Vortices and vortex states in Rashba spin-orbit-coupled condensates

The Rashba spin-orbit coupling is equivalent to the finite Yang-Mills flux of a static SU(2) gauge field. It gives rise to the protected edge states in two-dimensional topological band insulators, much like magnetic field yields the integer quantum Hall effect. An outstanding question is which collective topological behaviors of interacting particles are made possible by the Rashba spin-orbit coupling. Here we address one aspect of this question by exploring the Rashba SU(2) analogs of vortices in superconductors. Using the Landau-Ginzburg approach and conservation laws, we classify the prominent two-dimensional condensates of two- and three-component spin-orbit-coupled bosons, and characterize their vortex excitations. There are two prominent types of condensates that take advantage of the Rashba spin-orbit coupling. Their vortices exist in multiple flavors whose number is determined by the spin representation, and interact among themselves through logarithmic or linear potentials as a function of distance. The vortices that interact linearly exhibit confinement and asymptotic freedom similar to quarks in quantum chromodynamics. One of the two condensate types supports small metastable neutral quadruplets of vortices, and their tiles as metastable vortex lattices. Quantum melting of such vortex lattices could give rise to non-Abelian fractional topological insulators, SU(2) analogs of fractional quantum Hall states. The physical systems in which these states could exist are trapped two- and three-component bosonic ultracold atoms subjected to artificial gauge fields, as well as solid-state quantum wells made either from Kondo insulators such as ${\mathrm{SmB}}_6$ or conventional topological insulators interfaced with conventional superconductors.

Figure 1

The phase diagram of superfluid states of two-component bosons with Rashba spin-orbit coupling. m-C is a conventional uniform condensate with spin polarization along the $z$ axis. The type-I or “striped” condensate (T1) and type-II or “plain wave” condensate (T2) carry helical spin currents.

Figure 2

Topological defects of two-dimensional vector fields $\mathbf{v}\propto -\widehat{\mathbf{x}}sin\left(Q\varphi \right)+\widehat{\mathbf{y}}cos\left(Q\varphi \right)$, where $\varphi$ is the polar angle, can implement any total rotation angle $2\pi Q$ of the vector $\mathbf{v}$ around a loop that encloses the singularity. However, only the $Q=1$ case can describe a current field of a quantized U(1) vortex, $\mathbf{v}=\nabla \theta$. Any singular configuration of $\theta =k\varphi$, which winds by $2\pi k$, $k\ne 0$ on the loop around the singularity, still corresponds to the vector field $\mathbf{v}=k(-\widehat{\mathbf{x}}sin\varphi +\widehat{\mathbf{y}}cos\varphi )$ of a $Q=1$ vector vortex. Therefore, the curl of $\mathbf{v}$ can be zero only if $Q=0$ or $Q=1$.

Figure 3

(a) The structure of chiral $Q=Q^{\prime }=1$ vortices. The shaded circle at the center is a vortex core, and the solid oriented line at the core border shows the flow of chiral spin currents ${\mathbf{J}}^z\sim \nabla \theta$. The solid red lines indicate $\widehat{\theta }$, which is the orientation of the spin component that flows in a helical spin current. Dashed blue lines show the flow direction of the helical current ${\mathbf{J}}^{x,y}\sim \nabla \alpha$. (b) The structure of $Q=-1$, $Q^{\prime }=1$ vortices and the emergence of strings. The left panel shows the naive patterns of nonquantized helical spin currents $\left(\delta \alpha \right)$ based on the simple $\theta$ winding and the requirement (22). The right panel visualizes the formation of a string that keeps the same $\theta$ winding while fixing the quantization of helical currents (by removing their curls from the bulk). The entire shaded region is compressed into a filament to disguise this $Q=-1$ vortex as a $Q^{\prime }=+1$ vortex in most of the space.

Figure 4

The simplest neutral quadruplet of vortices. Vortex cores are shaded in gray, surrounded by thick oriented black lines that depict the flow of ${\mathbf{J}}^z\sim \nabla \theta$ spin currents. Solid thin red lines show the local orientation of the vector $\widehat{\theta }$. Dotted thin blue lines are the “field lines” of $\nabla \alpha$, hence showing the flow direction of helical spin currents ${\mathbf{J}}^{x,y}$. Far away from the quadruplet, the order parameter is reduced to that of a mean-field type-I condensate.

Figure 5

An example of a quadruplet that can combine the singularities of both ${\theta }_{\uparrow }$ and ${\theta }_{\downarrow }$. Going from left to right, the winding chiral angles can be $\uparrow \downarrow \uparrow \downarrow$, $\uparrow \uparrow \downarrow \downarrow$, and the two inverted combinations, among which the first one and its reflection have the lowest energy. The color and line convention is the same as in Fig. 4 (so that the configuration of $\theta ={\theta }_{\downarrow }-{\theta }_{\uparrow }$ is shown instead of either ${\theta }_{\downarrow }$ or ${\theta }_{\uparrow })$.

Figure 6

A domain-wall configuration of vortices separating two regions with different uniform type-I order parameters.

Figure 7

A TR-invariant lattice of type-I vortices.

Figure 8

The phase diagram of uniform currents in triplet Rashba spin-orbit-coupled condensates. The characteristic types of phases are type-I (T1) and type-2 (T2) condensates that carry spin currents, as well as conventional condensates (C) that carry no currents. A prefix “m” indicates finite magnetization in the $z$ direction. This plot was obtained by numerical minimization of the energy functional (33) with respect to the nine parameters in (35), at $m=1$, $v=1$, $t_{\mathrm{t}}=-2.5$, $a=-0.3$, $U_{\mathrm{t}}=2$ in arbitrary units. A qualitatively similar layout of condensates is obtained for all other combinations of coupling constants. As $a$ is increased, the conventional condensates are pushed to smaller values of $b_2$ by the expanding T1 and T2 phases.