Non-Abelian chiral spin liquid on a simple non-Archimedean lattice

We extend the scope of Kitaev spin liquids to non-Archimedean lattices. For the pentaheptite lattice, which results from the proliferation of Stone-Wales defects on the honeycomb lattice, we find an exactly solvable non-Abelian chiral spin liquid with spontaneous time-reversal symmetry breaking due to lattice loops of odd length. Our findings call for potential extensions of exact results for Kitaev models which are based on reflection positivity, which is not fulfilled by the pentaheptite lattice. We further elaborate on potential realizations of our chiral spin-liquid proposal in strained $\alpha \text{−}{\mathrm{RuCl}}_3$.

Figure 1

(a) Pentaheptite lattice with a unit cell highlighted in light yellow and the lattice vectors ${\mathit{e}}_1$ and ${\mathit{e}}_2$. Bond colors highlight the type of spin-spin coupling across a bond ${\sigma }_j^{\alpha }{\sigma }_k^{\alpha }$, $\alpha =x,y,z$ (violet for $z$, orange for $y$, and green for $x$). (b), (c) Site labels for the definition of plaquette operators in (2). (d) Energy per Ru atoms for $\alpha \text{−}{\mathrm{RuCl}}_3$ under strain with respect to the energy of the honeycomb configuration at $\theta ={120}^{\circ }$. In red for a honeycomb lattice, in blue for a Stone-Wales defect. The strain changes the angle $\theta$ between the lattice vectors and the Stone-Wales structure is preferred at $\theta <{105}^{\circ }$ and $\theta >{135}^{\circ }$.

Figure 2

(a) Phase diagram of the vortex-free sector of Hamiltonian (3). Phases $A_1$ and $A_2$ are topologically equivalent and realize the Abelian ${\mathbb{Z}}_2$ topological order. Phase $B$ realizes the non-Abelian Ising topological order. (b) Flux configuration of lowest excited state for a system on the torus $L{\mathit{e}}_1\times L{\mathit{e}}_2$ with $L=2$. Colored plaquettes have flux $\pi$. (c) Energy cost per vortex for a pair of vortices as a function of their separation on a torus with $L=10$ and $J_x=J_y=J_z=J$. For the green curve, vortices are in the pentagons. Vortices in heptagons are shown in orange. For the violet curve, one vortex is in a heptagon and the other in a pentagon.

Figure 3

(a) Boundary spectrum for a ribbon with open boundary conditions along ${\mathit{e}}_1$ and periodic along ${\mathit{e}}_2$, with $J_x=J_y=J_z=J$. The ribbon is composed of 50 unit cells along ${\mathit{e}}_1$. The Chern numbers of the three gaps are $C=-1,+1,-1$ from bottom to top. Modes localized at one edge are plotted in green and those localized at the other edge in orange. The inset shows the edge terminations of the ribbon with respective colors. (b) Non-Abelian chiral spin liquid (phase $B$) arising from weakly coupled spin chains. The Kitaev pentaheptite model can be deformed to an array of spin chains with alternating ${\sigma }^x\text{−}{\sigma }^x$ and ${\sigma }^z\text{−}{\sigma }^z$ coupling that are weakly coupled via ${\sigma }^y\text{−}{\sigma }^y$ interactions (orange lines). The dashed lines represent the ${\sigma }^y\text{−}{\sigma }^y$ interactions for the coupled wire construction of the honeycomb lattice.