Deconfined criticality and a gapless $\mathbb{Z}_2$ spin liquid in the square lattice antiferromagnet

TL;DR

This paper develops theoretical models for quantum phase transitions in the square lattice antiferromagnet, proposing deconfined critical SU(2) and U(1) gauge theories that describe the emergence of a gapless $ ext{Z}_2$ spin liquid and related phases.

Contribution

It introduces novel deconfined critical gauge theories for transitions into a gapless $ ext{Z}_2$ spin liquid, unifying observed phases in the $J_1$-$J_2$ model.

Findings

Proposes a deconfined SU(2) gauge theory with a dangerously irrelevant coupling.

Introduces a deconfined U(1) gauge theory with no dangerously irrelevant coupling.

Unifies phases and critical points in an SU(2) gauge theory with Higgs fields and fermionic spinons.

Abstract

The theory for the vanishing of N\'eel order in the spin $S=1/2$ square lattice antiferromagnet has been the focus of attention for many decades. A consensus appears to have emerged in recent numerical studies on the antiferromagnet with first and second neighbor exchange interactions (the $J_1$-$J_2$ model): a gapless spin liquid is present for a narrow window of parameters between the vanishing of the N\'eel 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 $\mathbb{Z}_2$ spin liquid with massless Dirac spinon excitations; on the other side the critical point, the SU(2) spin liquid (the `$\pi$-flux' phase) is presumed to be unstable to confinement to the N\'eel 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 N\'eel and valence bond solid order parameters. We also propose a distinct deconfined critical U(1) gauge theory for a transition into the same gapless $\mathbb{Z}_2$ 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 $z \neq 1$, 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 $J_2/J_1$: N\'eel, gapless $\mathbb{Z}_2$ spin liquid, and valence bond solid.