QED3theory of pairing pseudogap in cuprates: Fromd-wave superconductor to antiferromagnet via an algebraic Fermi liquid

High-$T_c$ cuprates differ from conventional superconductors in three crucial aspects: the superconducting state descends from a strongly correlated Mott-Hubbard insulator (as opposed to a Fermi liquid), the order parameter exhibits d-wave symmetry, and fluctuations play an all important role. We formulate an effective theory of underdoped cuprates within the pseudogap state by taking advantage of these unusual features. In particular, we introduce a concept of “pairing protectorate” and we seek to describe various phases within this protectorate by phase disordering a d-wave superconductor. The elementary excitations of the protectorate are the Bogoliubov–de Gennes quasiparticles and topological defects in the phase of the pairing field—vortices and antivortices—which appear as quantum and thermal fluctuations. The effective low-energy theory of these elementary excitations is shown to be, apart from intrinsic anisotropy, equivalent to the quantum electrodynamics in (2+1) spacetime dimensions $\left({\mathrm{QED}}_3\right).\mathrm{}$ A detailed derivation of this ${\mathrm{QED}}_3$ theory is given and some of its main physical consequences are inferred for the pseudogap state. As the superconducting order is destroyed by underdoping two possible outcomes emerge: (i) the system can go into a symmetric normal state characterized as an “algebraic Fermi liquid” (AFL) before developing antiferromagnetic (AF) order or (ii) a direct transition into the insulating AF state can occur. In both cases the AF order arises spontaneously through an intrinsic “chiral” instability of ${\mathrm{QED}}_3/\mathrm{AFL}.$ Here we focus on the properties of the AFL and propose that inside the pairing protectorate it assumes the role reminiscent of that played by the Fermi liquid theory in conventional metals. We construct a gauge-invariant electron propagator of the AFL and show that within the $1/N$ expansion it has a non-Fermi-liquid, Luttinger-like form with positive anomalous dimension ${\eta }^{\prime }=16/3\mathrm{}{\pi }^{2}N,$ where N denotes the number of pairs of nodes. We investigate the effects of Dirac anisotropy by perturbative renormalization group analysis and find that the theory flows into an isotropic fixed point. We therefore conclude that, at long length scales, the AFL is stable against anisotropy.