Motivated by recent experiments, we investigate the excitation energy of a proximitized Rashba wire in the presence of a position dependent pairing. In particular, we focus on the spectroscopic pattern produced by the overlap between two Majorana bound states that appear for values of the Zeeman field smaller than the value necessary for reaching the bulk topological superconducting phase. The two Majorana bound states can arise because locally the wire is in the topological regime. We find three parameter ranges with different spectral properties: crossings, anticrossings and asymptotic reduction of the energy as a function of the applied Zeeman field. Interestingly, all these cases have already been observed experimentally. Moreover, since an increment of the magnetic field implies the increase of the distance between the Majorana bound states, the amplitude of the energy oscillations, when present, gets reduced. The existence of the different Majorana scenarios crucially relies on the fact that the two Majorana bound states have distinct k-space structures. We develop analytical models that clearly explain the microscopic origin of the predicted behavior.

Electrical currents in a quantum spin Hall insulator are confined to the boundary of the system. The charge carriers can be described as massless relativistic particles, whose spin and momentum are coupled to each other. While the helical character of those states is by now well established experimentally, it is a fundamental open question how those edge states interact with each other when brought in spatial proximity. We employ a topological quantum point contact to guide edge channels from opposite sides into a quasi-onedimensional constriction, based on inverted HgTe quantum wells. Apart from the expected quantization in integer steps of 2e 2 /h, we find a surprising additional plateau at e 2 /h. We explain our observation by combining band structure calculations and repulsive electron-electron interaction effects captured within the Tomonaga-Luttinger liquid model. The present results may have direct implications for the study of one-dimensional helical electron quantum optics, Majorana-and potentially para-fermions. The quantum spin Hall effect has been predicted in several systems [1][2][3][4] and was first realized in HgCdTe/HgTe quantum wells [5]. Later, this phase was observed in other material systems such as InAs/GaSb double quantum wells [6] and in monolayers of WTe 2 and bismuthene [7,8]. The defining properties of this state, related to its helical nature, are well established by numerous experiments such as the observation of conductance quantization of two spin polarized edge channels G 0 = 2e 2 /h with e the electron charge and h the Planck's constant [5]. Additionally, non-local edge transport and spin-polarization of the edge channels were demonstrated by suitable transport experiments [9,10]. We instead target a still open question, namely how helical edge states interact with each other.A quantum point contact (QPC) can be used to guide * All three authors contributed equally to this work, email: Jonas.Strunz@physik.uni-wuerzburg.de edge channels from opposite boundaries of the sample into a constriction. Such a device allows for studies of charge and spin transfer mechanisms by, e.g., adjusting the overlap of the edge states [11][12][13][14][15][16][17][18][19][20]. Besides the general interest in the study of transport processes in such a device, the appropriate model to describe the essential physics and to capture interaction effects of helical edge states is still unclear. The one-dimensionality of the helical edge modes suggests a description in terms of the Tomonaga-Luttinger liquid when electron-electron interactions are taken into account. In this respect, the QPC setup provides an illuminating platform as it may give rise to particular backscattering processes.We present the realization of a QPC based on HgTe quantum wells as evidenced by the observation of the expected conductance steps in integer values of G 0 . The newly developed lithographic process allows the fabrication of sophisticated nanostructures based on topological materials without lowering the material quality. It t...

Topological superconductors give rise to unconventional superconductivity, which is mainly characterized by the symmetry of the superconducting pairing amplitude. However, since the symmetry of the superconducting pairing amplitude is not directly observable, its experimental identification is rather difficult. In our work, we propose a system, composed of a quantum point contact and proximity induced s-wave superconductivity at the helical edge of a two dimensional topological insulator, for which we demonstrate the presence of odd-frequency pairing and its intimate connection to unambiguous transport signatures. Notably, our proposal requires no time-reversal symmetry breaking terms. We discover the domination of crossed Andreev reflection over electron cotunneling in a wide range of parameter space, which is a quite unusual transport regime.

Parafermions are generalizations of Majorana fermions that may appear in interacting topological systems. They are known to be powerful building blocks of topological quantum computers. Existing proposals for realizations of parafermions typically rely on strong electronic correlations which are hard to achieve in the laboratory. We identify a novel physical system in which parafermions generically develop. It is based on a quantum point contact formed by the helical edge states of a quantum spin Hall insulator in vicinity to an ordinary s-wave superconductor. Interestingly, our analysis suggests that Z4 parafermions are emerging bound states in this setup -even in the weakly interacting regime.Introduction.-During the last decades, topological quantum physics has become one of the most active directions of modern condensed matter research. Especially, the formation of topological boundary excitations, such as Majorana fermions [1,2], has attracted a lot of attention, both theoretically as well as experimentally [3-10]. These robust bound states have been proposed in various host materials, ranging from vortices in p x + ip y superconductors [11,12] over ferromagnet-superconductor heterojunctions in quantum spin Hall insulators (QSHIs) [13][14][15][16][17][18][19] to spin-orbit coupled quantum wires [3,4]. Due to their non-Abelian statistics [20][21][22], the interest in those topological bound states is not only fundamental but also practical: They can potentially be used for protocols in topological quantum computation (TQC) [23]. Majorana fermions are the conceptually simplest representatives of non-Abelian particles. However, braiding of Majorana fermions is not able to generate all the operations needed for universal TQC. For this task, more complex anyonic particles, assigned in general to a Z n permutation group, are required [21,24]. Due to the high groundstate degeneracy of those Z n anyons, electron-electron interactions are essential in physical realizations thereof. In particular, Z n parafermions are concrete examples of topological states that are proposed to emerge in correlated topological systems.

The chiral anomaly is based on a non-conserved chiral charge and can happen in Dirac fermion systems under the influence of external electromagnetic fields. In this case, the spectral flow leads to a transfer of right-to left-moving excitations or vice versa. The corresponding transfer of chiral particles happens in momentum space. We here describe an intriguing way to introduce the chiral anomaly into real space. Our system consists of two quantum dots that are formed at the helical edge of a quantum spin Hall insulator on the basis of three magnetic impurities. Such a setup gives rise to fractional charges which we show to be sharp quantum numbers for large barrier strength. Interestingly, it is possible to map the system onto a quantum spin Hall ring in the presence of a flux pierced through the ring where the relative angle between the magnetization directions of the impurities takes the role of the flux. The chiral anomaly in this system is then directly related to the excess occupation of particles in the two quantum dots. This analogy allows us to predict an observable consequence of the chiral anomaly in real space. The physics of relativistic fermions, described by the Dirac equation, inspired generations of physicists and gave rise to many astonishing discoveries, such as a bound soliton state of fractional charge 1/2 in the presence of a kinked background mass term, first predicted by Jackiw and Rebbi [1]. It was subsequently shown by Kivelson and Schrieffer that this charge 1/2 can be understood as a sharp quantum observable [2]. Interestingly, Goldstone and Wilzcek extended the corresponding theory to a bound state of any fractional charge considering complex solitons [3]. The first connection to condensed matter physics was introduced by Su, Schrieffer, and Heeger considering domain walls in polyacetylen chains [4]. With the discovery of topological insulators [5,6], another broad connection from condensed matter physics to particle physics was established. As part of that, Qi, Hughes, and Zhang [7] demonstrated the analogy of the Jackiw-Rebbi model to a system of two magnetic impurities aligned along the helical edge of quantum spin Hall systems. These peculiar one-dimensional (1D) edge states of 2D topological insulators have recently been theoretically predicted [8,9] and soon after experimentally observed [10].Yet another prominent characteristic of relativistic fermions is the chiral anomaly, referring to a nonconservation of the chiral current, first studied by Adler [11], Bell, and Jackiw [12] to explain the observed decay of a pion into two photons. With the theoretical prediction [13][14][15][16][17][18] and experimental discovery of Weyl/Dirac semimetals [19][20][21][22], the chiral anomaly has become prominent in condensed matter physics. In fact, it is considered to be one of the key features for an unambiguous detection of Weyl points in Weyl semimetals. Since the non-conservation of right-and left-movers is intimately connected to momentum space, it is hard to get a direct experime...