Investigation On The Phase Transition And Transport Properties Of Low-dimensional Topological Superconducting Systems

As a tangible implementation of Majorana fermions in condensed matter systems,Majorana bound states(MBSs)are zero-energy quasiparticle excitations populating the boundaries or defects in topological superconductors.MBSs exhibit non-Abelian exchange statistics which makes them promising potential building blocks for fault-tolerant quantum computers.Typical devices to engineer MBSs include hybrid superconductorsemiconductor nanowire exposed to an applied magnetic field or the magnetic adatoms deposited on a superconductor.Despite the early success of these platforms,the searching for solid proofs of MBSs existence is still in ongoing dispute.Alternative origins of signatures similar to MBSs-induced ones cannot be completely excluded in relevant experiments.These undesired signatures can be produced by non-topological Andreev bound states(ABSs)or other zero-energy excitations in unconventional quantum phases.This dissertation is devoted to the theoretical and numerical study on these non-MBSs excitations and the possible distinctions between them and MBSs,particularly through transport measurements.The main content of the dissertation includes:1.A retrospective study on the development of realizing Majorana fermions in condensed matter systems via MBSs and the explorations of long-range phase in relevant superconducting systems are introduced.These literature reviews underline the urgent need for a conclusive evidence for MBSs.Thus,the Andreev reflection crossing the normal-superconductor interface is briefly reviewed since it facilitates the basic transport measurements of MBSs.Following the non-equilibrium Green’s function formalism,the derivations of tunneling currents flowing from a normal lead into a hybrid superconductorsemiconductor nanowire are outlined.2.The effect of long-range pairing interactions on the phase diagram and the transport properties in a semiconductor-superconductor hybrid nanowire.Both the power-law and exponential decay rates of long-range pairing are considered for comparison.It is found that a long-range topological phase hosting massive edge modes(MEMs)is induced when the exponent of the power-law decay is less than one.By diagonalizing the tight-binding model,we find that near-zero-energy ABSs could be induced in the topologically trivial phase for slowly decayed long-range pairing interactions.For a one-dimensional longrange Kitaev chain,the shot noise is suggested to serve as a probe to discriminate MBSs and topological massive modes.Our numerical results indicate that for realistic device parameters,the Fano factor of the shot noise is irrelevant to the topological property of the system,and it fails to distinguish the MBSs and ABSs.However,the noise Fano factor is generally consistent with the variation of the bound state energy,which can be used to detect the energy splitting of these bound states.3.The dc and ac Josephson currents between two long-ranged Majorana nanowires.By tunning the magnetic field and long-range pairing,the nanowires can be driven into different phases hosting MEMs,MBSs,or trivial ABSs,which considerably modulate the Josephson currents in different ways.For weakly linked Josephson junctions,we show that the MEMs emerging in the long-range phases can induce a zigzag profile in the dc current-phase relation featured by two sign reversals deviating from the point where the superconducting phase difference φ = π.In contrast,the Josephson current indicates a sharp sign reversal at φ = π in the short-range phase that hosts MBSs,while a smooth sinelike dc Josephson current appears in the presence of ABSs.In the nonequilibrium junction biased by a voltage,we identify a 4π ac Josephson current in the short-range phases,indicating a MBSs-mediated tunneling process dominating the supercurrents.Differently,when the system enters MEMs-hosted long-range phases,the ordinary ac Josephson current with2π periodicity can be restored.In addition,we find the ac Josephson current induced by MEMs is significantly suppressed under the strong long-range pairing interactions.These features can serve as a preliminary probe for the long-range superconducting transition in relevant experiments.4.A scheme to distinguish MBSs and ABSs in a hybrid superconductor-semiconductor nanowire in the presence of a sharp local potential barrier.The position and height of the local potential barrier could be modulated by a gate voltage applied to the nanowire.It is found that a new pair of MBSs appear at both sides of the barrier in case of enough barrier height,as a manifestation of the topological nature.Differently,the local potential barrier does not induce extra ABSs.Such a difference can be traced back to the evolution of conductance spectrum the local potential strength.When modulating the local potential barrier,the zero-bias conductance peak induced by MBSs can split into two-,three-,or four-peak structures,while the ABSs-induced zero-bias peak only splits once.