Nonholonomy of order parameters and su(3) vortices in spin-1 Bose-Einstein condensates

A generalized Mermin-Ho relation for a spin-1 Bose-Einstein condensate is derived, which is applicable to vortices regardless of the symmetry and spin polarization of the order parameter. The obtained relation implies an su(3) mass-current circulation and two classes of vortices corresponding to two different su(2) subalgebras.

Figure 1

Three basis functions corresponding to (i) $|x\rangle$, (ii) $|y\rangle$, and (iii) $|z\rangle$ in the Cartesian representation plotted in the spherical coordinate $(r,\theta ,\varphi )$, where $r,\theta$, and $\varphi$ represent the radial coordinate, the polar angle, and the azimuth angle, respectively. Here, $\varphi \in [0,2\pi )$ gives the angle between the $x$ axis and the vector projected onto the $xy$ plane. The basis functions are represented in terms of the rank-1 spherical harmonics as $u_x(\theta ,\varphi ):=\langle \theta ,\varphi |x\rangle =(-Y_1^1(\theta ,\varphi )+Y_1^{-1}(\theta ,\varphi ))/\sqrt{2},u_y(\theta ,\varphi ):=\langle \theta ,\varphi |y\rangle =i(Y_1^1(\theta ,\varphi )+Y_1^{-1}(\theta ,\varphi ))/\sqrt{2}$, and $u_z(\theta ,\varphi ):=\langle \theta ,\varphi |z\rangle =Y_1^0(\theta ,\varphi )$. The radial coordinate is given by $r(\theta ,\varphi )=|u_{\mu }(\theta ,\varphi )|$. The color gauge shown on the right indicates the value of $\varphi$.

Figure 2

(i) Physical meanings of the parameters $\alpha ,\beta$, and $\gamma$, giving the principal axes ${\mathit{e}}_1$ and ${\mathit{e}}_2$, and the normal direction ${\mathit{e}}_3={\mathit{e}}_1\times {\mathit{e}}_2$ through ${\mathit{e}}_1={(cos\alpha cos\beta ,sin\alpha cos\beta ,-sin\beta )}^T,{\mathit{e}}_2={(-sin\alpha ,cos\alpha ,0)}^T$, and ${\mathit{e}}_3={(cos\alpha sin\beta ,sin\alpha sin\beta ,cos\beta )}^T$. The parameters $\alpha$ and $\beta$ represent the azimuth and polar angles of the spin vector $\mathit{f}=\left|\mathit{f}\right|{\mathit{e}}_3$, which is indicated by the thick solid arrow parallel to ${\mathit{e}}_3$, and $\gamma$ is the rotation angle of the order parameter around ${\mathit{e}}_3$. In panel (a), $\gamma =0$ and the major and minor principal axes of the order parameter coincide with ${\mathit{e}}_1$ and ${\mathit{e}}_2$, respectively. In panel (b), the coordinate system is rotated about ${\mathit{e}}_3$ through $\gamma =-0.25\pi$, where ${\mathit{e}}_1^{\prime }$ and ${\mathit{e}}_2^{\prime }$ give the major and minor principal axes in the rotated frame of reference. (ii) Parameter $\vartheta$ dependence of the order parameter. Here, $\alpha =\beta =\gamma =0$. The polarization parameters (a) $\vartheta =0$, (b) $0.15\pi$, and (c) $0.25\pi$ give the unpolarized (polar) state, a partially polarized (broken-axisymmetry) state, and the fully polarized (ferromagnetic) state, respectively. The wire frames are plotted as a guide to the eye.

Figure 3

Fully polarized spin textures in (i) CAT and (ii) MH vortices. Each arrow represents the direction of a local spin vector with the color showing the value of $\beta$ according to the upper right gauge. The projection circle below represents the value of $\alpha$ according to the lower right gauge. In both (i) and (ii), $\alpha$ changes by $2\pi$ along the circumference of the vortex, while $\beta$ changes by $\pi$ in (i) and $\pi /2$ in (ii) along the radial direction.

Figure 4

Spatial distributions of (a) the order parameter and (b) the spin vector in spin-nematic vortices which are dual to the Chechetkin-Anderson-Toulouse vortex and the Mermin-Ho vortex in Fig. 3. In panel (a), the color of the order parameter indicates the value of $\vartheta$ according to the upper right gauge and the color on the projection circle of the wave-function texture represents the value of $\gamma$ according to the lower right gauge. In panel (b), the color maps of the spin texture and its projection circle are the same as those in Figs. 3–3. In a vortex dual to the Chechetkin-Anderson-Toulouse vortex, $\gamma$ and $\vartheta$ change by $\pi$ and $\pi /2$, respectively. In terms of the pseudospin vector ${\mathit{f}}^{\prime },{\alpha }^{\prime }$ and ${\beta }^{\prime }$ change by $2\pi$ and $\pi$, as in $\alpha$ and $\beta$ of the CAT vortex. (ii)–(a) In a vortex dual to the Mermin-Ho vortex, $\gamma$ and $\vartheta$ change by $\pi$ and $\pi /4$, which implies that ${\alpha }^{\prime }$ and ${\beta }^{\prime }$ change by $2\pi$ and $\pi /2$. On the other hand, $\alpha$ and $\beta$ stay constant in both cases.

Figure 5

Circumference of a vortex $\mathcal{C}$ (left) decomposed into infinitesimal rectangular loops (upper middle), one of which is enlarged and labeled as $OABC$ (right) with $R_1$ and $R_2$ being SO(3) rotations given in Eqs. (26) and (27), where $\mathcal{S}$ indicates the area enclosed by $\mathcal{C}$.

Figure 6

Order-parameter and spin-vector textures in the polar-core vortices in Fig. 1 of Ref. [13] and Figs. 6 and 6 in Ref. [11] displayed in the same manner as in Fig. 4. (i) and (ii) correspond to the vortices in Refs. [13] and [11], respectively.

Figure 7

Order-parameter texture of the polar-core half-quantum vortex characterized by the winding number of $(n_{\alpha },n_{\beta },n_{\gamma },n_{\vartheta })=(0,1,1,0)$. The color gauge for the polarization $\vartheta$ and that for the orientation of the order parameter $\gamma$ are the same as those in the upper panels of Fig. 4.