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In this work we investigate the supercurrent in a hybrid topological Josephson junction consisting of two planes of topological insulator (TI) in a specific configuration, which allows both local (LAR) and crossed (CAR) Andreev processes at the interfaces with two conventional s-wave superconductors. We describe the effects of gate voltage and magnetic flux controls applied to the edge states of each TI. In particular, we demonstrate that the voltage gating allows the manipulation of the entaglement symmetry of non-local Cooper pairs associated to the CAR process. We establish a connection between the Josephson current-phase relationship of the system and the action of the two external fields, finding that they selectively modify the LAR or the CAR contributions. Remarkably, we find that the critical current of the junction takes a very simple form which reflects the change in the symmetry occurred to the entangled state and allows to determine the microscopic parameters of the junction.

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The thermoelectric performance of a topological Josephson nonlocal heat engine is thoroughly investigated. The nonlocal response is obtained by using a normal metal probe coupled to only one of the proximized helical edges in the middle of the junction. In this configuration, we investigate how the flux and phase biases trigger the nonlocal thermoelectric effects under the application of a thermal difference between the superconducting terminals. Possible experimental nonidealities such as asymmetric proximized superconducting gaps are considered, showing how the nonlocal response can be affected. The interplay between Doppler shift, which tends to close gaps, and Andreev interferometry, which affects particle-hole resonant transport, are clearly identified for different operating regimes. Finally, we discuss the power and the efficiency of the topological thermoelectric engine which reaches maximum power at maximal efficiency for a well-coupled normal probe. We find quite high nonlocal Seebeck coefficients of the order of tenths of μV/K at a few Kelvin, a signal that would also be clearly detectable against any spurious local effects even with moderate asymmetry of the gaps.