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 $\pi$-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 nonlocal 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 $\widehat{z}$ 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 nonlocal.

Figure 1

Schematic of the model and the phase diagram: (a) single layer unit cell. Four different colors depict four types of bonds. There are two inequivalent plaquettes $p$ and $p^{\prime }$ in a unit cell. (b) Bilayer model with intralayer Kitaev parameters $K_{\nu }$ and an interlayer exchange, $J$. (c) The low-energy description of the model is a $\pi$-flux Hubbard model which exhibits a Mott transition at $U/t\sim 6$ (black). In terms of the original degrees of freedom, the Mott transition corresponds to a magnetic fragmentation transition where a local magnetic order coexists with a nonlocal topological order.

Figure 2

Illustration of the pseudospin and bond ($\eta -\rho$) configurations. (a) Effect of flux operator $W_{\nu p}$ on a state with fixed ${\rho }_{ij}^{\gamma }$ bonds. $W_{\nu p}$ changes the signs of the bonds marked in blue. It also changes the signs of ${\eta }^{x/y}$ pseudospins on the identity (dashed) bond. (b) String defect, with red lines indicating bonds with signs opposite to the background bonds, marked in black. The string shown here is created by operating with ${\tau }_{\nu i}^x$ on any orbital state along the sites marked with blue dots. When operating on the GS in Eq. (20), the $\tau$'s change the fluxes on the hashed plaquettes, since they ant-commute with $W_{\nu p/p^{\prime }}$. Consequently, this open string terminates with a pair of visons.