This was the first of two papers studying phase transitions between gapless \(\mathbb{Z}_2\) spin liquids and ordered phases on the square lattice. This was motivated by a series of numerical studies of the \(J_1 - J_2\) model on the square lattice, which indicated the existence of a gapless \(\mathbb{Z}_2\) spin liquid in a narrow parameter window between Neel antiferromagnetism and valence bond solid (VBS) order. There has been strong numerical evidence over the years for some form of non-magnetic state existing in this parameter regime, but the precise nature of the phase or phases has not been definitively settled - in particular, there have been differing conclusions reached on whether excitations are gapped or gapless.

We construct a critical theory describing phase transitions between this spin liquid and the nearby Neel and VBS phases. With the inclusion of the square lattice symmetries, there exist a large number of spin liquids classified by the manner in which these symmetries are realized on the fractionalized excitations. In order to fully specify our critical theory, we assume a particular spin liquid arising from a fermionic parton construction, which goes by the name Z2azz13 under Xiao-Gang Wen's PSG classification. This particular ansatz is used in variational wavefunction studies of the \(J_1 - J_2\) model and is found to have energies comparable to more unbiased studies, so we take this as our starting point.

Instabilities to Neel and VBS order are naturally captured by transitions into proximate spin liquid phases with larger gauge groups (U(1), SU(2)), which are then in turn unstable to these ordered phases through mechanisms like monopole proliferation and confinement. In particular, the \(\pi\)-flux phase, with an SU(2) gauge group, and a staggered flux phase with a U(1) gauge group, can both be accessed from the Z2azz13 phase via Higgs transitions and both are unstable to Neel and VBS order. Consideration of the \(\pi\)-flux phase is especially natural due to its dual description as the noncompact \(\text{CP}^1\) model describing a direct Neel/VBS transition.

Leveraging the PSG classification of these symmetries, we can write down an effective field theory describing the transition between our \(\mathbb{Z}_2\) spin liquid and the proximate unstable spin liquid phases. The \(\mathbb{Z}_2 \rightarrow \text{SU}(2)\) transition is studied in this paper. A surprising feature about this critical theory, studied to leading-order in a \(N_f^{-1}\) expansion with \(N_f\) the number of fermions, is that it possesses a set of subsystem symmetries, arising from the projective representation of translational and rotational symmetries. Regulating the divergences arising from zero modes of this subsystem symmetry leads to UV/IR mixing and correlation functions which decay as \( e^{-\ln^2(r)}\) rather than power law. This is a rather striking prediction, although we emphasize that this behavior is found to leading-order in the \(N_f^{-1}\) expansion - the stability of this at finite \(N_f\), and validity of the large-\(N_f\) expansion more generally, is a direction for future work.