Corrections to scaling at two-dimensional Ising transitions.

Abstract

We investigate the completeness of the set of irrelevant critical exponents predicted by conformal invariance, particularly the question of whether the set of operators in the conformal block is complete in general, or only in a subspace of the critical hypersurface. The leading correction-to-scaling exponent is determined numerically at the critical points of some versions of the two-dimensional Ising model that enhance vacancy excitations: the antiferromagnetic Ising model in a magnetic field and the Blume-Capel model. The presence of corrections to scaling with the vacancy exponent ${y}_{\mathrm{vac}=\mathrm{\ensuremath{-}}(4/3}$ would be consistent with conformal invariance, but would contradict completeness of the closed operator algebra. Only corrections to scaling with an irrelevant critical exponent ${y}_{\mathrm{ir}=\mathrm{\ensuremath{-}}2.0}$ are found, consistent with completeness. However, an investigation of the random-cluster-model representation of the q=2+\ensuremath{\epsilon} state Potts model shows that a vacancy correction to scaling with exponent ${y}_{\mathrm{vac}=\mathrm{\ensuremath{-}}(4/3}$ appears. The correction is of order \ensuremath{\epsilon}: its amplitude vanishes in the Ising case q=2.