A model of d-wave superconductivity, antiferromagnetism, and charge order on the square lattice

We describe the confining instabilities of a proposed quantum spin liquid underlying the pseudogap metal state of the hole-doped cuprates. The spin liquid can be described by a SU(2) gauge theory of N(f) = 2 massless Dirac fermions carrying fundamental gauge charges—this is the low-energy theory of a mean-field state of fermionic spinons moving on the square lattice with π-flux per plaquette in the ℤ(2) center of SU(2). This theory has an emergent SO(5)(f) global symmetry and is presumed to confine at low energies to the Néel state. At nonzero doping (or smaller Hubbard repulsion U at half-filling), we argue that confinement occurs via the Higgs condensation of bosonic chargons carrying fundamental SU(2) gauge charges also moving in π ℤ(2)-flux. At half-filling, the low-energy theory of the Higgs sector has N(b) = 2 relativistic bosons with a possible emergent SO(5)(b) global symmetry describing rotations between a d-wave superconductor, period-2 charge stripes, and the time-reversal breaking “d-density wave” state. We propose a conformal SU(2) gauge theory with N(f) = 2 fundamental fermions, N(b) = 2 fundamental bosons, and a SO(5)(f)×SO(5)(b) global symmetry, which describes a deconfined quantum critical point between a confining state which breaks SO(5)(f) and a confining state which breaks SO(5)(b). The pattern of symmetry breaking within both SO(5)s is determined by terms likely irrelevant at the critical point, which can be chosen to obtain a transition between Néel order and d-wave superconductivity. A similar theory applies at nonzero doping and large U, with longer-range couplings of the chargons leading to charge order with longer periods.