Effects of dissipation on quantum phase transitions

We discuss the effect of dissipation on quantum phase transitions. In particular, we concentrate on the superconductor to insulator and quantum Hall to insulator transitions. By invoking a phenomenological parameter $\alpha$ to describe the coupling of the system to a continuum of degrees of freedom representing the dissipative bath, we obtain phase diagrams for the quantum Hall and superconductor-insulator problems. Our main result is that, in two dimensions, the metallic phases observed in finite magnetic fields (possibly also strictly zero field) are adiabatically deformable from one to the other. This is plausible, as there is no broken symmetry that differentiates them.