Thermal Transport in the Kitaev Model

In conventional insulating magnets, heat is carried by magnons and phonons. In contrast, when the magnets harbor a quantum spin liquid state, emergent quasiparticles from the fractionalization of quantum spins can carry heat. Here, we investigate unconventional thermal transport yielded by such exotic carriers, in both longitudinal and transverse components, for the Kitaev model, whose ground state is exactly shown to be a quantum spin liquid with fractional excitations described as itinerant Majorana fermions and localized $Z_2$ fluxes. We find that the longitudinal thermal conductivity exhibits a single peak at a high temperature, while the nonzero frequency component has a peak at a low temperature, reflecting the spin fractionalization. On the other hand, we show that the transverse thermal conductivity is induced by the magnetic field in a wide temperature range up to the energy scale of the bare exchange coupling; while increasing temperature, the transverse response divided by temperature decreases from the quantized value expected for the topologically nontrivial ground state and shows nonmonotonic temperature dependence. These characteristic behaviors provide experimentally accessible evidence of fractional excitations in the proximity to the Kitaev quantum spin liquid.

Figure 1

(a) Longitudinal thermal conductivity, ${\kappa }^{xx}={\lim }_{\omega \to 0}{\kappa }^{xx}(\omega )$ and (b) ${\kappa }^{xx}/T$ as functions of $T$. In (a), we also plot the specific heat $C_v$ for $L=20$. $T^{*}$ and $T^{**}$ are two crossover temperatures, determined from the broad peaks in $C_v$. (c) Contour map of ${\kappa }^{xx}(\omega )$ on the $T\text{−}\omega$ plane calculated for the $L=20$ cluster. (d) Integrated intensity $I^{xx}={\int }_0^{\infty }{\kappa }^{xx}(\omega )d\omega$. The inset of (d) represents the honeycomb lattice on the $xy$ plane where the Kitaev model is defined in an applied magnetic field $h$ along the $z$ direction [Eq. (1)]. The different bond colors illustrate three different types of bonds in the Kitaev model.

Figure 2

(a) ${\kappa }^{xx}$, (b) ${\kappa }^{xx}/T$, (c) ${\kappa }^{xy}$, and (d) ${\kappa }^{xy}/T$ in the magnetic field $\tilde{h}$. The data are calculated for $L=8$, 10, and 12, but we show only the data for $L=12$ as they are indistinguishable within the statistical errors. In (d), we also plot ${\kappa }^{xy}/T$ calculated for the flux-free (dashed-dotted curve) and random flux (solid curve) cases at $\tilde{h}/J=0.048$.

Figure 3

$\tilde{h}$ dependences of (a) ${\kappa }^{xx}/T$ and (b) ${\kappa }^{xy}/T$ at several $T$ for the $L=12$ cluster.