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Weak localization corrections to the thermal conductivity in $s$-wave superconductors

We study the thermal conductivity in disordered $s$-wave superconductors. Expanding on previous works for normal metals, we develop a formalism that tackles particle diffusion as well as the weak localization (WL) and weak anti-localization (WAL) effects. Using a Green's functions diagrammatic technique, which takes into account the superconducting nature of the system by working in Nambu space, we identify the system's low-energy modes, the diffuson and the Cooperon. The time scales that characterize the diffusive regime are energy dependent; this is in contrast with the the normal state, where the relevant time scale is the mean free time $τ_e$, independent of energy. The energy dependence introduces a novel energy scale $\varepsilon_*$, which in disordered superconductors ($τ_e Δ\ll 1$, with $Δ$ the gap) is given by $\varepsilon_* = \sqrt{Δ/τ_e}$. From the diffusive behavior of the low-energy modes, we obtain the WL correction to the thermal conductivity. We give explicitly expressions in two dimensions. We determine the regimes in which the correction depends explicitly on $\varepsilon_*$ and propose an optimal regime to verify our results in an experiment.