Andreev and Majorana bound states in low-dimensional systems

The discovery of the quantum Hall effect [1] and its explanation in terms of topology opened up a new field in condensed matter theory [2–5]. Topology, in the original context of mathematics, classifies geometric structures and studies whether an object can be continuously deformed into another one. If such a continuous deformation exists, then the two objects belong to the same topological class. Different topological classes are distinguished by the value of a topological invariant. For example, the tea cup and the torus both have one hole and therefore share the same topology. The topological invariant is in this example the number of holes which is called genus [6–9].
In 1982, Thouless, Kohmoto, Nightingale, and den Nijs related the quantized Hall conductance to a topological invariant [2]. This work among others [4, 10–13] paved the way towards the research of topological insulators (TIs) and superconductors (TSCs), which has become one of the most active research fields in condensed matter physics [6–8, 14–16]. TIs and TSCs are characterized by an energy gap in the bulk and subgap states at the surface/edge. The analogue of the continuous deformation of a geometrical object is here the continuous deformation of the Hamiltonian, under which the energy gap of the bulk does not close.
The quantum Hall system is one example of a TI, and it requires an external magnetic field, which explicitly breaks time reversal symmetry (TRS). Another class, namely time-reversal invariant TIs based on spin-orbit interaction (SOI), has been intensively studied in theory [10–13, 17] and experiment [18, 19]. The next important step in the research field of TIs was the description [20–23] and discovery [24– 26] of three-dimensional topological insulators.
Topological superconductors have more exotic edge states, so-called Majorana modes, named after Ettore Majorana who proposed the existence of a particle that is its own antiparticle in the context of particle physics [27]. In two dimensions, chiral Majorana modes move around the one-dimensional boundary of the TSC, similar to the conducting edge states in a quantum Hall sample. A pair of vortices, in contrast, can host a pair of Majorana bound states (MBSs), with one MBS in each vortex [28–31]. These states with exactly zero energy are their own antiparticles and, even more interestingly, they obey non-Abelian braiding statistics [30, 32, 33]. This makes them highly attractive for quantum computation, since information could be protected against local perturbations due to the topology of the system. Usual Majorana qubit proposals rely on fermion parity conservation. One pair of MBSs forms a nonlocal fermionic state, which can be empty or occupied by one fermion. Therefore, two pairs of MBSs, so in total four MBSs, can define a qubit with conserved fermion parity [30, 32]: If only one fermion is present, then it can either occupy the first or the second pair of MBSs and the fermion parity does not change when the fermion is transferred from one pair of MBSs to the other one.
Another famous system hosting MBSs apart from vortices in two-dimensional px + ipy superconductors is the Kitaev chain [34], which is a one-dimensional spinless p-wave superconductor. In this model, the MBSs appear at the ends of the chain and are topologically protected against disorder, as long as the length of the chain is much longer than the localization length of the MBSs. In the opposite limit, the MBSs lose their topological protection, however, they can still exist for a fine-tuned choice of parameters. Some works on so-called Poor Man’s MBSs, namely MBSs which are not topologically protected, propose how to realize this short length limit of the Kitaev chain in real systems, for example with the help of strong local magnetic fields that polarize electrons of two quantum dots along different axes [35, 36]. First experiments on Poor Man’s MBSs agree to some extent with the theory [37].
Most proposals for the realization of MBSs in experiments are based on the interplay of several ingredients which can mediate an effective superconducting p-wave pairing. One prominent example of an engineered platform is the Rashba nanowire [38, 39], in which the interplay of spin-orbit interaction, an external Zeeman field, and conventional s-wave superconductivity gives rise to an effective p-wave pairing. In 2012, Mourik et al. performed the first experiment looking for MBSs on this platform and reported zero-energy states at one end of the nanowire, in agreement with the theory predicting MBSs [40]. Many subsequent differential conductance experiments using different materials and slightly changed device designs confirmed the existence of the zero bias peak (ZBP) [41–45]. However, soon it was realized that also trivial states like Andreev bound states (ABSs) can accidentally appear at zero energy [46–54]. Therefore, the measurement of a single ZBP is not a sufficient indicator for the presence of an MBS. Subsequently, in order to clarify the origin of the ZBPs, other signatures were under consideration, for example, oscillations of the MBS energy as a function of the magnetic field [55], an almost quantized differential conductance value of 2e2/h [56] and the simultaneous appearance of ZBPs on both ends of the nanowire [57]. However, it turned out that trivial ABSs can mimic all these features, and the problem remained unsolved [57–60]. More recently, non-local conductance has been proposed as another indicator revealing the bulk gap closing and reopening which accompanies a topological phase transition [61, 62]. Potential braiding experiments that could confirm the non-Abelian braiding statistic of MBSs, however, go beyond the current experimental capabilities.
Another platform for MBSs is based on chains of magnetic impurities deposited on s-wave superconductors [63–76]. The exchange coupling between the magnetic moments of the impurities and the itinerant electrons plays a similar role as the external magnetic field in the Rashba nanowire. Scanning tunneling microscopy (STM) measurements reported ZBPs at the ends of such impurity chains [77–80]. However, this platform has a limited tunability compared to the Rashba nanowire, since the exchange coupling is fixed. Therefore, and because trivial subgap states might cause similar signatures in experiments, the unambiguous identification of MBS remains challenging also in this platform.
Josephson junctions [81] are also a promising platform for the observation of MBSs, the phase difference between two superconductors serves here as an additional control knob which can drive the system into the topological phase [82–87]. The original proposal for Josephson junctions hosting MBSs requires an external magnetic field which is in-plane, in contrast to the usual out-of-plane configuration for topological superconductivity in two dimensions [88]. In fact, in SNSNS junctions, with three superconducting sections (S) and two normal sections (N), a magnetic field is not required for the realization of MBSs, if the Fermi velocity at the inner and outer Fermi surface differ [89, 90].
In general, the application of topology in condensed matter physics goes far beyond TIs and TSCs. For example, the fractional quantum Hall effect has an intrinsic topological order due to strongly correlated electrons [91, 92]. Moreover, magnetic textures like skyrmions [93–96] have a non-trivial topology in the sense that their local magnetization wraps one time around a sphere such that one can assign a topological invariant, also called topological charge, to this winding [97]. This realspace topology in combination with manipulation schemes based on external fields, currents, and temperature makes skyrmions good candidates for future information carriers [98–101].
This thesis is organized as follows: in the introduction, chapter 1 of this thesis, we recapitulate the concept of symmetry protected topological order and discuss it using the examples of the Kitaev chain, the Rashba nanowire, and magnetic chains deposited on superconductors. Moreover, we summarize the scattering matrix formalism in the context of differential conductance simulations. In chapter 2, we study the local and non-local conductance in short Rashba nanowires. In particular, we show that a single Andreev bound state can lead to ZBPs at both ends of the nanowire, if the length of the nanowire is comparable to the localization length of the Andreev bound state. Next, in chapter 3, we propose a mechanism based on overlapping Andreev bound states, that can lead to a signature reminiscent of the bulk gap closing and reopening of a topological phase transition in non-local conductance. In chapter 4, we study trivial zero-energy states in helical spin chains. In particular, we analyze spin configurations in which the angle between adjacent spins smoothly changes close to the ends of the chain. Last, in chapter 5, we analyze the superconducting diode effect for magnetic domain walls and skyrmions moving on a racetrack that is sandwiched by two superconductors forming a Josephson junction. Additionally, we classify magnetic textures and predict which textures support a superconducting diode effect without the need for an extra Rashba SOI in the two-dimensional electron gas beneath the Josephson junction.