Order parameter for two-dimensional critical systems with boundaries

TL;DR

This paper derives a universal order parameter profile for two-dimensional critical systems with boundaries, using conformal transformations, and validates it through numerical simulations of various models.

Contribution

It extends the conformal approach to include rectangular geometries and demonstrates how to determine the critical exponent eta from order parameter profiles.

Findings

Universal order parameter profile for rectangles at criticality

Accurate determination of eta from profile fitting

Validation through simulations of Ising, percolation, and planar rotor models

Abstract

Conformal transformations can be used to obtain the order parameter for two-dimensional systems at criticality in finite geometries with fixed boundary conditions on a connected boundary. To the known examples of this class (such as the disk and the infinite strip) we contribute the case of a rectangle. We show that the order parameter profile for simply connected boundaries can be represented as a universal function (independent of the criticality model) raised to the power eta/2. The universal function can be determined from the Gaussian model or equivalently a problem in two-dimensional electrostatics. We show that fitting the order parameter profile to the theoretical form gives an accurate route to the determination of eta. We perform numerical simulations for the Ising model and percolation for comparison with these analytic predictions, and apply this approach to the study of the planar rotor model.