Thermal transport across a Josephson junction in a dissipative environment

At zero temperature, a Josephson junction coupled to an ohmic environment displays a quantum phase transition between superconducting and insulating phases, depending on whether the resistance of the environment is below or above the resistance quantum. This so-called Schmid transition, representative of the effect in a broad class of quantum impurity problems, turns into a crossover at finite temperatures. We determine the conditions under which the temperature dependence of the thermal conductance, which characterizes heat flow from a hot to cold resistor across the Josephson junction, displays a universal scaling characteristic of the Schmid transition. We also discuss conditions for heat rectification to happen in the circuit. Our work can serve as a guide for identifying signatures of the Schmid transition in heat transport experiments.