Magnetic fragmentation and fractionalized Goldstone modes in a bilayer quantum spin liquid

We study the phase diagram of a bilayer quantum spin liquid model with Kitaev-type interactions on a square lattice. We show that the low energy limit is described by a π-flux Hubbard model with an enhanced SO(4) symmetry. The antiferromagnetic Mott transition of the Hubbard model signals a magnetic fragmentation transition for the spin and orbital degrees of freedom of the bilayer. The fragmented “N´eel order” features a non-local string order parameter for an in-plane N´eel component, in addition to an anisotropic local order parameter. The associated quantum order is characterized by an emergent Z₂ × Z₂ gauge field when the N´eel vector is along the z-direction, and a Z₂ gauge field otherwise. We underpin these results with a perturbative calculation, which is consistent with the field theory analysis. We conclude with a discussion on the low energy collective excitations of these phases and show that the Goldstone boson of the Z₂×Z₂ phase is fractionalized and non-local.