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

Abstract

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 $\ddot{I}€$-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 "Néel order" features a non-local string order parameter for an in-plane Néel component, in addition to an anisotropic local order parameter. The associated quantum order is characterized by an emergent ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ gauge field when the Néel vector is along the $\hat{z}$ direction, and a ${\mathbb{Z}}_2$ 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 ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ phase is fractionalized and non-local.

26 pages, 7 figures