The underdoped cuprates as fractionalized Fermi liquids: transition to superconductivity

We model the underdoped cuprates using fermions moving in a background with local antiferromagnetic order. The antiferromagnetic order fluctuates in orientation, but not in magnitude, so that there is no long-range antiferromagnetism, but a ‘topological’ order survives. The normal state is described as a fractionalized Fermi liquid (FL*), with electron-like quasiparticles coupled to the fractionalized excitations of the fluctuating antiferromagnet. The electronic quasiparticles reside near pocket Fermi surfaces enclosing total area x (the dopant density), centered away from the magnetic Brillouin zone boundary. The violation of the conventional Luttinger theorem is linked to a ‘species doubling’ of these quasiparticles. We describe phenomenological theories of the pairing of these quasiparticles, and show that a large class of mean-field theories generically displays a nodal-anti-nodal `dichotomy’: the interplay of local antiferromagnetism and pairing leads to a small gap near the nodes of the d-wave pairing along the Brillouin zone diagonals, and a large gap in the anti-nodal region.