Renormalization-group proof of Josephson's relation andεexpansion for the helicity modulus (superfluid density)

We show that Josephson's relation, υ=2β-ην=2-α-2ν, connecting the critical exponent υ, describing the vanishing of the helicity modulus (superfluid density), to other thermodynamic and correlation exponents is an exact scaling law within the renormalization-group framework. We use the standard isotropic n-vector model with continuous spins; when $d<4$ hyperscaling presumably holds, and we have υ=(d-2)ν. Secondly, we derive the first two leading terms of the helicity modulus in an ε=4-d expansion; this allows second-order evaluation of universal amplitude ratios associated with the superfluid transition.