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Motivated by recent transport measurements on the candidate spin-liquid phaseof the organic triangular lattice insulator EtMe$_3$Sb[Pd(dmit)$_2$]$_2$, weperform a controlled calculation of the thermal conductivity at intermediatetemperatures in a spin liquid system where a spinon Fermi surface is coupled toa $U(1)$ gauge field. The present computation builds upon the double expansionapproach developed by Mross \emph{et al.} [Phys. Rev. B \textbf{82}, 045121(2010)] for small $\epsilon=z_b -2$ (where $z_b$ is the dynamical criticalexponent of the gauge field) and large number of fermionic species $N$. Usingthe so-called memory matrix formalism that most crucially does not assume theexistence of well-defined quasiparticles at low energies in the system, wecalculate the temperature dependence of the thermal conductivity $\kappa$ ofthis model due to non-critical Umklapp scattering of the spinons for a finite$N$ and small $\epsilon$. Then we discuss the physical implications of suchtheoretical result in connection with the experimental data available in theliterature.

Controlled calculation of the thermal conductivity for a spinon Fermi surface coupled to a $U(1)$ gauge field