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Quantum fluctuations of the N\'eel state of the square latticeantiferromagnet are usually described by a $\mathbb{CP}^1$ theory of bosonicspinons coupled to a U(1) gauge field, and with a global SU(2) spin rotationsymmetry. Such a theory also has a confining phase with valence bond solid(VBS) order, and upon including spin-singlet charge 2 Higgs fields, deconfinedphases with $\mathbb{Z}_2$ topological order possibly intertwined with discretebroken global symmetries. We present dual theories of the same phases startingfrom a mean-field theory of fermionic spinons moving in $\pi$-flux in eachsquare lattice plaquette. Fluctuations about this $\pi$-flux state aredescribed by 2+1 dimensional quantum chromodynamics (QCD$_3$) with a SU(2)gauge group and $N_f=2$ flavors of massless Dirac fermions. It has recentlybeen argued by Wang et al. (arXiv:1703.02426) that this QCD$_3$ theorydescribes the N\'eel-VBS quantum phase transition. We introduce adjoint Higgsfields in QCD$_3$, and obtain fermionic dual descriptions of the phases with$\mathbb{Z}_2$ topological order obtained earlier using the bosonic$\mathbb{CP}^1$ theory. We also present a fermionic spinon derivation of themonopole Berry phases in the U(1) gauge theory of the VBS state. The globalphase diagram of these phases contains multi-critical points, and our resultsimply new boson-fermion dualities between critical gauge theories of thesepoints.

Fermionic spinon theory of square lattice spin liquids near the N\'eel state