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.