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

Abstract

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 <i>N</i>_<i>f</i> = 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 <i>π</i>-flux per plaquette in the ℤ₂ center of SU(2). This theory has an emergent SO(5)_<i>f</i> global symmetry and is presumed to confine at low energies to the Néel state. At nonzero doping (or smaller Hubbard repulsion <i>U</i> at half-filling), we argue that confinement occurs via the Higgs condensation of bosonic chargons carrying fundamental SU(2) gauge charges also moving in <i>π</i> ℤ₂-flux. At half-filling, the low-energy theory of the Higgs sector has <i>N</i>_<i>b</i> = 2 relativistic bosons with a possible emergent SO(5)_<i>b</i> global symmetry describing rotations between a <i>d</i>-wave superconductor, period-2 charge stripes, and the time-reversal breaking "<i>d</i>-density wave" state. We propose a conformal SU(2) gauge theory with <i>N</i>_<i>f</i> = 2 fundamental fermions, <i>N</i>_<i>b</i> = 2 fundamental bosons, and a SO(5)_<i>f</i>×SO(5)_<i>b</i> global symmetry, which describes a deconfined quantum critical point between a confining state which breaks SO(5)_<i>f</i> and a confining state which breaks SO(5)_<i>b</i>. 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 <i>d</i>-wave superconductivity. A similar theory applies at nonzero doping and large <i>U</i>, with longer-range couplings of the chargons leading to charge order with longer periods.