Persistent currents in ferromagnetic condensates

Persistent currents in Bose condensates with a scalar order parameter are stabilized by the topology of the order parameter manifold. In condensates with multicomponent order parameters it is topologically possible for supercurrents to “unwind” without leaving the manifold. We study the energetics of this process in the case of ferromagnetic condensates using a long wavelength energy functional that includes both the superfluid and spin stiffnesses. Exploiting analogies to an elastic rod and rigid body motion, we show that the current carrying state in a 1D ring geometry transitions between a spin helix in the energy minima and a solitonlike configuration at the maxima. The relevance to recent experiments in ultracold atoms is briefly discussed.

Figure 1

Two branches of the dispersion for $s=1$, with the solid and dashed lines corresponding to $p=0,1$ in (19). We use the soliton dispersion (74) at $\mathrm{\Delta }=0.1$ for simplicity; see Sec. 3d.

Figure 2

Elastic rod described by a space curve $\mathbf{\gamma }\left(x\right)$ as a function of arc length $x$, with unit tangent vector $\mathbf{n}\equiv {\mathbf{\gamma }}^{\prime }\left(x\right)$. For this curve, the corresponding $\mathbf{n}\left(x\right)$ is shown in Fig. 6.

Figure 3

Soliton dispersion (74) for $s=1/2$ and $\mathrm{\Delta }=0.1$.

Figure 4

(Solid line) Values of the Weierstrass invariants $g_2$, $g_3$ for which the quantization conditions (50) are satisfied for $m=0.3$. (Dashed lines) Curves of vanishing modular discriminant (87): the top curve corresponds to the helix solution; the lower to the soliton. For larger values of $m$ the solid curve would lie very close to the soliton line, though it always starts on the helix line.

Figure 5

Numerically computed ${\mathcal{E}}_{\sigma }$ for $m=0.4$ (dotted line), 0.6 (dot dashed), and 0.7 (dashed). Solid line shows the helix dispersion relation ${\mathcal{E}}_{\sigma }\left(\mathrm{\Omega }\right)=\frac{2\pi \alpha \mathrm{\Omega }}{L}\left(1-\frac{\mathrm{\Omega }}{4\pi }\right)$.

Figure 6

Example of a solitonlike configuration of $\mathbf{n}\left(x\right)$ for $m=0.9$, $\mathrm{\Omega }=1.1\pi$.

Figure 7

Full dispersion showing metastable minimum for the spin-1/2 case with $m=0.7$. The end point of the curve corresponds to $\mathcal{E}=2{\pi }^2\rho /L$, and corresponds to the energy of a fully polarized state with a single flux quantum.