Critical Scaling Properties at the Superfluid Transition of $^4$He in Aerogel

Abstract

We study the superfluid transition of $^4$He in aerogel by Monte Carlo simulations and finite size scaling analysis. Aerogel is a highly porous silica glass, which we model by a diffusion limited cluster aggregation model. The superfluid is modeled by a three dimensional XY model, with excluded bonds to sites on the aerogel cluster. We obtain the correlation length exponent $ν=0.73 \pm 0.02$, in reasonable agreement with experiments and with previous simulations. For the heat capacity exponent $α$, both experiments and previous simulations suggest deviations from the Josephson hyperscaling relation $α=2-dν$. In contrast, our Monte Carlo results support hyperscaling with $α= -0.2\pm 0.05$. We suggest a reinterpretation of previous experiments, which avoids scaling violations and is consistent with our simulation results.