Correlation functions of twist operators applied to single self-avoiding loops

TL;DR

This paper studies the correlation functions of twist operators in a 2D loop model related to the O(n) spin model, deriving differential equations and explicit formulas for loop separation probabilities, and connecting them to SLE processes.

Contribution

It introduces a novel analysis of twist operator correlators in self-avoiding loop models, deriving PDEs and explicit expressions for loop separation probabilities, and links these to SLE theory.

Findings

Correlation functions satisfy specific partial differential equations.

Explicit formulas for the expected number of loops separating points.

Connection established between correlators and Schramm-Loewner evolution (SLE) processes.

Abstract

The O(n) spin model in two dimensions may equivalently be formulated as a loop model, and then mapped to a height model which is conjectured to flow under the renormalization group to a conformal field theory (CFT). At the critical point, the order n terms in the partition function and correlation functions describe single self-avoiding loops. We investigate the ensemble of these self-avoiding loops using twist operators, which count loops which wind non-trivially around them with a factor -1. These turn out to have level two null states and hence their correlators satisfy a set of partial differential equations. We show that partly-connected parts of the four point function count the expected number of loops which separate one pair of points from the other pair, and find an explicit expression for this. We argue that the differential equation satisfied by these expectation values should have an interpretation in terms of a stochastic(Schramm)-Loewner evolution (SLE_kappa) process with kappa=6. The two point function in a simply connected domain satisfies a closely related set of equations. We solve these and hence calculate the expected number of single loops which separate both points from the boundary.