Suppressing the Kibble-Zurek Mechanism by a Symmetry-Violating Bias

The formation of topological defects in continuous phase transitions is driven by the Kibble-Zurek mechanism. Here we study the formation of single- and half-quantum vortices during transition to the polar phase of ${}^3\mathrm{He}$ in the presence of a symmetry-breaking bias provided by the applied magnetic field. We find that vortex formation is suppressed exponentially when the length scale associated with the bias field becomes smaller than the Kibble-Zurek length. We thus demonstrate an experimentally feasible shortcut to adiabaticity—an important aspect for further understanding of phase transitions as well as for engineering applications such as quantum computers or simulators.

Figure 1

Experimental principles. (a) The cubic $4\times 4\times 4{\mathrm{mm}}^3$ sample container is surrounded by NMR coils and filled with solid strands oriented along the vertical axis with the average diameter $d_1=9\mathrm{nm}$ [39] and average separation $d_2\approx 35\mathrm{nm}$. The space between the strands is filled with liquid ${}^3\mathrm{He}$. The magnetic field $\mathbf{H}\parallel \widehat{\mathbf{z}}$ can be applied in any direction in the plane transverse to the NMR coil axes (angle $\mu$ is represented by the light red sector). The orbital anisotropy vector $\widehat{\mathbf{m}}$ is pinned along the confining strands, and the spin anisotropy vector $\widehat{\mathbf{d}}$ is locked to the $\mathbf{x}\mathbf{y}$ plane (light blue sector represents angle $\alpha$) by $\mathbf{H}$. The sample can be rotated about the vertical axis with the angular velocity $\mathbf{\Omega }$ up to $3\mathrm{rad}{\mathrm{s}}^{-1}$. (b) The arrows represent the winding of $\widehat{\mathbf{d}}$ by angle $\alpha$ (highlighted by the arrow colors) in the vicinity of two HQV cores (green cylinders). On a loop around an HQV core the $\widehat{\mathbf{d}}$ vector rotates by $\pi$. For a large applied bias (top) pairs of HQVs are connected by narrow $\widehat{\mathbf{d}}$ solitons (highlighted with the background color) and the width of the soliton, giving the characteristic length scale of the applied bias field ${\xi }_{\mathrm{bias}}$, is much smaller than the KZ length, ${\xi }_{\mathrm{bias}}\ll l_{\mathrm{KZ}}$, resulting in suppression of the HQV formation in the phase transition to the superfluid phase. For a vanishing bias (bottom) ${\xi }_{\mathrm{bias}}\gg l_{\mathrm{KZ}}$, the winding of the $\widehat{\mathbf{d}}$ vector is nearly uniform, and formation of HQVs in the phase transition is not suppressed.

Figure 2

Suppression of the HQV density created by the KZM as a function of the applied bias. Filled red circles, magenta triangles, and black diamonds correspond to quench rates of ${\tau }_Q\approx 3.8\times {10}^2\mathrm{s}$, ${\tau }_Q\approx 1.4\times {10}^3\mathrm{s}$, and ${\tau }_Q\approx 7.7\times {10}^3\mathrm{s}$, respectively, while applying a constant $H=11\mathrm{mT}$ magnetic field. The field is rotated to achieve different bias fields $H_{\perp }=H\sin \mu$. Open blue squares (${\tau }_Q\approx 6.0\times {10}^2\mathrm{s}$) correspond to measurements with zero axial field component, $H_{\perp }=H$. Vortex density is constant for $H_{\perp }<H_{\perp t}$ and suppressed for higher bias fields. The suppression starts when the characteristic length scale of the bias field ${\xi }_{\mathrm{bias}}(H_{\perp })$ becomes smaller than the relevant Kibble-Zurek length. Solid lines correspond to theoretical model (see text for details). The dashed line shows where the intervotex distance becomes comparable with the container size. The inset shows the extracted threshold bias length ${\xi }_t$ as a function of $l_{\mathrm{KZ}}$ with the same symbols. The dashed line is ${\xi }_t=l_{\mathrm{KZ}}$. The patterned gray diamond is the same measurement as the black diamond, but with $l_{\mathrm{KZ}}$ on the horizontal axis replaced with an estimation of the transition front thickness $l_F$ [15]. For other measurements, $l_F$ lies beyond the right border of the plot.

Figure 3

Suppression of SQV density as a function of the applied bias. The measured magnon BEC relaxation rate (magenta diamonds) includes contributions from HQVs and SQVs. The HQV contribution (red circles) is separated using linear NMR measurements of the satellite intensity and calibration from Ref. [41]. The remaining contribution to relaxation (blue triangles) we attribute to SQVs. The observed relaxation rates are compared with the suppression model [solid lines, Eq. (7)], and the red line corresponds to the same threshold as in Fig. 2. In these measurements the quench rate is ${\tau }_Q\sim 4\times {10}^2\mathrm{s}$, and the magnitude of the magnetic field is kept constant while its direction is varied. The largest transverse field value corresponds to $\mu =\pi /2$. Constant BEC relaxation rate not related to vortices has been subtracted.