Helicity Modulus, Superfluidity, and Scaling in Isotropic Systems

The ordered state of a $d$-dimensional isotropic system with an $n$-vector ($n\ge 2$) order parameter is considered. By the imposition of suitable boundary conditions it is shown how to define explicitly a helicity modulus $\Upsilon \mathrm{}\left(T\right)\mathrm{}$ 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 ${\rho }_s\mathrm{}\left(T\right)={\left(\frac{m}{\hslash }\right)}^2\Upsilon \mathrm{}\left(T\right)\mathrm{}$. A critical exponent $v$ is defined by $\Upsilon \mathrm{}\left(T\right)\mathrm{}\sim {|T-T_c|}^v$ as $T\to T_c$; for an ideal Bose gas and spherical model ($n\to \infty$), $v=1\mathrm{}$ is an exact result for all $d>\mathrm{}2\mathrm{}$. 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 $v=2\mathrm{}\beta -\eta \nu =2-\alpha -2\mathrm{}\nu$. This reduces to $v=(d-2\mathrm{})\mathrm{}\nu$ (used by some authors with $d=3\mathrm{}$), only if one accepts $d$-dependent, "hyperscaling" relations such as $d\nu =2-\alpha$; however, both these latter relations fail for the ideal Bose gas when $d>\mathrm{}4\mathrm{}$. An alternative derivation of the formula $v=2-\alpha -2\mathrm{}\nu$ is based on the scaling theory for systems with a large but finite dimension.