Theoretical Study On Topological Superconductor And Inverse Proximity Effect

In the past two decades,condensed matter physicists have devoted themselves to finding Majorana fermions for their non-Abelian exchange statistics and their essential role as building blocks for topological quantum computation.The Majorana fermion here is not a basic particle but exists in the topological superconductors in the form of quasiparticle excitation.The theoretical predictions of topological superconductors and Majorana quasiparticles have been first proposed by Kitaev and Green et al..They proved that edge states in one-dimensional and two-dimensional p wave superconductors are the Majorana quasiparticles(Majorana zero mode,MZMs).However,intrinsic p wave superconductors are rare in nature.This impasse was not broken until the emergence of the Fu-Kane proximity scheme in 2008.Inspired by this scheme,physicists have designed a variety of schemes to realize MZMs in condensed matter systems in recent ten years,and observed many signs of the existence of MZMs,such as zerobias conductance peaks and single-electron Coulomb blocking peaks.However,these signals are not unique to MZMs.The most convincing evidence may come from verifying its non-Abelian statistics by braiding operation,which has not yet been observed in experiments.Current theoretical and experimental progress can be roughly divided into four aspects:(1)the similarity between trivial states and MZMs,and more trivial explanations,including the influence from Andreev bound states(ABSs),disorder,Kondo effect,weak localization and so on;(2)Based on the original scheme,consider more influencing factors and establish a more accurate theoretical model;(3)propose a new detection scheme;(4)find new proximity schemes or intrinsic topological superconductors.Based on the above theoretical and experimental progress,we study the similarity of MZMs and ABSs in topological superconductors,topological superconductors with unconventional pair density wave(PDW)and inverse proximity effect.The main research works are stated as follows:1.Majorana zero mode and Andreev bound state in a nanowire-superconductor systemIn the past few years,the similarity between ABSs and MZMs has been widely studied,but it has been focused on the correlation behavior of zero modes.There are few studies on the properties of non-zero energy ABSs.Here,we investigate the MSMs and ABSs in a nanowire in partially proximity to a superconductor.In the gapped topological superconducting phase,it plays the role of a topological barrier,which confines some localized edge modes in the QD region.These low-lying localized modes exhibit linear spectra with equal energy level spacing.We demonstrate these features using a spinless nanowire in proximity to a p-wave superconductor and a spin-orbit coupled semiconductor nanowire in proximity to a s-wave superconductor.A simple picture is proposed to understand the behavior of these results.However,in the trivial superconducting phase when both bands are occupied in the spin-orbit coupled mode,we observe some ABSs,with complicated low-lying excited spectra.We show quantitative theoretical and numerical results for the low energy excitation spectra of the ABSs.These scaling behaviors can be measured experimentally,so our work provides theoretical support for the experiment to distinguish between the MZMs and ABSs and verify their relationship.2.Topological pair-density-wave states with spin-orbit coupling and Zeeman fieldThe hybrid system integrated with magnetism,spin-orbit coupling(SOC)and superconductivity is an important platform to realize topological superconductors in recent ten years.At present,people not only focus on the conventional s-wave topological superconductor,but also predict the possibility of the existence of unconventional pairdensity-wave topological states and MZMs.Here we consider a one-dimensional superconducting system with spin-orbit coupling,Zeeman field and PDW △0+△1 cos(qx).By calculating the pfaffian polynomial at the high symmetry points in the Brillouin region,we obtain the phase diagram of the system.We find a new kind of phase transition process,termed as sudden emerged phase transition,which takes place in an interior point of a different phase,instead of their phase boundaries.It arises from the root structure of the Pfaffian polynomials in the Bloch bands,which can be emerged or disappeared suddenly at any point in the parameter space by the fundamental theorem of algebra.We examine the appearance,fusion and disappearance of these phases,and determine their phase patterns in the parameter space.These results may open remarkable prospects in exploring more intriguing phase patterns with periodic potentials,including the possible phases in different topological classes and even in models with many-body interaction,which represent a broad class of models with unpredictable phase transitions in theories.3.Inverse proximity effect in a topological insulator-superconductor hybrid structureThe coupling between superconductor(SC)and topological insulator(TI)is of particularly important in the context of the proximity effect,in which the pairings can penetrate into the insulator,tuning the insulator into a topological SC,which has been an important paradigm for the realization of MZMs.However,much less attention has been paid to the inverse proximity effect(IPE),i.e.,the effect that the TI has on the SC pairing.Based on the recent experimental progress of the topological proximity effect and two-dimensional superconducting thin films,we study the IPE in the twodimensional superconductor-topological insulator hybrid structure,and propose a new picture to understand the influence of the substrate on superconductivity.Starting from the two-dimensional SC model with strong SOC,we make a self-consistent calculation of the SC pairing potential,and understand the results of the pairing potential from the density of states(DOS)on the Fermi surface.Then,considering the coupling between the ultrathin SC thin film(L=1-10)and topological surface state,we also use a selfconsistent calculation method to determine the SC pairing potential.Under the control of parameters such as SOC strength,interface coupling strength,the chemical potential of SC film and topological surface state,the SC pairing potential will be significantly enhanced or suppressed.We show the effect of the renormalized electron-phonon(eph)coupling constant on pairings is determined by two important mechanisms:the modified DOS from the occupied bands near the Fermi surfaces and the renormalized interaction from the unoccupied bands.Our results not only provide a theoretical basis for tuning the pairing in the SC by chemical potential and interface coupling,but also provide some new insight into the pairings in a large number of SC-substrate hybrid systems via the IPE.