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

摘要: We show that Josephson's relation, $\ensuremath{\upsilon}=2\ensuremath{\beta}\ensuremath{-}\ensuremath{\eta}\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$, connecting the critical exponent $\ensuremath{\upsilon}$, describing vanishing of helicity modulus (superfluid density), to other thermodynamic and correlation exponents is an exact scaling law within renormalization-group framework. use standard isotropic $n$-vector model with continuous spins; when $dl4$ hyperscaling presumably holds, we have $\ensuremath{\upsilon}=(d\ensuremath{-}2)\ensuremath{\nu}$. Secondly, derive first two leading terms in $\ensuremath{\epsilon}=4\ensuremath{-}d$ expansion; this allows second-order evaluation universal amplitude ratios associated superfluid transition.