Stability of Half-Quantum Vortices in $p_x+i{p}_y$ Superconductors

We consider the stability conditions for half-quantum vortices in a quasi-two-dimensional $p_x+i{p}_y$ superconductor (such as ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ is believed to be). The predicted exotic nature of these excitations has recently attracted much attention, but they have not been observed yet. We emphasize that an isolated half-quantum vortex has a divergent energy cost in the bulk due to its unscreened spin current, which requires two half-quantum vortices with opposite spin winding to pair. We show that the stability of such a pair is enhanced when the ratio of spin superfluid density to superfluid density ${\rho }_{\mathrm{sp}}/{\rho }_{\mathrm{s}}$ is small. We propose using various mesoscopic geometries to stabilize and observe these exotic excitations.

Figure 1

Left: a schematic representation of phase winding in (a) a $1/2$ QV and (b) a full QV. Right: the energy difference (see text) as a function of pair size, using $\xi =0.4\lambda$ for ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ for different values of ${\rho }_{\mathrm{sp}}/{\rho }_{\mathrm{s}}$: ${\rho }_{\mathrm{sp}}/{\rho }_{\mathrm{s}}=0.3$ (the blue solid curve), ${\rho }_{\mathrm{sp}}/{\rho }_{\mathrm{s}}=0.4$ (the purple dashed curve), and ${\rho }_{\mathrm{sp}}/{\rho }_{\mathrm{s}}=0.7$ (the red dotted curve). The energy per unit length is in units of ${\Phi }_0^2/16{\pi }^2{\lambda }^2$.

Figure 2

(a) A slab of thickness $L=2\lambda =0.304\mu \mathrm{m}$ in an external field $H$. (b) $G(x)$ [see Eq. (9)] in units of ${\Phi }_0^2/(4\pi \lambda {)}^2$ for a full-QV (blue solid curve) and for a $\frac{1}{2}$ QV (red dashed curve) at $H=212\mathrm{G}\sim 3.0{\Phi }_0/4\pi {\lambda }^2$, for $\xi =0.4\lambda$ and ${\rho }_{\mathrm{sp}}/{\rho }_{\mathrm{s}}=0.4$.