Electric probe for the toric code phase in Kitaev materials through the hyperfine interaction

Abstract: The Kitaev model is a remarkable spin model with gapped and gapless spin liquid phases, which are potentially realized in iridates and $\alpha$-RuCl$3$. In the recent experiment of $\alpha$-RuCl$_3$, the signature of a nematic transition to the gapped toric code phase, which breaks the $C_3$ symmetry of the system, has been observed through the angle dependence of the heat capacity. We here propose a mechanism by which the nematic transition can be detected electrically. This is seemingly impossible because $J\textrm{eff}=1/2$ spins do not have an electric quadrupole moment (EQM). However, in the second-order perturbation the virtual state with a nonzero EQM appears, which makes the nematic order parameter detectable by nuclear magnetic resonance and M\"ossbauer spectroscopy. The purely magnetic origin of EQM is different from conventional electronic nematic phases, allowing the direct detection of the realization of Kitaev's toric error-correction code.