Thermodynamics in topological Josephson junctions

We study the thermodynamic properties of topological Josephson junctions using a quantum spin Hall (QSH) insulator-based junction as an example. In particular, we propose that phase-dependent measurements of the heat capacity offer an alternative to Josephson-current measurements to demonstrate key topological features. Even in an equilibrium situation, where the fermion parity is not conserved, the heat capacity exhibits a pronounced double peak in its phase dependence as a signature of the protected zero-energy crossing in the Andreev spectrum. This double-peak feature is robust against changes of the tunneling barrier and thus allows one to distinguish between topological and trivial junctions. At short time scales fermion parity is conserved and the heat capacity is $4\pi$-periodic in the superconducting phase difference. We propose a dispersive setup coupling the Josephson junction to a tank LC circuit to measure the heat capacity of the QSH-based Josephson junction sufficiently fast to detect the $4\pi$-periodicity. Although explicitly calculated for a short QSH-based Josephson junction, our results are also applicable to long as well as nanowire-based topological Josephson junctions.