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

Early studies proposed a connection between cuprate superconductivity and fractionalized spin liquid states. But the low temperature phase diagram is dominated by states without fractionalization, with a competition between superconductivity and charge-ordered states which break translational symmetry. Our theory uncovers novel features associated with a particular spin-liquid presumed to underlie the pseudogap metal, and shows that it has multiple nearly-degenerate instabilities to confinement of fractionalized excitations, leading to antiferromagnetism, $d$-wave superconductivity, and/or charge order. Our theory provides routes to resolving a number of open puzzles on the cuprate phase diagram. The spin liquid is described by a SU(2) gauge theory of $N_f=2$ massless fundamental Dirac fermions, has an emergent SO(5)$_f$ global symmetry, and is presumed to confine at low energies to the N\'eel state. At non-zero doping (or smaller Hubbard repulsion at half-filling) we argue that confinement occurs via the Higgs condensation of bosonic chargons carrying fundamental SU(2) gauge charges moving in $\pi$ flux. At half-filling, the low energy 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 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 N\'eel order and $d$-wave superconductivity. A similar theory applies at non-zero doping and large $U$, with longer-range couplings of the chargons leading to charge order with longer periods.