Phase diagram and the strong-coupling fixed point in the disordered O(n) loop model

TL;DR

This study explores the phase diagram and critical behavior of the disordered two-dimensional O(n) loop model, revealing lines of fixed points and their properties across different n values using numerical methods.

Contribution

It identifies and characterizes lines of random and multicritical fixed points in the disordered O(n) loop model, including strong randomness effects and connections to known universality classes.

Findings

Line of random fixed points for n_c < n < 1 with matching perturbative results.

Line of multicritical fixed points at strong randomness for n > n_c.

Fixed point at n=1 with central charge suggesting Nishimori universality class.

Abstract

We numerically study the phase diagram and critical properties of the two-dimensional disordered O(n) loop model by using the transfer matrix and the worm Monte Carlo methods. The renormalization group flow is extracted from the landscape of the effective central charge obtained by the transfer matrix method based on the Zamolodchikov's C-theorem. We find a line of random fixed points (FPs) for $n_c < n <1$, with $n_c \sim 0.5$, for which the central charge and critical exponents agree well with the results of the $1-n$ perturbative expansion. Furthermore, for $n> n_c$, we find a line of multicritical FPs at strong randomness. The FP at $n=1$ has $c=0.4612(4)$, which suggests that it belongs to the universality class of the Nishimori point in the $\pm J$ random-bond Ising model. For $n>2$, we find another critical line that connects the hard-hexagon FP in the pure model to a finite-randomness zero-temperature FP.