  This work is dedicated to the study of both large-$N$ and perturbative
quantum behaviors of Lifshitz nonlinear sigma models with dynamical critical
exponent $z=2$ in 2+1 dimensions. We discuss renormalization and
renormalization group aspects with emphasis on the possibility of emergence of
Lorentz invariance at low energies. Contrarily to the perturbative expansion,
where in general the Lorentz symmetry restoration is delicate and may depend on
stringent fine-tuning, our results provide a more favorable scenario in the
large-$N$ framework. We also consider supersymmetric extension in this
nonrelativistic situation.
