Parity anomaly cancellation in a three-dimensional QED with single massless Dirac fermion

We study a three-dimensional non-compact QED with a single two-component massless fermion and two infinitely massive regulator fermions of half the charge using lattice overlap formalism. The parity anomaly is expected to cancel exactly between the massless and regulator fermions in the continuum, but this cancellation is inexact on lattice akin to lattice chiral gauge theories. We show non-perturbatively that parity-breaking terms vanish in the continuum limit at any finite volume. We present numerical evidences that the resulting parity-invariant theory spontaneously breaks parity in the infinite volume limit.