Properties of a nematic spin vortex in an antiferromagnetic spin-1 Bose-Einstein condensate

A spin-1 condensate with antiferromagnetic interactions supports nematic spin vortices in the easy-plane polar phase. These vortices have a $2\pi$ winding of the nematic director with a core structure that depends on the quadratic Zeeman energy. We characterize the properties of the nematic spin vortex in a uniform quasi-two-dimensional system. We also obtain the vortex excitation spectrum and use it to quantify its stability against dissociating into two half-quantum vortices, finding a parameter regime where the nematic spin vortex is dynamically stable. These results are supported by full dynamical simulations.

Figure 1

The basic structure of the NSV at the origin in a quasi-2D EPP phase condensate. (a) Normal-core NSV and (b) a polar-core NSV. Cylindrical rods indicate the orientation of the nematic director $\vec{d}$. Far from the vortex core, the director lies in the plane and undergoes a $2\pi$ rotation as we complete a closed loop around the core. The background shaded colors indicate the total density.

Figure 2

Stationary-state component densities $|{\chi }_m{\left(\rho \right)|}^2$ of a NSV with (a) a normal core and (b) a polar core. (c) The central (peak) density of the $m=0$ component. (d) The core radius ${\rho }_{\text{core}}$ defined as the radius where $|{\chi }_{\pm 1}\left({\rho }_{\text{core}}\right){|}^2=\frac{1}{4}{n}_b$ [see arrow in subplot (a)]. The red dotted line shows $\frac{1}{\sqrt{2}}{\xi }_q$. (e) The effective potential $c_0{\left|{\chi }_{\mathrm{v}}\right|}^2$ for the core localized state ${\psi }_{\mathrm{core}}$ used to identify the critical point (see text).

Figure 3

Unstable (splitting) mode of the NSV. (a) Phase diagram showing the imaginary part of the eigenvalue of the unstable mode with $\eta =\pm 1$. Dotted line indicates the simple model for instability boundary based on countersuperflow instability. Several slices through the phase diagram showing the strength of instability as (b) $q$ varies and (c) as $c_1$ varies. In (b), the dashed line shows the zero mode for $c_1=0.15c_0$. The inset shows the behavior of the $c_1=0.15c_0$ modes near where the instability vanishes. (d) The nonzero $u$ components of the $\eta =-1$ unstable mode for $q=-0.3{\mu }_b$ and $c_1=0.05c_0$. Note for $q>q_c$ the $m=0$ component of the unstable mode is also nonzero. (e) The $z$ component of the spin density of the NSV after the unstable mode shown in (d) is added to the stationary vortex state, revealing that the vortices of the two components spatially separate.

Figure 4

The nontrivial $u$ components of a $\eta =-1$ zero energy mode and the functions ${\mathrm{\Lambda }}_{\pm }$. The same scaling factor is applied to the $u$ amplitudes. Note the nontrivial $v$ components for the zero-energy mode are given by ${\tilde{v}}_{\pm }={\tilde{u}}_{\mp }$. Parameters are as in Fig. 3.

Figure 5

(a)–(f) Time-evolution of the $z$ component of the spin density of a normal-core NSV in a circular flat-bottomed trap. The vortices in the $m=1$ (white cross) and $m=-1$ (black cross) components are indicated. Subplots (a) and (b) zoom in to reveal the vortex dynamics in the central region, whereas (c)–(f) show the full simulation domain and indicate the trap boundary (black dashed line). In (f), the vortex trajectories are also shown for the vortices in the $m=1$ (white line) and $m=-1$ (black line) components. Other simulation parameters are $q=-0.3{\mu }_b, c_1=0.05c_0, U_0=100{\mu }_b$, and $R=95{\xi }_b$ is the radius.

Figure 6

(a) Vortex separation $\mathrm{\Delta }r$ of a normal-core NSV, obtained with parameters $q=-0.3{\mu }_b, c_1=0.05c_0$. (b) Quadratic Zeeman dependence of the separation time $t_{\text{sep}}$ for $c_1=0.05c_0$. Black dots indicate numerical results. The red line is a fit to the BdG results, given by $4.07{\mu }_b/Im\left(E_{\text{un}}\right)$, where $Im\left(E_{\text{un}}\right)$ is shown in Fig. 3.