Topological Order and Semions in a Strongly Correlated Quantum Spin Hall Insulator

We provide a self-consistent mean-field framework to study the effect of strong interactions in a quantum spin Hall insulator on the honeycomb lattice. We identify an exotic phase for large spin-orbit coupling and intermediate Hubbard interaction. This phase is gapped and does not break any symmetry. Instead, we find a fourfold topological degeneracy of the ground state on the torus and fractionalized excitations with semionic mutual braiding statistics. Moreover, we argue that it has gapless edge modes protected by time-reversal symmetry but a trivial $Z_2$ topological invariant. Finally, we discuss the experimental signatures of this exotic phase. Our work highlights the important theme that interesting phases arise in the regime of strong spin-orbit coupling and interactions.

Figure 1

(a) Phase diagram obtained in the $Z_2$-mean-field approximation. QSH is a quantum spin Hall insulator and VBS a valence-bond solid. The focus of the current Letter is the strongly correlated QSH* at intermediate $U$ and large spin-orbit coupling $t_2$. (b) Topological ground-state degeneracy in the QSH*: logarithm of the energy difference between solutions with and without a global $Z_2$ flux on a $L\times L$ torus for $U=26t$ and $t_2=t$.

Figure 2

Invisibility of the string in the QSH*. (a) The (unphysical) bond variables $g_{ij}$ in the presence of two $Z_2$ flux excitations (doublon-holon pair). Dash-dotted (blue) lines are negative. (b) The kinetic bond energy $g_{ij}{J}_{ij}$ for the same configuration as in (a) along with the deviation of the charge density from half filling (dark disks denote excess and bright deficit charge). The string is invisible in physical quantities. The parameters are $U=21t$, $t_1=0.2t$, and $t_2=0.7t$.

Figure 3

Spin-charge separation in the QSH*. Shown is the self-consistent local electron density $\rho$ and the square of the local spin density, $⟨\overrightarrow{S}{⟩}^2/S^2$, averaged over the Wigner-Seitz cell of the honeycomb lattice. The parameters are $U=21t$, $t_1=0.2t$, and $t_2=0.8t$, and a single hole has been doped into the half filled system.

Figure 4

Excitations in the QSH*. (a) The fermionic pseudoparticles carry an electronic charge and a spin as well as a $Z_2$ charge. (b) Spinon excitations carry a (generalized) spin but no charge and are bosons. (c) Doublons and holons have charge but no spin and are also bosons. The mutual braiding statistics is semionic.