase diagram and the strong-coupling fixed point in the disordered O(n) loop model