Helicity Modulus, Superfluidity, and Scaling in Isotropic Systems

The ordered state of a d-dimensional isotropic system with an n-vector (n≥2) order parameter is considered. By the imposition of suitable boundary conditions it is shown how to define explicitly a helicity modulus Υ(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 ρₛ(T)=(m)²Υ(T). A critical exponent v is defined by Υ(T)~|T-Tc|ᵛ as T→Tc; 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=2β-ην=2-α-2ν. This reduces to v=(d-2)ν (used by some authors with $d=3$), only if one accepts d-dependent, "hyperscaling" relations such as dν=2-α; however, both these latter relations fail for the ideal Bose gas when $d>4$. An alternative derivation of the formula v=2-α-2ν is based on the scaling theory for systems with a large but finite dimension.