FINITE-SIZE DEPENDENCE OF THE HELICITY MODULUS WITHIN THE MEAN SPHERICAL MODEL

The validity of the finite-size scaling prediction about the existence of logarithmic corrections in the helicity modulus UPSILON of three-dimensional O(n)-symmetric order parameter systems in confined geometries is studied for the three-dimensional mean spherical model of geometry L3-d' x infinity(d'), 0 less-than-or-equal-to d' < 3. For a fully finite geometry the general case of d(p) greater-than-or-equal-to 0 periodic, d(a) greater-than-or-equal-to 0 antiperiodic, d0 greater-than-or-equal-to 0 free, and d1 greater-than-or-equal-to 0 fixed (d(p) + d(a) + 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 temperature T(c) and the shifted critical temperature T(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 -[(d0-d1)/4pi+2(da-1)/pi2]ln L/L is obtained only for (T(c) - T) L much greater than In L.