A formula for boundary correlations of the critical Ising model

TL;DR

This paper derives a simple matrix formula for boundary spin correlations in the critical Ising model on polygonal regions, revealing explicit trigonometric sums for regular polygons, advancing understanding of boundary behaviors in statistical physics.

Contribution

It provides a novel, explicit matrix formula for boundary correlations in the critical Ising model, applicable to polygonal regions and simplifying to trigonometric sums for regular polygons.

Findings

Matrix formula for boundary correlations derived

Explicit trigonometric sums for regular polygons obtained

Invariance under star-triangle transformations demonstrated

Abstract

Given a finite rhombus tiling of a polygonal region in the plane, the associated critical $Z$-invariant Ising model is invariant under star-triangle transformations. We give a simple matrix formula describing spin correlations between boundary vertices in terms of the shape of the region. When the region is a regular polygon, our formula becomes an explicit trigonometric sum.