Skyrmion quantum spin Hall effect

The quantum spin Hall effect is conventionally thought to require a strong spin-orbit coupling, producing an effective spin-dependent magnetic field. However, spin currents can also be present without transport of spins, for example, in spin-waves or skyrmions. In this paper, we show that topological skyrmionic spin textures can be used to realize a quantum spin Hall effect. From basic arguments relating to the single-valuedness of the wave function, we deduce that loop integrals of the derivative of the Hamiltonian must have a spectrum that is integer multiples of $2\pi$. By relating this to the spin current, we form a new quantity called the quantized spin current which obeys a precise quantization rule. This allows us to derive a quantum spin Hall effect, which we illustrate with an example of a spin-1 Bose-Einstein condensate.

Figure 1

Two types of spin current: (a) spin current due to spatial displacement of spin momenta (large arrows indicate movement) and (b) spin current due to spin waves (spins are fixed in space). Example of a spin texture in two dimensions: (c) the reference state (“blank canvas”) $|\mathrm{\Psi }\rangle ={\otimes }_{\mathit{x}}|{\psi }_0\rangle$ and (d) the spin texture parametrized by the generator $G\left(\mathit{x}\right)$.

Figure 2

Examples of the quantum spin Hall effect using spin textures. The current distribution in a spin-1 BEC for (a) the conventional current ${\mathit{j}}^{\left(0\right)}\left(\mathit{x}\right)$, (b) the first-order spin current ${\mathit{j}}^{\left(1\right)}\left(\mathit{x}\right)$, and (c) the second-order spin current ${\mathit{j}}^{\left(2\right)}\left(\mathit{x}\right)$. The chosen spin texture is $f\left(\mathit{x}\right)=m\theta$ and $g\left(\mathit{x}\right)=m^{\prime }\theta n\left(\mathit{x}\right)$, with $m=1$, $m^{\prime }=-2$, $\mathit{u}=(1,1,1)/\sqrt{3}$, and $n\left(\mathit{x}\right)=1+\gamma {\sum }_i\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{|\mathit{x}-{\mathit{x}}_i{|}^2}{2{\sigma }^2}}$ is the noise function with $\gamma =0.5$ and $\sigma =1$. The noise centers are at $(-2,0)$, (3,2), and $(-3,-3)$. (d) The $l\mathrm{th}$-order integrated spin current $J_x^{\left(l\right)}=\int j_x^{\left(l\right)}\left(\mathit{x}\right)dxdy$ versus the asymmetry parameter $A_y$. (e) The total quantized spin current $J_{\text{Q}}^x$ versus the asymmetry parameter $A_y$ for various spin textures for the parameters as marked. (f) Error in the current due to the inclusion of noise for the case $m=1$ and $m^{\prime }=2$ for the noise strengths as marked. The Gaussian density profile $\rho \left(\mathit{x}\right)=e^{-y^2/\left(2{\sigma }_y^2\right)}$ is assumed with ${\sigma }_y=100$.