Finite-size dependence of the helicity modulus within the mean spherical model

Abstract

The validity of the finite-size scaling prediction about the existence of logarithmic corrections in the helicity modulusγ of three-dimensional O(n)-symmetric order parameter systems in confined geometries is studied for the three-dimensional mean spherical model of geometryL 3/s-d′×∞d′, 0⩽d′<3. For a fully finite geometry the general case ofd p⩾0 periodic,d a⩾0 antiperiodic,d 0⩾0 free, andd 1⩾0 fixed (d p+da+d0+d1=d, d=3) boundary conditions is considered, whereas for film (d′=2) and cylinder (d′=1) geometries only the case of antiperiodic and/or periodic boundary conditions is investigated. The corresponding expressions for the finite-size scaling function of the helicity modulus and its asymptotics in the vicinity, below, and above the bulk critical temperatureT c and the shifted critical temperatureT c,L are derived. The obtained results are not in agreement with the hypothesis of the existence of a log(L) correction term to the finite-size behavior of the helicity modulus in the finite-size critical region if d=3. In the case of film and cylinder geometries there are no logarithmic corrections. In the case of a fully finite geometry a universal logarithmic correction term −[(d 0 −d 1)/4π−2da−1/π2] lnL/L is obtained only for (T c-T) L≫lnL.