Fermion-Vortex Interactions in Axion Electrodynamics

A relativistic action for scalar condensate-fermion mixture is considered where both the scalar boson and the fermion fields are coupled to a $U(1)$ gauge field. The dynamics of the gauge field is governed by a linear combination of the Maxwell term, and the Lorentz invariant $\mathbf{E}\cdot \mathbf{B}$ term with a constant coefficient $\theta$. We obtain an effective action describing an emergent fermion-fermion interaction and fermion-vortex tube interaction by using the particle-string duality, and find that the $\theta$ term can significantly affect the interaction of fermions and vortices. We also perform a dimensional reduction to show a $\theta$ dependent flux attachment to the itinerant fermions.