Specific Heat of Disordered Superfluid ${}^3\mathrm{H}\mathrm{e}$

The specific heat of superfluid ${}^3\mathrm{H}\mathrm{e}$, disordered by a silica aerogel, is found to have a sharp discontinuity marking the thermodynamic transition to superfluidity at a temperature reduced from that of bulk ${}^3\mathrm{H}\mathrm{e}$. The magnitude of the discontinuity is also suppressed. This disorder effect can be understood from the Ginzburg-Landau theory which takes into account elastic quasiparticle scattering suppressing both the transition temperature and the amplitude of the order parameter. We infer that the limiting temperature dependence of the specific heat is linear at low temperatures in the disordered superfluid state, consistent with predictions of gapless excitations everywhere on the Fermi surface.

Figure 1

Pressure temperature phase diagram for bulk ${}^3\mathrm{H}\mathrm{e}$ and disordered ${}^3\mathrm{H}\mathrm{e}$ in aerogel. The smooth curve is an interpolation from Greywall’s data [4] for bulk ${}^3\mathrm{H}\mathrm{e}$. The data points are high resolution measurements of the transition temperatures obtained during slow warming shown in the inset at $P=20.06\mathrm{\text{bars}}$. Two heat capacity anomalies are evident. The solid fit curve defines the bulk transition (dotted vertical line, open circles). The superfluid transition temperature in aerogel is at a lower temperature (solid vertical line, solid circles), $\sim 80\mu \mathrm{K}$ wide.

Figure 2

Heat capacity of ${}^3\mathrm{H}\mathrm{e}$ in aerogel. On the left side we show the heat capacity as a function of temperature for bulk ${}^3\mathrm{H}\mathrm{e}$ and ${}^3\mathrm{H}\mathrm{e}$ in a 98% porous silica aerogel contained together in the same calorimeter with addendum subtracted. On the right side we show the heat capacity divided by the temperature for the disordered superfluid alone. The curves represent the known temperature dependence of the heat capacity for bulk ${}^3\mathrm{H}\mathrm{e}$ (dashed lines) and a smooth fit to the data in the disordered superfluid with the constraint for entropy conservation (dotted lines) described in the text.

Figure 3

Heat capacity addendum. The temperature and pressure dependence of the heat capacity addendum are shown for bulk ${}^3\mathrm{H}\mathrm{e}$ in aerogel with an estimated accuracy of $0.25\mathrm{m}\mathrm{J}/\mathrm{K}$.

Figure 4

Heat capacity jump and density of states. The pressure dependence of the heat capacity jump (open circles, left axis) and the density of states (filled circles, right axis) at zero energy (the Fermi energy) is shown for bulk ${}^3\mathrm{H}\mathrm{e}$ in a 98% porous silica aerogel. The error bars arise from a possible systematic error in the choice of temperature dependence for the heat capacity just below $T_{c,a}$. Other statistical errors are of the size of the data points. Calculations from the homogeneous-isotropic-scattering model (HISM) are shown as curves for various values of the transport-mean-free path.