Protected superconductivity at the boundaries of charge-density-wave domains

Solid Helium-4 may acquire superfluid characteristics due to the frustration of the solid phase at grain boundaries. I will discuss an analogous effect in systems with competition among charge-density-waves (CDWs) and superconductivity in the presence of disorder. The theoretical phase diagram will be used to interpret magnetotransport experiments in cuprates. At intermediate temperatures the magnetic field drives the system from a superconductor to a CDW as expected for a clean quantum critical point (QCP). Lowering the temperature the clean QCP is avoided and a filamentary phase appears analogous to ``glassy'' supersolid phenomena in Helium-4. Our results explain the occurence of characteristic "foots" in the phase diagram with two associated QCP critical points. I will discuss the role of doping and, time allowing, strongly underdoped systems.