Topological superconductivity and Majorana states in low-dimensional systems

Abstract

We discuss the properties of topologically nontrivial superconducting phases and the conditions for their realization in condensed matter, the criteria for the appearance of elementary Majorana-type excitations in solids, and the corresponding principles and experimental methods for identifying Majorana bound states (MBSs). Along with the well-known Kitaev chain and superconducting nanowire (SW) models with spin-orbit coupling in an external magnetic field, we discuss models of quasi-two-dimensional materials in which MBSs are realized in the presence of noncollinear spin ordering. For finite-length SWs, we demonstrate a cascade of quantum transitions occurring with a change in the magnetic field, accompanied by a change in the fermion parity of the ground state. The corresponding anomalous behavior of the magnetocaloric effect can be used as a tool for identifying MBSs. We devote considerable attention to the analysis of the transport characteristics of devices that contain topologically nontrivial materials. The results of studying the conductance of an Aharonov-Bohm ring whose arms are connected by an SW are discussed in detail. An important feature of this device is the appearance of Fano resonances in the dependence of conductance on the magnetic field when the SW is in a topologically nontrivial phase. We establish a relation between the characteristics of such resonances and the spatial structure of the lowest-energy SW state. The conditions for the occurrence of an MBS in the phase of the coexistence of chiral d+id superconductivity and 120-degree spin ordering are determined in the framework of the t-J-V model on a triangular lattice. We take electron-electron interactions into account in discussing the topological invariants of low-dimensional superconducting materials with noncollinear spin ordering. The formation of Majorana modes in regions with an odd value of a topological ℤ invariant is demonstrated. The spatial structure of these excitations in the Hubbard fermion ensemble is determined.

Electron transport in quantum channels with spin-orbit interaction: Effects of the sign of the Rashba coupling and applications to nanowires

“…While panel (a) refers to the case where αL, αR > 0, panel (b) refers to the case of coupling changing sign across the junction, αL < 0 and αR > 0. Insets: the spatial behavior of the spin-orbit angle (4) describing the rotation of the effective magnetic texture problem (5).…”

Aspects of topological superconductivity in 2D systems: noncollinear magnetism, skyrmions, and higher-order topology

“…where a is the chain parameter and t is the hopping amplitude of electrons between the nearest sites of the chain. A more detailed description of the correspondence between the above mentioned systems can be found in [19]. In general, noncollinear magnetic structures include a spin helix, spin spiral, cycloidal texture etc., which will be considered later.…”

“…At present, topological superconductivity has become an extensive and intensively developed area of condensed matter physics [15][16][17][18][19][20]. For example, superconducting pairing of electrons with the same spin projections (triplet pairing) was found to play an important role in the realization of TP with MMs.…”

“…Most of the detailed reviews concerning the topological superconductivity and MMs [15][16][17][18][19][20] are devoted to topological systems with spin-orbit coupling (SOC). In this review, we will discuss the aspects of 2D TSC where the topological order appears when SOC can be neglected (namely, points 2 and 3 from the list above) and 2D higher-order topological superconductors for which there is another component in addition to the spin-orbital one for inducing strongly localized MMs in the 2D case.…”

Majorana vortex modes in spin-singlet chiral superconductors with noncollinear spin ordering: Local density of states study

“…Majorana modes that are localized at the ends of quasi-1D quantum wires have been actively studied in recent years both theoretically and experimentally [1][2][3][4]. It is believed that such spatially separated modes can be used for the realization of topologically protected quantum computations [5].…”