Equilibrium axial current due to a static localized spin in Weyl semimetals

We theoretically study the equilibrium axial current j^eq₅, which is the difference between the charge current of left-handed and right-handed helicity of the Weyl fermions, in the junction of a magnetic insulator attached to Weyl semimetals with time-reversal breaking and inversion symmetry. The charge current of each helicity j^eq_γ=± is induced by the localized spin S of the magnetic insulator. j^eq_γ=± is proportional to ∇ × S and is independent of the sign of γ. As a result, the difference between j^eq_γ=+ and j^eq_γ=− are canceled out, and j^eq₅ is zero