On the Symmetry of Universal Finite-Size Scaling Functions in Anisotropic Systems

In this work a symmetry of universal finite-size scaling functions under a certain anisotropic scale transformation is postulated. This transformation connects the properties of a finite two-dimensional system at criticality with generalized aspect ratio $\rho > 1$ to a system with $\rho < 1$. The symmetry is formulated within a finite-size scaling theory, and expressions for several universal amplitude ratios are derived. The predictions are confirmed within the exactly solvable weakly anisotropic two-dimensional Ising model and are checked within the two-dimensional dipolar in-plane Ising model using Monte Carlo simulations. This model shows a strongly anisotropic phase transition with different correlation length exponents $\nu_{||} \neq \nu_\perp$ parallel and perpendicular to the spin axis.