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.

• Received 6 June 2012

DOI:https://doi.org/10.1103/PhysRevD.86.065024