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Signatures of Rashba spin-orbit interaction in the superconducting proximity effect in helical Luttinger liquids

We consider the superconducting proximity effect in a helical Luttinger liquid at the edge of a 2D topological insulator, and derive the low-energy Hamiltonian for an edge state tunnel-coupled to a s-wave superconductor. In addition to correlations between the left and right moving modes, the coupling can induce them inside a single mode, as the spin axis of the edge modes is not necessarily constant. This can be induced controllably in HgTe/CdTe quantum wells via the Rashba spin-orbit coupling, and is a consequence of the 2D nature of the edge state wave function. The distinction of these two features in the proximity effect is also vital for the use of such helical modes in order to split Cooper-pairs. We discuss the consequent transport signatures, and point out a long-ranged feature in a dc conductance measurement that can be used to distinguish the two types of correlations present and to determine the magnitude of the Rashba interaction.