Quantum Valley Hall Effect in Massive Dirac Systems Coupled to a Scalar Field

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We use Pseudo Quantum Electrodynamics to study massive (2+1)D Dirac systems interacting electromagnetically via a U(1) gauge field in (3+1)D. It was recently found in Ref. [1], that an interaction-induced Quantum Hall Effect (QHE) and Quantum Valley Hall Effect (QVHE) occur in these systems, when considering a two-component fermion representation. Here, we study the corrections to these effects when coupling the fermions to a (2+1)D massive scalar field via a quartic interaction. We find no correction to the QHE and a non-universal correction to the QVHE, which depends on the ratio of the fermion and scalar-field masses.