Pre-logarithmic and logarithmic fields in a sandpile model

TL;DR

This paper analyzes the boundary effects in a 2D Abelian sandpile model, linking lattice operators to boundary logarithmic conformal field theory, and identifies the scaling limits of key operators.

Contribution

It provides a detailed lattice analysis of boundary-changing operators and identifies the mass insertion operator as a logarithmic field in the conformal limit.

Findings

The boundary change operator is a weight -1/8 primary field.

The mass insertion operator scales as a weight zero logarithmic field.

Fusion rules match lattice calculations.

Abstract

We consider the unoriented two-dimensional Abelian sandpile model on the half-plane with open and closed boundary conditions, and relate it to the boundary logarithmic conformal field theory with central charge c=-2. Building on previous results, we first perform a complementary lattice analysis of the operator effecting the change of boundary condition between open and closed, which confirms that this operator is a weight -1/8 boundary primary field, whose fusion agrees with lattice calculations. We then consider the operators corresponding to the unit height variable and to a mass insertion at an isolated site of the upper half plane and compute their one-point functions in presence of a boundary containing the two kinds of boundary conditions. We show that the scaling limit of the mass insertion operator is a weight zero logarithmic field.