  We construct fermionic Lagrangians with anisotropic scaling z=2, the natural
counterpart of the usual z=2 Lifshitz field theories for scalar fields. We
analyze the issue of chiral symmetry, construct the Noether axial currents and
discuss the chiral anomaly giving explicit results for two-dimensional case. We
also exploit the connection between detailed balance and the dynamics of
Lifshitz theories to find different z=2 fermionic Lagrangians and construct
their supersymmetric extensions.
