Symmetry In Topological Insulators And Topological Superconductors

Topological properties of condensed matter systems give rise to a surge of interest in recent years.Topological insulators are novel states of quantum matter which are chacacterized by a gap in the bulk and topological gapless edge states on the surface.Topological supercoductors have a full pairing gap in the bulk and indicate the appear-ance of zero-energy Majorana surface modes.The gapped states can be characterized by topological invariants which are constructed from the wave functions.In general,a topologically nontrivial phase can not adiabatically deform into a topologically trivial one,and it is robust under perturbations unless the bulk gap closes.In topological super-conductors,a Majorana fermion,being its own antiparticle,can appear at the defects.Majorana fermion exhbits non-Abelian statistics,which opens a way to fault-tolerant quantum computing.The signatures of Majorana bound states in topological supercon-ductors,such as zero bias conductance peaks and fractional Josephson effects,may be detected.For topological phases protected by global symmetries,an exhaustive classifi-cation has been obtained for arbitrary dimensions.The Hamiltonian of global symmetric model have time-reversal symmetry,particle-hole symmetry,or chiral symmetry.The crystalline symmetries can give rise to new topological phases of matter besides global symmetries.Among the crystalline symmetries,the mirror or rotation symmetries are used to define new topological invariants.Much efforts have been devoted to classi-fying the topological phases of matter protected by crystalline symmetries.A famous example of a topological material protected by mirror symmetry is the topological crys-talline insulator SnTe.The topological phase with an even number of Dirac cones on the surface is characterized by the non-zero mirror Chern number.Therefore,the crys-talline symmetries play an essential role in forming a new class of topological phases.The doctoral dissertation is organized as follows.In the first chapter,I give a brief review on the topological insulators and topologi-cal superconductors.In the aspect of topological insulator,we discuss quantum anoma-lous Hall effect(Haldane model)and quantum spin Hall effect(Kane-Mele model).The Kane-Mele model can be considered as the combination of two Haldane models.In the aspect of topological superconductor,we discuss one dimendional hybrid semiconductor-superconductor nanowires with strong spin-orbital coupling under magnetic field,and three-dimensional noncentrosymmetric superconductors.Besides Majorana edge states there exist vortex-bound states in the noncentrosymmetric superconductors.A research on one dimendional topological insulator in CII class is presented in chapter two.A brief introduction on one dimendional topological ladder insulators is given in the first section,such as double-ladder system.In the second section,we pro-pose a model of two-leg ladder belonging to CII class and study its topological edge states.In this model,spin-orbital couplings are presented in both intra-chain and inter-chain interactions.The ladder system is proved to be a topological insulator character-ized by 2Z invariant.We find that there are four-fold degenerate edge states protected by time-reversal and chiral symmetries.In contrast to two edge states spatially localized at one end of the ladder being not distinguished,these two edge states can be detected by the momentum density.The momentum density can be measured in time-of-flight images of ultracold atoms.The two peaks of momentum density become the signature of two edge states localized at one end of ladder.Additionally,these edge states do not obey non-Abelian statistics under braiding,because one unitary operator can not exchange two left edge states and two right edge states simultaneously.This is quite different from one-half charged fermions that exhibit non-Abelian statistics.With a uni-versal set of gates implemented,the quantum information can be transferred between the spin qubit and Majorana fermion qubit.However,our model does not support quan-tum computation with hybrid qubits formed by coupling soliton qubit and a quantum dot spin qubit.In the third section,We focus on the fractional charge that the edge state carries using both effective field theory and numerical calculation.Single edge state carries one half charge.In the fourth section,We propose the experimental scheme in the optical lattice to realize the ladder model.The spin-conserved hopping in the y direction is suppressed when there is a huge potential barrier along the y direction.The spin-flip hopping terms can be achieved using the Raman transitions.In the fifth section,when the magnetic field in x direction is applied,there exist two distinct topo-logical phases that exhibit four degenerate edge states and two degenerate edge states in the gap respectively in AIII class.And we also find that the edge state carries a half charge in both topological phases.Our work provides a complement to the research of one dimendional topological insulator.It reveals the distinct topological edge states,and gives the relation of topological insulators in CII class and in AIII class.In chapter three,we study the topological crystalline phases and edge states in two dimensional superconductors.In the first section,I give a brief introduction on the topological crystalline insulators and topological crystalline superconductors.In addi-tion to nonspatial symmetries in the Altland-Zirnbauer symmetry classes,crystalline symmetries,including mirror,inversion and rotation symmetry,play an essential role in forming a new class of topological phases.A typical example of topological crys-talline insulator is the semiconductor SnTe,in which the mirror symmetry allows one to introduce the non-zero mirror Chem number to characterize the topological phase.In the second section,we propose the topological classication for two dimensional time-reversal invariant superconductors with crystalline symmetries.The topological edge states can be classified as Majorana fermions and quasiparticles without particle-hole symmetry.In the third section,we investigate the possibility of realizing topological crystalline phases in two dimensional supercoductors preserving time-reversal symme-try with interorbital pairing.Four distinct topological phases and the corresponding gapless edge states are protected by mirror and mirror-inversion symmetry.The mirror-inversion symmetry denotes the combination of mirror and inversion symmetry.The topological edge modes in the AIII class rather than Majorana edge modes appear with a at flat dispersion in mirror-inversion parity subspace due to the absence of particle-hole symmetry within each subspace.On the contrary,Majorana edge modes arise in mirror parity subspace.The Majorana edge modes in each plane mirror parity subspace are identified through the diagnoses of the topology along either mirror invariant line or mirror-inversion invariant line.To understand the role of the mirror-inversion sym-metry alone played in topological superconductivity,we propose another model which possesses the mirror-inversion symmetry rather than other crystalline symmetries.In this model,there appear the topological edge modes in the AIII class along the mirror-inversion invariant line and no Majorana edge modes are permitted to exist.There-fore,the diffrence between mirror and mirror-inversion symmetry in classification of the topological superconductor depends on whether or not the particle-hole symmetry remains to be preserved in the subspace.We calculate the the momentum resolved local density of states(LDOS)of both the BdG Hamiltonian and the Hamiltonian in each subspace by using iterative Green function method on the surface layer of a semi-infinite system.Since the presented LDOS is not sufficient for characterizing the most important physics coming from the nontrivial topology,we check the robustness of the topological edge states under random perturbations in the fourth section,.In fact,the topological edge states is robust against the random perturbations.In the fifth section,we calculate the differential conductance of the NS junction in the mirror subspace us-ing non-equilibrium Green function method.The quantized value of 2e2/h at zero-bias voltage can verify the existence of the Majorana modes.Our work provides a direction to the material search of the topological crystalline superconductors.In the last chapter,we present a conclusion and the outlook for the effect of sym-metry on the formation of topological insulator and superconductor.