Talk presented by Martin Weigel

Parallel Session: Spin Models II

Thursday July 1st, 17.20 - 17.40, Room B

Universal amplitude ratios in finite-size scaling: three-dimensional O($n$) spin models

Abstract: Within the framework of conformal field theory (CFT) many systems of statistical mechanics can be solved exactly in the sense that the full algebra of scaling operators and the mass spectrum is given explicitly. This, however, only applies for two-dimensional geometries, where the constraint of conformal invariance at criticality is strong enough. For one prominent result, namely the linear relation of the scaling amplitudes of the masses in a strip geometry to the corresponding scaling dimensions, Cardy [1] has proposed a formal generalization to the $S^d\times R$ geometry. Successful numerical studies for this case have up to now been prevented by the fact that higher-dimensional spheres do not allow for a regular triangulation. In the same spirit, a transfer matrix calculation for a 3D geometry of (not conformally flat) slabs $S^1\times S^1\times R$ by Henkel [2] seems to indicate that the 2D result still applies in the 3D case with suitably adjusted boundary conditions for the special case of the Ising model. In a high-precision Monte Carlo simulation we investigate the scaling properties of the correlation lengths of the O($n$) symmetric classical spin models on 3D geometries with periodic and antiperiodic boundary conditions and find strong evidence for linear relations between FSS amplitudes and scaling dimensions that hint at the possibility to extract at least parts of the spectrum of these models by an analytical treatment.

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