Intersecting Loop Models on Z^D: Rigorous Results

TL;DR

This paper studies intersecting loop models in multiple dimensions, revealing phase transitions and symmetry breaking phenomena, especially for specific parameters, and clarifies their relation to spin models.

Contribution

It provides rigorous results on phase transitions and symmetry breaking in intersecting loop models across various dimensions and parameters.

Findings

Existence of a phase transition for n=2, D=2 model with divergent loops.

No phase transition for large n and arbitrary D.

Broken translational symmetry observed in D=2, large n case.

Abstract

We consider a general class of (intersecting) loop models in D dimensions, including those related to high-temperature expansions of well-known spin models. We find that the loop models exhibit some interesting features - often in the ``unphysical'' region of parameter space where all connection with the original spin Hamiltonian is apparently lost. For a particular n=2, D=2 model, we establish the existence of a phase transition, possibly associated with divergent loops. However, for n >> 1 and arbitrary D there is no phase transition marked by the appearance of large loops. Furthermore, at least for D=2 (and n large) we find a phase transition characterised by broken translational symmetry.