  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.
