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 E· B term with a constant coefficient θ. 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 θ term can significantly affect the interaction of fermions and vortices. We also perform a dimensional reduction to show a θ dependent flux attachment to the itinerant fermions.
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Kantha et al. (2024) studied this question.
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