Low-Energy Spin Dynamics of the Honeycomb Spin Liquid Beyond the Kitaev Limit

We investigate the generic features of the low energy dynamical spin structure factor of the Kitaev honeycomb quantum spin liquid perturbed away from its exact soluble limit by generic symmetry-allowed exchange couplings. We find that the spin gap persists in the Kitaev-Heisenberg model, but generally vanishes provided more generic symmetry-allowed interactions exist. We formulate the generic expansion of the spin operator in terms of fractionalized Majorana fermion operators according to the symmetry enriched topological order of the Kitaev spin liquid, described by its projective symmetry group. The dynamical spin structure factor displays power-law scaling bounded by Dirac cones in the vicinity of the $\mathrm{\Gamma }$, $K$, and $K^{\prime }$ points of the Brillouin zone, rather than the spin gap found for the exactly soluble point.

Figure 1

(a) The honeycomb lattice for the Kitaev model. $A$ and $B$ denote the sublattice index. A spin ${\sigma }_6^z$ (in green) on the ground state changes the 1-6 bond operator eigenvalue, creating fluxes in two plaquettes sharing that bond. The symmetries $T_1$, $T_2$, $C_6$, $\sigma$, and $h_x$ are displayed. The ($x$, $y$) coordinate system is shown on the upper right corner. (b) Spin operators schematically expanded in terms of Majorana fermion operators. Each site contains four Majorana modes (small dots), where the central one is $c_i$ and the surrounding three are $c_i^{\mu }$’s. Each black dot represents a corresponding Majorana operator present.

Figure 2

The spectral function along the high symmetry line at the isotropic point. The simplest form as listed in Eq. (6) is used for the calculation performed on a honeycomb lattice with $240\times 240$ unit cells. The dashed line marks the gap $\mathrm{\Delta }=0.262J_K$ of flux excitations, above which the leading contribution from the unperturbed Kitaev spin liquid will dominate.

Figure 3

The spectral function versus frequency at (a) the $\mathrm{\Gamma }$ point $\mathit{q}=0$ and (b) the $K^{\prime }$ point $\mathit{q}\sim 2{\mathit{q}}_0$. The cube root of $A_S(K^{\prime },\omega )$ is plotted in (b) to demonstrate the scaling behavior $A_S(K^{\prime },\omega )\sim {\omega }^3$. The dashed line marks the flux gap $\mathrm{\Delta }=0.262J_K$.