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We study the phase diagram of the Yao-Lee model with Kitaev-type spin-orbital interactions in the presence of Dzyaloshinskii-Moriya interactions and external magnetic fields. Unlike the Kitaev model, the Yao-Lee model can still be solved exactly under these perturbations due to the enlarged local Hilbert space. Through a variational analysis, we obtain a rich ground-state phase diagram that consists of a variety of vison crystals with periodic arrangements of background Z2 flux (i.e., visons). With an out-of-plane magnetic field, these phases have gapped bulk and chiral edge states, characterized by a Chern number ν and an associated chiral central charge c=ν/2 of edge states. We also find helical Majorana edge states that are protected by magnetic mirror symmetry. For the bilayer systems, we find that interlayer coupling can also stabilize new topological phases. Our results spotlight the tunability and the accompanying rich physics in exactly solvable spin-orbital generalizations of the Kitaev model. 

Motivated by the recent developments in moiré superlattices of van der Waals magnets and the desire to control the magnetic interactions of α-RuCl3, here we present a comprehensive theory of the long-range ordered magnetic phases of twisted bilayer α-RuCl3. Using a combination of first-principles calculations and atomistic simulations, we show that the stacking-dependent interlayer exchange gives rise to an array of magnetic phases that can be realized by controlling the twist angle. In particular, we discover a complex hexagonal domain structure in which multiple zigzag orders coexist. This multidomain order minimizes the interlayer energy while enduring the energy cost due to domain wall formation. Further, we show that quantum fluctuations can be enhanced across the phase transitions. Our results indicate that magnetic frustration due to stacking-dependent interlayer exchange in moiré superlattices can be exploited to tune quantum fluctuations and the magnetic ground state of α-RuCl3. 

Abstract We determine the phase diagram of a bilayer, Yao-Lee spin-orbital model with inter-layer interactions (J), for several stackings and moiré superlattices. For AA stacking, a gapped$${{\mathbb{Z}}}_{2}$$ $Z_2$quantum spin liquid phase emerges at a finiteJ_c. We show that this phase survives in the well-controlled large-Jlimit, where an isotropic honeycomb toric code emerges. For moiré superlattices, a finite-qinter-layer hybridization is stabilized. This connects inequivalent Dirac points, effectively ‘untwisting’ the system. Our study thus provides insight into the spin-liquid phases of bilayer spin-orbital Kitaev materials.