Chiral invariance and lattice fermions with minimal doubling

A few years ago some attention has been given to a fermionic action on the lattice, with a Wilson-like term which is chirally invariant but breaks the hypercubic space-time lattice symmetry. This action describes two Dirac fields in the continuum limit, provided the coefficient $\lambda$ of the Wilson-like term satisfies $\lambda > \frac{1}{2}$. In this letter it is shown that for $\frac{1}{2} < \lambda \leq 1$ the theory is link-reflection positive. The propagator has the expected real energy poles. Modulo a phase shift on the fermions, the only relevant terms which can be added to the action respecting its symmetries have dimension $4$.