  In this paper we critically reconsider the Casimir repulsion between surfaces
that carry the Chern-Simons interaction (corresponding to the Hall type
conductivity). We present a derivation of the Lifshitz formula valid for
arbitrary planar geometries and discuss its properties. This analysis allows us
to resolve some contradictions in the previous literature. We compute the
Casimir energy for two surfaces that have constant longitudinal and Hall
conductivities. The repulsion is possible only if both surfaces have Hall
conductivities of the same sign. However, there is a critical value of the
longitudinal conductivity above which the repulsion disappears. We also
consider a model where both parity odd and parity even terms in the
conductivity are produced by the polarization tensor of surface modes. In
contrast to the previous publications L. Chen and S.-L. Wan, Phys. Rev. B84,
075149 (2011); B85, 115102 (2012), we include the parity anomaly term. This
term ensures that the conductivities vanish for infinitely massive surface
modes. We find that at least for a single mode regardless of the sign and value
of its mass, there is no Casimir repulsion.

Mesoscale and Nanoscale Physics;High Energy Physics - Theory;Quantum Physics