From QED₃ to Self-Dual Multicriticality in the Fradkin-Shenker Model

We consider the Fradkin-Shenker {mathbb Z}₂ gauge-Higgs lattice model in 2+1 dimensions, i.e. the toric code deformed by an in-plane magnetic field. Its phase diagram contains a multicritical CFT with gapless, mutually non-local electric and magnetic particles, exchanged by a {mathbb Z}₂^{mathsf{D}} self-duality symmetry. We introduce a staggered generalization of the model in which these particles carry global U(1)_e and U(1)_m charges, respectively, and we propose a continuum QFT description in terms of QED₃ with N_f = 2 Dirac fermion flavors and a charge-two Higgs field with Yukawa couplings. The conjectured phase diagram harbors a multicritical CFT with (O(2)_e times O(2)_m)rtimesmathbb{Z}₂^mathsf{D} symmetry, some of which is emergent in the QFT description. We compute the scaling dimensions of some operators using a large-N_f expansion and find agreement with the emergent selection rules. The staggered model admits a deformation to the original Fradkin-Shenker model, which maps to unit-charge monopole operators in Higgs-Yukawa-QED₃ that break the U(1)_e times U(1)_m symmetry. We show explicitly that this deformation reproduces all features of the Fradkin-Shenker phase diagram. Finally, we propose a multicritical duality between Higgs-Yukawa-QED₃ and the easy-plane mathbb{ CP}¹ model i.e. two-flavor scalar QED$₃$ with a suitable potential, which describes spin-1/2 anti-ferromagnets on a square lattice. This duality implies a first-order line of Néel-VBS transitions ending in a deconfined quantum multicritical point, described by the same O(2)_e times O(2)_m symmetric CFT that arises in the staggered Fradkin-Shenker model, which separates it from a gapped {mathbb Z}₂ spin liquid phase.