Critical Scaling Properties at the Superfluid Transition of ${}^4\mathrm{He}$ in Aerogel

We study the superfluid transition of ${}^4\mathrm{He}$ 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 $\nu =0.73\pm 0.02$, in reasonable agreement with experiments and with previous simulations. For the heat capacity exponent $\alpha$, both experiments and previous simulations suggest deviations from the Josephson hyperscaling relation $\alpha =2-d\nu$. In contrast, our Monte Carlo results support hyperscaling with $\alpha =-0.2\pm 0.05$. We suggest a reinterpretation of the experiments, which avoids scaling violations and is consistent with our simulation results.

Figure 1

MC data for the heat capacity $c$ (main figure) and the dimensionless helicity modulus $LY/T$ (inset) as a function of temperature $T$ for different system sizes $L$, with $m=5\%$ aerogel volume fraction.

Figure 2

Finite size scaling data collapses of MC data showing the scaling functions for the helicity modulus $Y$ and the heat capacity $c$ for $m=5\%$. The data collapses giving the scaling functions are obtained from a joint fit to Eqs. (3, 4) for system sizes $L\ge 20$, assuming that $\alpha$ is given by the hyperscaling relation $\alpha =2-d\nu$.

Figure 3

Experimental data of the heat capacity for three aerogel volume fractions, 0.5%, 2%, and 5%, reproduced from Ref. 2 (curves with symbols), together with fits to the scaling function (thick curves) given by Eq. (6).