Competition between spin density wave order and superconductivity in the underdoped cuprates

We describe the interplay between $d$-wave superconductivity and spin density wave (SDW) order in a theory of the hole-doped cuprates at hole densities below optimal doping. We describe quantum and thermal fluctuations in the orientation of the local SDW order, which lead to $d$-wave superconductivity. We also describe the back-action of the superconductivity on the SDW order, showing that SDW order is more stable in the metal. Finally, we propose a finite temperature crossover phase diagram for the cuprates. In the metallic state, these are controlled by a `hidden' quantum critical point near optimal doping involving the onset of SDW order in a metal. However, the onset of superconductivity results in a decrease in stability of the SDW order, and consequently the actual SDW quantum critical point appears at a significantly lower doping.