Deconfined criticality and gapless Z2 spin liquids in the square lattice antiferromagnet

The theory for the vanishing of Néel order in the spin S=1/2 square lattice antiferromagnet has been the focus of attention for many decades. Recent numerical studies on the antiferromagnet with first and second neighbor exchange interactions (the J1−J2 model) suggest the presence of a gapless Z2 spin liquid for a narrow window of parameters between the vanishing of the Néel order and the onset of a gapped valence bond solid state. We propose a deconfined critical SU(2) gauge theory for a transition into a stable Z2 spin liquid with massless Dirac spinon excitations; on the other side of the critical point, the SU(2) spin liquid (the 'π-flux' phase) is presumed to be unstable to confinement to the Néel phase. We identify a dangerously irrelevant coupling in the critical SU(2) gauge theory, which contributes a logarithm-squared renormalization. This critical theory is also not Lorentz invariant and weakly breaks the SO(5) symmetry which rotates between the Néel and valence bond solid order parameters. We also propose a distinct deconfined critical U(1) gauge theory for a transition into the same gapless Z2 spin liquid; on the other side of the critical point, the U(1) spin liquid (the 'staggered flux' phase) is presumed to be unstable to confinement to the valence bond solid. This critical theory has no dangerously irrelevant coupling, dynamic critical exponent z≠1, and no SO(5) symmetry. All of these phases and critical points are unified in a SU(2) gauge theory with Higgs fields and fermionic spinons which can naturally realize the observed sequence of phases with increasing J2/J1: Néel, gapless Z2 spin liquid, and valence bond solid.