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In a recent paper [V. Aji and C.M. Varma, Phys. Rev. Lett. {\bf 99}, 167003(2007) \cite{aji1}] we have shown that the dissipation driven quantum phasetransition of the 2D xy model represents a universality class where thecorrelations at criticality is local in space and power law in time. Here weprovide a detailed analysis of the model. The local criticality is broughtabout by the decoupling of infrared singularities in space and time. The formerleads to a Kosterlitz Thouless transition whereby the excitations of thetransverse component of the velocity field (vortices) unbind in space. Thelatter on the other hand leads to a transition among excitations (warps) in thelongitudinal component of the velocity field, which unbind in time. The quantumAshkin-Teller model, with which the observed loop order in the Cuprates isdescribed maps in the critical regime to the quantum xy model. We also discussother models which are expected to have similar properties.

Quantum-Criticality in Dissipative Quantum Two-Dimensional XY and Ashkin-Teller Models: Application to the Cuprates