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-d imensional O(n)-symmetric order parameter systems in confined geometri es is studied for the three-dimensional mean spherical model of geomet ry L3-d' x infinity(d'), 0 less-than-or-equal-to d' < 3. For a fully f inite geometry the general case of d(p) greater-than-or-equal-to 0 per iodic, d(a) greater-than-or-equal-to 0 antiperiodic, d0 greater-than-o r-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 antiperio dic and/or periodic boundary conditions is investigated. The correspon ding 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 hypot hesis 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 logari thmic 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 obtain ed only for (T(c) - T) L much greater than In L.