  The role of magnetic and electric perturbations to the quantum Lifshitz model
in 2+1 dimensions are examined in this paper. The quantum Lifshitz model is an
effective field theory for quantum multicritical systems, that include
generalized 2D quantum dimer models in bipartite lattices and their
generalizations. It describes a class of quantum phase transitions between
ordered and topological phases in 2+1 dimensions. Magnetic perturbations break
the dimer conservation law. Electric excitations, whose condensation lead to
ordered phases, have been studied extensively both in the classical 3D model
and in the quantum 2D model. However, the role of magnetic vortex excitations
whose condensation drive these systems into a $\mathbb{Z}_2$ topological phase
has been largely ignored. To study the interplay of both excitations, we
perform a perturbative renormalization group study to one loop order and study
the stability of the theory away from quantum multi criticality. This is done
by generalizing the operator product expansion to anisotropic models. The
relation with recent classical Monte Carlo simulations in Rokhsar-Kivelson wave
functions will be discussed.
