Spin supercurrent in the canted antiferromagnetic phase

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The spin and layer (pseudospin) degrees of freedom are entangled coherently in the canted antiferromagnetic phase of the bilayer quantum Hall system at the filling factor $\nu =2$. There emerges a complex Goldstone mode describing such a combined degree of freedom. In the zero tunneling-interaction limit ($\Delta_{\text{SAS}}\rightarrow 0$), its phase field provokes a supercurrent carrying both spin and charge within each layer. The Hall resistance is predicted to become anomalous precisely as in the $\nu =1$ bilayer system in the counterflow and drag experiments. Furthermore, it is shown that the total current flowing in the bilayer system is a supercurrent carrying solely spins in the counterflow geometry. It is intriguing that all these phenomena occur only in imbalanced bilayer systems.