Helicity Modulus, Superfluidity, and Scaling in Isotropic Systems

The ordered state of a d-dimensional isotropic system with an n-vector (n2) order parameter is considered. By the imposition of suitable boundary conditions it is shown how to define explicitly a helicity modulus (T) which measures the free-energy increment associated with "twisting" the direction of the order parameter. For a Bose system the superfluid density is seen to be ₒ (T) = (m{) }^2 (T). A critical exponent v is defined by (T) |T-{T₂|}^v as TT₂; for an ideal Bose gas and spherical model (n), v=1 is an exact result for all d>2. The difficulties of defining a correlation length in the ordered phase are discussed. A full scaling theory of the correlations avoids these problems and may be linked to a phenomenological hydrodynamic approach, to clarify and rederive Josephson's relation v=2-=2--2. This reduces to v= (d-2) (used by some authors with d=3), only if one accepts d-dependent, "hyperscaling" relations such as d=2-; however, both these latter relations fail for the ideal Bose gas when d>4. An alternative derivation of the formula v=2--2 is based on the scaling theory for systems with a large but finite dimension.