Nonlinear field dependence and $f$-wave interactions in superfluid ${}^3$He

We present results of transverse acoustics studies in superfluid ${}^3$He-$B$ at fields up to 0.11 T. Using acoustic cavity interferometry, we observe the acoustic Faraday effect for a transverse sound wave propagating along the magnetic field, and we measure Faraday rotations of the polarization as large as 1710${}^{\circ }$. We use these results to determine the Zeeman splitting of the imaginary squashing mode, an order-parameter collective mode with total angular momentum $J=2$. We show that the pairing interaction in the $f$-wave channel is attractive at a pressure of $P=6$ bars. We also report nonlinear field dependence of the Faraday rotation at frequencies substantially above the mode frequency not accounted for in the theory of the transverse-acoustic dispersion relation formulated for frequencies near the mode. Consequently, we have identified the region of validity of the theory allowing us to make corrections to the analysis of Faraday rotation experiments performed in earlier work.

Figure 1

Frequencies of the ISQ (blue line) and pair-breaking edge (green dashed line) as a function of reduced temperature. The grey shaded area is the region supporting transverse sound. The red arrow indicates the path of the sound frequency, normalized to the pressure-dependent gap, during a typical decreasing pressure sweep. Inset: A representative acoustic response versus pressure measured between the ISQ and pair breaking, taken on a path similar to that indicated by the red arrow, at $H=0.04$ T, with the visible minima, indicated by black arrows, corresponding to AFE rotations of ${90}^{\circ }$, ${270}^{\circ }$, and ${450}^{\circ }$.

Figure 2

AFE rotation angle $\theta$ and sound velocity $c_t$ data at all experimental fields, as a function of both shift and magnetic field. The vertical dashed lines in (a) and (c) indicate the shift values for which data are presented in (b) and (d). (a) Data for $\theta$ as a function of shift. The solid circles correspond to minima in the acoustic trace. (b) A subset of $\theta$ data at representative values of $({\omega }^2-{\Omega }_0^2)/{\omega }^2$ as a function of field, taken at shifts from $({\omega }^2-{\Omega }_0^2)/{\omega }^2=0.035$ to 0.21 at intervals of 0.025, demonstrating nonlinear behavior and a decrease in $\theta$ as a function of distance from the ISQ. The dotted lines directly join the data points, and are presented solely as guides to the eye. The solid lines indicate the $\theta$ values expected from the dispersion for linear splitting at constant $g$ at the lowest (light grey, 0.035) and highest (black, 0.21) shifts presented, exhibiting less than two-thirds the decrease with field shown in the data. (c) Data for $c_t$ as a function of shift; the data points for different fields overlap at each shift value. (d) A subset of $c_t$ data at the same shift values as in (b), demonstrating a very slight field dependence of $c_t$.

Figure 3

Field dependence parameters $A$, $B$, and $C$ for Eq. (5), normalized to appropriate orders of $\omega$ to make them dimensionless, plotted against shift. (a) $A/\omega$ quantifies the linear field dependence, with the results from this work given by red circles, and data from Davis et al. (Ref. 4) at the fixed pressure of $P=4.7$ bars shown as open blue diamonds, which we have reanalyzed as discussed later in the text. At low shift the dominant contribution to $A$ is from the Zeeman splitting of the ISQ for which the theory[7] should be applicable. This splitting is predicted[6] to change slightly with $T$ and $P$ for a typical pressure sweep as shown by the solid grey curve. Clearly there are important contributions to $A$ for frequencies well away from the ISQ mode not described by the theory. The dark red dashed line indicates the extrapolation of our data to $\omega ={\Omega }_0$, where the theory is valid. (b) $B$ identifies an upper bound on the quadratic field dependencies, green triangles, with the shaded area showing the possible range of the magnitude of $B$. The result from Movshovich et al. (Ref. 10) is shown as a black square at $\omega ={\Omega }_0$, consistent with our analysis. (c) The cubic field parameter $C\omega$ is shown as solid orange diamonds.

Figure 4

A comparison of the relative importance of linear, quadratic, and cubic field dependencies on, (a) ${\theta }_{\text{calc}}$ and (b) $\delta c_t/c_{t0}=(c_{t\text{calc}}-c_{t0})/c_{t0}$ at $H=0.1$ T as a function of shift. Here ${\theta }_{\text{calc}}$ and $c_{t\text{calc}}$ are the rotation angle and sound velocity calculated by solving the dispersion for $q_{\pm }$ and inserting the result into Eqs. (6) and (7); $c_{t0}$ is calculated at zero field. These calculations use Eq. (5) with only the linear term, red open symbols; the linear plus quadratic terms, green lines; and linear, quadratic, plus cubic field terms, blue solid symbols. Adding the quadratic dependence is seen to change ${\theta }_{\text{calc}}$ very little, and $\delta c_t/c_{t0}$ a significant amount. Note that the quadratic field dependence, the $B$ term, is just an upper bound from our measurement of the field dependence of $c_t$ dictated by the precision of our measurement. Adding the cubic dependence changes ${\theta }_{\text{calc}}$ significantly, and $\delta c_t/c_{t0}$ very little, apart from very near the ISQ.

Figure 5

Calculated $g$ and $x_3^{-1}$ results as a function of pressure. (a) Extrapolated zero-shift $g$ values for this work (solid red circle, error bars inside the data point) and reanalyzed data of Davis et al. (Ref. 4) (solid blue diamonds). In the absence of Fermi liquid interaction effects, the expected weak-coupling value of $g$ for these temperatures and pressures varies between 0.033 and 0.04, well below most of these results (Ref. 6). (b) $f$-wave pairing parameter values calculated from $g$ values in (a). Data from this work (open red circle) and Davis et al. (open blue diamonds) are presented. Positive values correspond to repulsive interactions, and negative values to attractive interactions; the data generally indicate attractive $f$-wave pairing.