Phase-dependent charge and heat current in thermally biased short Josephson junctions formed at helical edge states

We explore the phase-dependent charge and heat current in the short Josephson junctions with two normal metal regions attached at opposite ends, formed at helical edge states of two-dimensional topological insulators (TIs). For all finite phases, an asymmetry appears around the zero energy in the transmission spectra except for $\phi\!=\!n\phi_0$, where $n$ is a half-integer and $\phi_0$ ($=2\pi$) is the flux quantum. The phase-induced asymmetry plays a key role in inducing charge and heat current through the thermally biased junction. However, the current amplitudes are sensitive to the size of the junction. We show that in the short Josephson junction when subject to a temperature gradient, the charge current shows an odd-symmetry in phase. It indicates that the phase-tunable asymmetry around the zero-energy is not sufficient to induce a dissipative thermoelectric current in the junction. This is in contrast to the behavior of long Josephson junction as shown in the literature. The phase-tunable heat currents are obtained with amplitudes set by the phase difference, base temperature, and system size.