Transport in honeycomb lattice with random $\pi$-fluxes: implications for low-temperature thermal transport in the Kitaev spin liquids

Motivated by the thermal transport problem in the Kitaev spin liquids, we consider a nearest-neighbor tight-binding model on the honeycomb lattice in the presence of random uncorrelated $\pi$-fluxes. We employ different numerical methods to study its transport properties near half-filling. The zero-temperature DC conductivity away from the Dirac point is found to be quadratic in Fermi momentum and inversely proportional to the flux density. Localization due to the random $\pi$-fluxes is observed and the localization length is extracted. Our results imply that, for realistic system size, the thermal conductivity of a pure Kitaev spin liquid diverges as ${\kappa }_{\text{K}}\sim T^3{e}^{{\mathrm{\Delta }}_v\mathrm{/}{k}_{B}T}$ when $k_{B}T\ll {\mathrm{\Delta }}_v$, and suggest the possible occurrence of strong Majorana localization ${\kappa }_{\text{K}}\mathrm{/}T\ll k_B^2\mathrm{/}2\pi \mathrm{\hslash }$ when $k_{B}T\sim {\mathrm{\Delta }}_v$, where ${\mathrm{\Delta }}_v$ is the vison gap.