Quantum criticality on a compressible lattice

As an example of quantum criticality on a compressible lattice we study the Lorentz invariant Φ4 theory with an N-component field Φ, where strain couples to the square of the order parameter. In three spatial dimensions this coupling as well as the self-interaction of the Φ field are both marginal on the tree-level. We compute the one-loop renormalization group equations treating the Φ field as well as the phonons on the same footing. We find that the velocities of the Φ field as well as of the phonons are renormalized yielding an effective dynamical exponent z > 1. The renormalization group flow is found
to depend on the number of components N. Whereas we find run-away flow for N < 4 a new fixed-point emerges for N >= 4. We discuss the relation to known results for classical criticality. Our findings are directly relevant to insulating quantum critical antiferromagnets.