Thermal conductivity in the mixed state of a superconductor at low magnetic fields

We evaluate accurate low-field/low-temperature asymptotics of the thermal conductivity perpendicular to the magnetic field for one-band and two-band s-wave superconductors using the Keldysh-Usadel formalism. We show that heat transport in this regime is limited by tunneling of quasiparticles between adjacent vortices across a number of local points and therefore a widely used approximation of averaging over the circular unit cell is not valid. In the single-band case, we obtain a parameter-free analytical solution which provides a theoretical lower limit for heat transport in the mixed state. In the two-band case, we show that the heat transport is controlled by the ratio of gaps and diffusion constants in different bands. The presence of a weaker second band strongly enhances the thermal conductivity at low fields.