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We theoretically explore the behavior of thermal transport in the topologicalSQUIPT, in the linear and nonlinear regime. The device consists of atopological Josephson junction based on a two-dimensional topological insulatorin contact with two superconducting leads, and a probe tunnel coupled to thetopological edge states of the junction. We compare the performance of a normalmetal and a graphene probe, showing that the topological SQUIPT behaves as apassive thermal rectifier and that it can reach a rectification coefficient ofup to $~ 145\%$ with the normal metal probe. Moreover, the interplay betweenthe superconducting leads and the helical edge states leads to a uniquebehaviour due to a Doppler shift like effect, that allows one to influencequasi-particle transport through the edge channels via the magnetic flux thatpenetrates the junction. Exploiting this effect, we can greatly enhance therectification coefficient for temperatures below the critical temperature$T_\text{C}$ in an active rectification scheme.

Phase-tunable thermal rectification in the topological SQUIPT