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...

One dimensional free-fermions and hard-core bosons are often considered to be equivalent. Indeed, when restricted to nearest-neighbor hopping on a chain the particles cannot exchange themselves, and therefore hardly experience their own statistics. Apart from the off-diagonal correlations which depends on the so-called Jordan-Wigner string, real-space observables are similar for free-fermions and hard-core bosons on a chain. Interestingly, by coupling only two chains, thus forming a twoleg ladder, particle exchange becomes allowed, and leads to a totally different physics between free-fermions and hard-core bosons. Using a combination of analytical (strong coupling, field theory, renormalization group) and numerical (quantum Monte Carlo, density-matrix renormalization group) approaches, we study the apparently simple but non-trivial model of hard-core bosons hopping in a two-leg ladder geometry. At half-filling, while a band insulator appears for fermions at large interchain hopping t ⊥ > 2t only, a Mott gap opens up for bosons as soon as t ⊥ = 0 through a Kosterlitz-Thouless transition. Away from half-filling, the situation is even more interesting since a gapless Luttinger liquid mode emerges in the symmetric sector with a non-trivial filling-dependent Luttinger parameter 1/2 ≤ Ks ≤ 1. Consequences for experiments in cold atoms, spin ladders in a magnetic field, as well as disorder effects are discussed. In particular, a quantum phase transition is expected at finite disorder strength between a 1D superfluid and an insulating Bose glass phase.

We study a one-dimensional helical system with random Rashba spin-orbit coupling. Using renormalization group methods, we derive a consistent set of flow equations governing the important control parameters of the backscattering process. Thereby, we prove the existence of disorder-induced two-particle backscattering that can even be non-local in space. This analysis allows us to derive the scaling form of the conductance at low temperatures. We find that two-particle backscattering due to random spin-orbit coupling differs from the one off a single Rashba impurity by both the scaling of the conductance with the temperature and the relevance of the backscattering operators.

We study the properties of a topological Josephson junction made of both edges of a two-dimensional topological insulator. We show that, due to fermion parity pumping across the bulk, the global parity of the junction has a clear signature in the periodicity and critical value of the Josephson current. In particular, we find that the periodicity with the flux changes from 4π in a junction with an even number of quasiparticles to 2π in the odd sector. In the case of long junctions, we exhibit a rigorous mathematical connection between the spectrum of Andreev bound states and the fermion parity anomaly, through bosonization. Additionally, we discuss the rather quantitative effects of Coulomb interactions on the Josephson current.

Majorana bound states can emerge as zero-energy modes at the edge of a two-dimensional topological insulator in proximity to an ordinary s-wave superconductor. The presence of an additional ferromagnetic domain close to the superconductor can lead to their localization. We consider both N-S and S-N-S junctions based on helical liquids and study their spectral properties for arbitrary ferromagnetic scatterers in the normal region. Thereby, we explicitly compute Andreev wave-functions at zero energy. We show under which conditions these states form localized Majorana bound states in N-S and S-N-S junctions. Interestingly, we can identify Majorana-specific signatures in the transport properties of N-S junctions and the Andreev bound levels of S-N-S junctions that are robust against external perturbations. We illustrate these findings with the example of a ferromagnetic double barrier (i.e. a quantum dot) close to the N-S boundaries.

We develop a Gaussian variational approach in replica space to investigate the phase diagram of a one-dimensional interacting disordered topological superconducting wire in the strong-coupling regime. This method allows for a nonperturbative treatment in the disorder strength, electron-electron interactions, and the superconducting pairing amplitude. We find only two stable phases: a topological superconducting phase and a glassy, nontopological localized phase, characterized by replica symmetry breaking.