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

We show that Josephson's relation, $\upsilon =2\mathrm{}\beta -\eta \nu =2-\alpha -2\mathrm{}\nu$, connecting the critical exponent $\upsilon$, 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 $d<\mathrm{}4\mathrm{}$ hyperscaling presumably holds, and we have $\upsilon =(d-2\mathrm{})\mathrm{}\nu$. Secondly, we derive the first two leading terms of the helicity modulus in an $\epsilon =4-d$ expansion; this allows second-order evaluation of universal amplitude ratios associated with the superfluid transition.