Two-dimensional O(n) model in a staggered field

TL;DR

This paper investigates a staggered field in a two-dimensional O(n) loop model, revealing phase transitions, critical behaviors, and mappings to other models, with implications for universality classes and fractal dimensions.

Contribution

It introduces a detailed analysis of the effects of a staggered field on the O(n) model, including phase diagram mapping and connections to Potts and loop models.

Findings

At T=0, for w>1, the model maps to a ferromagnetic Potts model with q=n^2.

The dense and dilute loop universality classes merge in the presence of the staggered field.

RGB loops exhibit a fractal dimension of 3/2.

Abstract

Nienhuis' truncated O(n) model gives rise to a model of self-avoiding loops on the hexagonal lattice, each loop having a fugacity of n. We study such loops subjected to a particular kind of staggered field w, which for n -> infinity has the geometrical effect of breaking the three-phase coexistence, linked to the three-colourability of the lattice faces. We show that at T = 0, for w > 1 the model flows to the ferromagnetic Potts model with q=n^2 states, with an associated fragmentation of the target space of the Coulomb gas. For T>0, there is a competition between T and w which gives rise to multicritical versions of the dense and dilute loop universality classes. Via an exact mapping, and numerical results, we establish that the latter two critical branches coincide with those found earlier in the O(n) model on the triangular lattice. Using transfer matrix studies, we have found the renormalisation group flows in the full phase diagram in the (T,w) plane, with fixed n.<br> Superposing three copies of such hexagonal-lattice loop models with staggered fields produces a variety of one or three-species fully-packed loop models on the triangular lattice with certain geometrical constraints, possessing integer central charges 0 <= c <= 6. In particular we show that Benjamini and Schramm's RGB loops have fractal dimension D_f = 3/2.