Strong coupling corrections to the Ginzburg-Landau theory of superfluid ${}^3\mathrm{He}$

In the Ginzburg-Landau theory of superfluid ${}^3\mathrm{He}$, the free energy is expressed as an expansion of invariants of a complex order parameter. Strong coupling effects, which increase with increasing pressure, are embodied in the set of coefficients of these order-parameter invariants [A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975); E. V. Thuneberg, Phys. Rev. B 36, 3583 (1987); J. Low Temp. Phys. 122, 657 (2001)]. Experiments can be used to determine four independent combinations of the coefficients of the five fourth-order invariants. This leaves the phenomenological description of the thermodynamics near $T_c$ incomplete. Theoretical understanding of these coefficients is also quite limited. We analyze our measurements of the magnetic susceptibility and the NMR frequency shift in the $B$ phase which refine the four experimental inputs to the phenomenological theory. We propose a model based on existing experiments, combined with calculations by Sauls and Serene [Phys. Rev. B 24, 183 (1981)] of the pressure dependence of these coefficients, in order to determine all five fourth-order terms. This model leads us to a better understanding of the thermodynamics of superfluid ${}^3\mathrm{He}$ in its various states. We discuss the surface tension of bulk superfluid ${}^3\mathrm{He}$ and predictions for novel states of the superfluid such as those that are stabilized by elastic scattering of quasiparticles from a highly porous silica aerogel.

Figure 1

${\widehat{g}}_z$ obtained from magnetization measurements by NMR. Closed circles are the measurements by Haard (Ref. 5) and open circles by Hoyt et al. (Ref. 21) and Scholz (Ref. 29). The results from both measurements are consistent and give approximately unity for ${\widehat{g}}_z$.

Figure 2

(Color online) Comparison of four known $\beta$ combinations from the experiments (dashed lines) and calculation of Sauls and Serene (Ref. 3) (solid lines). The pressure dependences are in good agreement, but the absolute values are not as close.

Figure 3

(Color online) Zero pressure values of the ${\beta }_i$’s parametrized by ${\beta }_1\left(0\right)$. The numbers on each line correspond to the subscript $i$ of ${\beta }_i$. The ${\beta }_i\left(0\right)$’s clearly converge around ${\beta }_i\left(0\right)∕{\beta }_i^{\mathrm{wc}}=1$, which is an indication that the ${\beta }_i$’s tend toward their weak coupling values at low pressure.

Figure 4

(Color online) Osheroff and Cross’s measurements of the surface tension (Ref. 16) in comparison with the calculation of Thuneberg (Ref. 15) for various ${\beta }_1$ choices and $2∕{\beta }_2^{\left(0\right)}={\beta }_{245}^{-1}+{({\beta }_{12}+{\beta }_{345}∕2)}^{-1}$. The comparison of the two with the choice of ${\beta }_1∕{\beta }_1^{\mathrm{wc}}\sim 1$ is consistent, but the measurement lacks the resolution to conclusively determine the value of ${\beta }_1$.

Figure 5

(Color online) Phase diagram for axial and axiplanar states with ${\beta }_1$ as a parameter. The choice of ${\beta }_1$ with our model (dashed line) places the $A$ phase in the region of the axial state.

Figure 6

(Color online) The asymmetry ratio $r$ for the $A_1\text{−}{A}_2$ splitting was calculated for the axial state (solid red line) and the robust phase (dashed line) in aerogel, where we used the IISM of Sauls and Sharma (Ref. 36) with a transport mean free path for ${}^3\mathrm{He}$ quasiparticles $\lambda =150\mathrm{nm}$ and strand-strand correlation length ${\xi }_a=40\mathrm{nm}$, which match well to phase diagram measurements on the same sample by Gervais et al. (Ref. 40). The measurements of Choi et al. (Ref. 48) (closed circles) are more consistent with the $A$-like phase of aerogel ${}^3\mathrm{He}$ being the axial state than the robust state.

Figure 7

(Color online) Phase diagram for the isotropic phase and the robust phase for ${}^3\mathrm{He}$ in 98% porosity aerogel with ${\beta }_1$ (bulk) as a parameter. Our choice of ${\beta }_1$ (dashed line) makes the isotropic phase more stable than the robust phase.