Superfluid quantum criticality in liquid ${}^3\mathrm{He}$ in anisotropic aerogel

In the novel superfluid polar phase realized in liquid ${}^3\mathrm{He}$ in highly anisotropic aerogels, a quantum transition to the polar-distorted A (PdA) phase may occur at low but finite pressure $P_c\left(0\right)$. It is shown that the nontrivial quantum dynamics of the critical fluctuation of PdA order is induced by the presence of both columnar impurity scattering leading to Anderson's theorem for the polar phase and the line node of the quasiparticle gap in the state, and that, in contrast to the situation of the normal to B-phase transition in isotropic aerogels, a weakly divergent behavior of the compressibility appears in the quantum critical region close to $P_c\left(0\right)$.

Figure 1

(a) Resulting $\overline{\omega }$ ($\equiv \left|\omega \right|/\left|\mathrm{\Delta }\right|$) vs $\mathrm{\Delta }{C}_{20}\left(\omega \right)$ $[\equiv C_{20}\left(\omega \right)-C_{20}\left(0\right)]$ curves for $\tau \left|\mathrm{\Delta }\right|=7.85$. Data directly follow from Eq. (16), while the solid curve is the corresponding result of Eq. (17). (b) Corresponding results for $\tau \left|\mathrm{\Delta }\right|=0.785$. We note that, according to Ref. [6], the experimental phase diagram is explained by using the value of $\tau \left|\mathrm{\Delta }\right|$ of order unity (see also the discussion related to Fig. 2). The figures indicate that the frequency range over which Eq. (17) is valid is wider than in the case (a).

Figure 2

Computed $I(x;c_2)$ curves for a moderately impure case with $c_2=1.0$ [upper two figures (a) and (b)] and for a less impure case with $c_2=-1.0$ [lower two figures (c) and (d)]. Each solid curve is the $\sqrt{|ln|x\left|\right|}$ curve best-fitted to the numerical data (symbols) of $I(x;c_2)$ for each case. To improve the convergence, the frequency cutoff ${\overline{\omega }}_c$ has been changed for the two cases and was chosen to be 1.0 for (a) and (b) and to be 0.1 for (c) and (d). In both cases, at low enough $x$, the $\sqrt{|ln|x\left|\right|}$ behavior is well satisfied.

Figure 3

The curve of the computed $I(x;1.0)$ in Fig. 2 extended over larger $x$-values. The data (symbols) nicely obey the $|ln|x\left|\right|$ form (the solid curve), implying that the $ln\left|\omega \right|$ correction in $\mathrm{\Delta }{C}_{20}$ is ineffective, so that the ordinary $z=1$ 3D GL action is valid, in the range $0.1<x<1.0$.

Figure 4

Schematic phase diagram near the QC point $P_c\left(0\right)$ of the polar to PdA second-order transition curve $P_c\left(T\right)$ (thick red solid curve). The thin solid curve and the blue solid curve denote the normal to polar second-order transition line and the PdA to PdB first-order transition line, respectively. The dashed curves imply the crossover lines on entering the thermal fluctuation region in $P>P_c\left(T\right)$ and the quantum disordered region in $P<P_c\left(0\right)$, which is typically expressed as $1-P/P_c\left(0\right)\sim 3.6{(T/T_{c0})}^2$. The divergent quantum critical behavior of $\kappa$ is expected to be seen on cooling along the downward arrow. The behavior Eq. (33) or $\kappa \simeq \sqrt{|ln(P_c-P\left)\right|}$ can be seen in the hatched region.