Topologically-Protected Remnant Vortices in Confined Superfluid $^3$He

Abstract

Symmetry breaking phase transitions from less to more ordered phases will typically produce topological defects in the ordered phase. Kibble-Zurek theory predicts that for any second-order phase transition, such as the early universe, the density of defects that form should be determined by the scaling law for the system coherence time and the phase transition quench time. We have performed measurements of fourth sound dissipation due to vortex mutual friction in thin channels of superfluid $^3$He where one spatial dimension is smaller than a characteristic length scale predicted by the Kibble-Zurek theory. Our measurements suggest that remnant vortices form after the normal to superfluid second-order phase transition, and that the density of vortices is correlated with the size of the channel, but crucially, is independent of quench time. We propose a modified picture of defect formation, where closely spaced walls prevent the ends of vortex lines from reconnecting into loops. This leads to a mean vortex separation set by the wall spacing, which can result in much higher defect densities than in bulk systems.

Figures (4)

• Figure 1: Domain Illustration. (a), (b) Scaling of coherence time, $\tau$, and length, $\xi$, respectively, as a function of time. Here, it is assumed that the temperature is changed linearly as a function of time. Both quantities diverge when the critical temperature is reached. The freeze-out time $\hat{t}$ is indicated by the dashed line. This time is defined as the moment when $\tau(\hat{t}) = \tau_Q = T_c(dT/dt)^{-1}$. At this moment, there is insufficient time for the domains to grow significantly, so the coherence length is assumed to be "frozen" to first order at the value $\hat{\xi}$. (c) During a rapid quench, domains with average size $\hat{\xi}$ form, giving rise to defects. Each domain will independently select a random phase $\phi$. If the phases $\phi_{1,2,3}$ increase sequentially, with a total change in phase close to $2\pi$ (clockwise or counter-clockwise), the resulting phase winding at the vertex will form a vortex. The density of vortices predicted to form is of order $\hat{\xi}^{-2}$. (d) The average domain size in $^3$He for a temperature ramp rate of 0.12 mK/hour is predicted by KZ theory to vary between 60 and 160 $\mu$m as a function of pressure SI.
• Figure 2: Helmholtz Resonator Data.(a) Resonance curves of the three different devices, with varying nanoscale confinements, measured during a temperature sweep carried out at 15 bar. Inset shows the Lorentzian resonance shape at a single temperature. From the center frequency, the superfluid density can be extracted Shook2020. (b) Additionally, the linewidth of each device can be extracted, plotted here as a function temperature. (c) Resulting vortex densities calibrated from the Helmholtz resonator linewidths and the mutual friction, see main text. Note the lack of temperature dependence, and the dependence on confinement.
• Figure 3: Vortex Density Calibration Fit.(a) Plot of the experimental ratio $\Delta f/f_0^2$ as a function of temperature at 15 bar for the 636 nm device. This data has been fit to our model and plotted as a solid orange line. The dashed line displays the same curve if the offset is omitted in order to highlight the modification. (b) The vortex density is computed from the fit parameters $C_1$ and $C_2$.
• Figure 4: Comparison of the KZ theory and truncation model.(a) The prediction of the KZ theory, for a ramp rate of $dT/dt = 0.12$ mK/hr, is plotted as a dashed black curve as a function of pressure. The superfluid coherence length, and therefore also $\hat{\xi}$, changes with pressure. This curve is compared to the experimental vortex densities --- shown as a function of pressure for 636 nm (red), 805 nm (blue), and 1067 nm (green) devices --- and found to differ by more than three orders of magnitude. The truncated length-scale prediction of $L$ is shown as colored dashed lines. The x markers represent the fits to the raw data, and the circles are after correction for the artificial broadening of the Helmholtz resonances (see the discussion in the supplementary material SI). (b) Cartoon diagram of a domain that propagates from an initial point to a distance $\hat{\xi} > 2H$ (diameter), The walls of the domain become plane-like due to the confining walls. (c) A domain which has propagated a distance $\hat{\xi} = 2H$. This is when vortex lines joining the two walls first become possible.