  Robust and tunable topological Josephson junctions (TJJs) are highly
desirable platforms for investigating the anomalous Josephson effect and
topological quantum computation applications. Experimental demonstrations have
been done in hybrid superconducting-two dimensional topological insulator
(2DTI) platforms, sensitive to magnetic disorder and interactions with phonons
and other electrons. In this work, we propose a robust and electrostatically
tunable TJJ by combining the physics of the integer quantum Hall (IQH) regime
and of superconductors. We provide analytical insights about the corresponding
Andreev bound state spectrum, the Josephson current and the anomalous current.
We demonstrate the existence of protected zero-energy crossings, that can be
controlled through electrostatic external gates. This electrostatic tunability
has a direct advantage to compensate for non-ideal interfaces and undesirable
reflections that may occur in any realistic samples. TJJs in the IQH regime
could be realized in graphene and other 2D materials. They are of particular
relevance towards scalable and robust Andreev-qubit platforms, and also for
efficient phase batteries.
