$CP^{N-1}$ Models at a Lifshitz Point

We consider $CP^{N-1}$ models in $d+1$ dimensions around Lifshitz fixed points with dynamical critical exponent $z$, in the large-N expansion. It is shown that these models are asymptotically free and dynamically generate a mass for the $CP^{N-1}$ fields for all $d=z$. We demonstrate that, for $z=d=2$, the initially nondynamical gauge field acquires kinetic terms in a way similar to usual $CP^{N-1}$ models in 1+1 dimensions. Lorentz invariance emerges generically in the low-energy electrodynamics, with a nontrivial dielectric constant given by the inverse mass gap and a magnetic permeability which has a logarithmic dependence on scale. At a special multicritical point, the low-energy electrodynamics also has $z=2$, and an essentially singular dependence of the effective action on $B=\epsilon_{ij}\partial_iA_j$.