The occurrence of two-particle inelastic backscattering has been conjectured in helical edge states of topological insulators and is expected to alter transport. Here, by using a renormalization group approach, we provide a microscopic derivation of this process, in the presence of a time-reversal invariant Rashba impurity potential. We are able to prove that such an effect only occurs in the presence of electron-electron interactions. Furthermore, we find that the linear conductance as a function of temperature exhibits a crossover between two scaling behaviors, T 4K for K > 1/2 and T 8K−2 for K < 1/2, with K the Luttinger parameter. PACS number(s): 72.15.Nj, 72.25.−b, 85.75.−d Introduction. Since the prediction of the quantum spin Hall phase 1,2 in HgTe quantum wells, 3 transport measurements on these compounds have shown evidence of a quantized edge conductance G = 2e 2 /h, thereby paving the way for nonlocal dissipationless transport in semiconductors at zero external magnetic field. 4-6 In the simplest case of quantum wells with inversion symmetry, transport occurs through two counterpropagating edge channels that carry opposite spin-1/2 quantum numbers. Such helical liquids form a new class of one-dimensional (1D) quantum liquids in the sense that they are protected by time-reversal symmetry against single-particle elastic backscattering. 2,7,8 However, deviations from the quantized conductance arise in various situations, involving either a breaking of time-reversal symmetry-by a magnetic impurity, for instance-or the interplay between a time-reversal invariant (TRI) external potential and a source of inelastic scattering. Inelastic single-particle backscattering 9,10 and two-particle backscattering 2,7,8,11 are two examples of the latter. In this Rapid Communication, we focus on twoparticle backscattering off a TRI impurity and report results regarding the temperature scaling of conductance corrections. Our purpose is to derive the Hamiltonian for such a process starting with a minimal model of an interacting helical liquid coupled to a TRI potential. In particular, we focus on a Rashba spin-orbit potential, 9,11 which can originate from fluctuations of an electric field perpendicular to the two-dimensional (2D) electron gas, 12 and acts as a TRI effective magnetic field that couples right and left movers. In the recent literature, inelastic two-particle backscattering off an impurity was mostly studied phenomenologically, by postulating the generic form of the Hamiltonian due to symmetry considerations, namely, TRI and the Pauli principle, 2,7,8,13

Non-local pairing processes at the edge of a two-dimensional topological insulator in proximity to an s-wave superconductor are usually suppressed by helicity. However, additional proximity of a ferromagnetic insulator can substantially influence the helical constraint and therefore open a new conduction channel by allowing for crossed Andreev reflection (CAR) processes. We show a oneto-one correspondence between CAR and the emergence of odd-frequency triplet superconductivity. Hence, non-local transport experiments that identify CAR in helical liquids yield smoking-gun evidence for unconventional superconductivity. Interestingly, we identify a setup -composed of a superconductor flanked by two ferromagnetic insulators -that allows us to favor CAR over electron cotunneling which is known to be a difficult but essential task to be able to measure CAR. PACS numbers: 74.45.+c, 74.78.Na, 71.10.Pm, 74.20.Rp Introduction.-The influence of strong spin-orbit coupling and the constraint of time reversal symmetry are responsible for the appearance of helical electronic channels at the edge of two-dimensional (2D) topological insulators [1][2][3]. Ample evidence for the experimental detection of these edge states has been seen in transport [4][5][6] and scanning SQUID experiments [7]. Proximity of an ordinary s-wave superconductor can induce pairing at the helical edge [8,9]. Interestingly, the interplay of helicity and superconducting order gives rise to unconventional proximity-induced superconductivity (SC) in these systems. However, it is fair to say that it is very difficult to unambiguously probe the emergence of unconventional SC because -often times -conventional and unconventional signatures of SC look alike. Several recent experiments have demonstrated (as a first step towards the detection of unconventional SC) that helical liquids as boundary states of quantum spin Hall systems can indeed be brought in proximity to s-wave superconductors [10] and serve, for instance, as conducting channels of a Josephson junction [11,12]. Importantly, helicity guarantees perfect local Andreev reflection [13] -the conversion of an electron into a hole with opposite spin -at the interface between the normal and the proximity-induced superconducting region called NS junction. Evidently, perfect local Andreev reflection can give rise to sub-gap Andreev states, for instance, in Josephson junction setups. Among these sub-gap states, the one with zero excitation energy is its own chargeconjugate state. It has been coined Majorana (bound) state and has attracted a lot of attention because of potential applications of this anyonic state for topological quantum computing [14]. If a ferromagnet (FM) is placed in vicinity to the NS interface the Majorana (bound) state can be localized in the region between the FM and the SC [9]. This localization allows us to make a formal connection to the physics of a finite-size 1D p-wave topological superconductor [15], and its potential realizations in spin-orbit nanowires [16,17]. In this Letter, ...

Spin-momentum locking in a semiconductor device with strong spin-orbit coupling (SOC) is thought to be an important prerequisite for the formation of Majorana bound states 1-3 . Such a helical state is predicted in one-dimensional (1D) nanowires subject to strong Rashba SOC and spin-mixing 4 -its hallmark being a characteristic re-entrant behaviour in the conductance. Here, we report direct experimental observations of the re-entrant conductance feature, which reveals the formation of a helical liquid, in the lowest 1D subband of an InAs nanowire. Surprisingly, the feature is very prominent also in the absence of magnetic fields. This behaviour suggests that exchange interactions have a substantial impact on transport in our device. We attribute the opening of the pseudogap to spin-flipping two-particle backscattering 5-7 . The all-electric origin of the ideal helical transport could have important implications for topological quantum computing.A 1D conductor with strong SOC is predicted 1,2,8 to represent a viable host for Majorana bound states. These zero-energy states feature characteristic non-Abelian exchange statistics 8 and can be created by mimicking spinless p-wave Cooper pairing using a semiconductor nanowire with a helical state and inducing s-wave superconductivity. InAs and InSb nanowires are promising host materials to explore the existence and nature of Majorana bound states 9,10 . To this end, it is essential to both establish transport in 1D subbands and induce a helical state in the nanowire. The usual mechanism that is considered to open a helical gap involves an external Zeeman field oriented perpendicular to the uniaxial spinorbit field 4 . The magnitude of the spin-orbit energy relative to the Zeeman energy is partly responsible for the size of the topological energy gap that will protect the zero-energy Majorana modes 11 . However, Oreg et al. 2,12 and Stoudenmire et al. 13 have pointed out that such an energy gap can also result from strong electronic correlations. Several mechanisms have been proposed along these lines: for example, spin-flipping two-particle backscattering 7 and hyperfine interaction between nuclear spins and a Luttinger liquid 14 , both of which can open a gap. The latter mechanism has been invoked to explain a conductance reduction by a factor of two at low temperatures in a GaAs quantum wire 15 , but no re-entrant behaviour is predicted within this framework.Other than Quay et al. 3 , we report on a re-entrant conductance feature in the lowest subbands of InAs nanowire quantum point contacts (QPCs), which offer the desired strong SOC (see Supplementary Section 1). Moreover, our proposed spin-mixing mechanism does not necessarily rely on external time-reversal symmetry-breaking terms: while the effect is pronounced in the presence of an external magnetic field, it persists also in its absence. Guided by the observation 16 of the Landé g factor enhancement for the lowest subband 17 and by signatures of the 0.7 anomaly 18 , we identify the important role of exchange int...