Topological Quantum Phases in Layered Systems with Rashba Spin-Orbit Interaction

The exploration of topological phases of quantum matter has attracted considerable interest of various research communities: the condensed matter physics community, the quantum information community, and importantly also industry. The interest of fundamental research in these phases is manifold. For one, the use of the abstract notions of topology and its powerful statements in the context of quantum many-body systems extends the tools of conventional condensed matter theory. Moreover, the theory predicts new exotic phases with protected boundary states, which can have fractional quantum numbers and anyonic statistics. Non-Abelian anyons, Majorana bound states being the most prominent example, are especially interesting in the context of quantum computation, since their topological nature makes qubits formed from such states immune against decoherence. It is thus of fundamental interest to study those systems, especially experimentally, in order to shed light on the existence of these phases in Nature. At the same time, it will be important to check to what extent the aforementioned mathematical tools provide an accurate description of actual physical systems.
A lot of experimental achievements were reported in the last two decades, ranging from the first signatures of the quantum spin Hall effect and the topological surface states of three-dimensional (3D) topological insulators (TIs) to the zero-bias conductance peak associated with Majorana bound states (MBSs) in one-dimensional (1D) topological superconductors. Despite the great advances since these first experimental findings, there are still open questions, especially regarding MBSs, which need to be answered in order to decisively conclude whether or not the observed signatures actually are of topological origin.
At the same time, the search for alternative materials with topological phases continues. In this Thesis it is theoretically shown that there exists a class of materials, namely two-dimensional (2D) layers with strong Rashba spin-orbit interaction (SOI) (subsequently called Rashba layers for short), which can be used to fabricate heterostructures realizing a wide range of topological phases.
In particular, it is shown (see Chapter 3) how a stack of Rashba layers, can be used to build 3D topological phases. In the first part, it is proven that the system can be a 3D strong TI. Starting from this topological phase, it is demonstrated that by including strong electron-electron interactions the system fractionalizes and realizes a fractional 3D TI. The use of rotational symmetry and energy condensation arguments provides an intuitive way to reduce the problem to a set of interacting quasi-1D systems. This is one of the first concrete material candidates for such a phase which does not rely on a coupled-wires construction. In the last part of Chapter 3, it is shown that if the 2D layers not only have Rashba SOI, but a combination of Rashba and Dresselhaus SOI, the stack of layers can realize a Weyl semimetal.
Moreover, it is shown (see Chapter 4) how a Rashba bilayer can be used to generate different 2D topological superconducting phases. The setup consists of a conventional $s$-wave superconductor sandwiched between two Rashba layers. Due to the proximity to the superconductor, the Rashba layers themselves become superconducting by the virtue of two competing pairing mechanisms: $(i)$ direct Andreev pairing -- a process where a Cooper pair as a whole tunnels into one of the layers and $(ii)$ crossed Andreev pairing -- a process where a Cooper pair splits and the electrons tunnel into opposite layers. The competition of these two processes leads to a 2D time-reversal invariant (TRI) topological superconducting phase when crossed Andreev pairing dominates. By applying a Zeeman, field the setup can be brought either into a chiral topological superconducting phase or a 2D gapless superconducting phase with unidirectional edge states. In essence, a Rashba bilayer system is a versatile platform which allows the realization of various 2D topological superconducting phases.
In Chapter 5, a system consisting of two tunnel-coupled Rashba layers which are proximitized by a top and bottom $s$-wave superconductor with a phase difference $\phi$ close to $\pi$ is studied. This system is predicted to be a 2D TRI topological superconductor if the tunnel coupling is stronger than the proximity induced superconducting pairing strength. By breaking time-reversal symmetry with an inplane magnetic field, the system can be brought into the recently discovered second-order topological superconducting phase. In this phase, the bulk as well as the edges of the system are gapped, while in a square geometry two MBSs appear on opposite corners. Numerical results show that this finding even holds true if deviations from the $\phi=\pi$ phase difference between the parent superconductors is as large as $\delta \phi \approx \pi/3$. This implies that in an experiment, the phase difference does not need to be fine tuned.
In Chapter 6, an alternative way to fabricate a 1D TRI topological superconducting phase in a Josephson bijunction is presented. Concretely, the setup consists of a thin superconductor~-~insulator~-~superconductor ($SIS$) $\pi$-Josephson junction sandwiched between Rashba layers with opposite SOI. Due to the proximity effect, the Rashba layers themselves form a superconductor~-~normal conductor~-~superconductor ($SNS$) junction. The $SIS$ junction is assumed to be thin enough such that electron tunneling between the Rashba layers is possible. This leads to a hybridization between the Andreev bound state bands (ABSBs) that emerge in the normal regions in the Rashba layers. Roughly speaking, the 1D channels formed by the ABSBs mimick the physical situation in tunnel coupled nanowires, and it is shown that indeed a topological phase with a Kramers pair of MBSs can emerge at the end of the normal regions.