Order parameter for two-dimensional critical systems with boundaries

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 $\frac{1}{2}\ensuremath{\eta}.$ 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 $\ensuremath{\eta}.$ 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.