Disorder, Low-Energy Excitations, and Topology in the Kitaev Spin Liquid

The Kitaev model is a fascinating example of an exactly solvable model displaying a spin-liquid ground state in two dimensions. However, deviations from the original Kitaev model are expected to appear in real materials. In this Letter, we investigate the fate of Kitaev’s spin liquid in the presence of disorder—bond defects or vacancies—for an extended version of the model. Considering static flux backgrounds, we observe a power-law divergence in the low-energy limit of the density of states with a nonuniversal exponent. We link this power-law distribution of energy scales to weakly coupled droplets inside the bulk, in an uncanny similarity to the Griffiths phase often present in the vicinity of disordered quantum phase transitions. If time-reversal symmetry is broken, we find that power-law singularities are tied to the destruction of the topological phase of the Kitaev model in the presence of bond disorder alone. However, there is a transition from this topologically trivial phase with power-law singularities to a topologically nontrivial one for weak to moderate site dilution. Therefore, diluted Kitaev materials are potential candidates to host Kitaev’s chiral spin-liquid phase.

Figure 1

Extended Kitaev model in Eq. (2) in the presence of bond disorder. (a) Links $x$, $y$, and $z$ in the honeycomb lattice and the hopping between second ($\kappa$) and third neighbors $(K_3^{\prime })$. (b) DOS as function of the energy for $\delta K=0.8$. Inset: log-log plot showing the power-law divergence at low $E$. (c) Bott index as a function of disorder. (d) Static uniform spin susceptibility as a function of the temperature in a log-log plot for $\delta K=0.8$. In (b)–(d) full (dashed) curves correspond to random (0) flux. We consider the same parameters in (b) and (d). The dot dashed curves in (b) and (d) are power-law fits, shifted with respect to the original curves, with $\alpha =0.455(5),0.222(3),0.209(4)$. We considered $L=30$ and $3\times {10}^3$ realizations of disorder.

Figure 2

Diluted extended Kitaev model, Eq. (2). (a) Bound-state flux configuration. The shaded $l=12$ plaquettes show the binding of a flux by each vacancy. (b) DOS as function of the energy for $\kappa =0.1K$ and $K_3^{\prime }=0$. Inset: log-log plot showing the power-law divergence at low $E$ for $x\ge 0.06$. The dot-dashed curves are power-law fits, shifted with respect to the original curves, with $\alpha =0.496(1),0.445(3)$. (c) Bott index as a function of the dilution for the bound flux (0 flux) in the upper (lower) panel. (d) Log-log plot of $\chi \times T$ in the bound-flux sector for $x=0.04$. We use the same parameters as in Fig. 1.