Critical behavior of loops and biconnected clusters on fractals of dimension d < 2

TL;DR

This paper provides an exact solution for the O(n) loop model on a fractal lattice, revealing critical points, phases, and the effects of perturbations like vertex defects and bending rigidity.

Contribution

It introduces an exact real space renormalization group solution for the O(n) model on fractals and explores the impact of vertex defects and bending rigidity on critical behavior.

Findings

Existence of a critical point as void density decreases

Power-law correlations in the dense phase with n-dependent exponents

Bending rigidity does not change the universality class

Abstract

We solve the O(n) model, defined in terms of self- and mutually avoiding loops coexisting with voids, on a 3-simplex fractal lattice, using an exact real space renormalization group technique. As the density of voids is decreased, the model shows a critical point, and for even lower densities of voids, there is a dense phase showing power-law correlations, with critical exponents that depend on n, but are independent of density. At n=-2 on the dilute branch, a trivalent vertex defect acts as a marginal perturbation. We define a model of biconnected clusters which allows for a finite density of such vertices. As n is varied, we get a line of critical points of this generalized model, emanating from the point of marginality in the original loop model. We also study another perturbation of adding local bending rigidity to the loop model, and find that it does not affect the universality class.