| The quantum Hall(QH)effect in a two-dimensional(2D)electron gas under a strong magnetic field provided the first example of topological state of matter in con-densed matter physics,which cannot be described by the Landau theory of symmetry breaking.Thouless,Kohmoto,Nightingale,and Nijs(TKNN)revealed that the essen-tial character of a QH insulator,different from an ordinary insulator,is a topological invariant of occupied electron states or many-body wavefunctions.They related the Hall conductivity of the system to the first Chern number(or TKNN number),which is quantized when the Fermi level lies in an energy gap between Landau levels.In such systems,topological phase transitions can happen only by closing the energy gap.Gapless edge states must appear on the boundary between a QH insulator and an or-dinary insulator,which is ensured by the topological invariant.Interestingly,Haldane proposed a spinless electron model on a 2D honeycomb lattice with staggered magnetic fluxes to realize the topological QH effect without Landau levels.The quantum spin Hall(QSH)effect was first theoretically predicted by Kane and Mele and by Bernevig and Zhang,and then experimentally observed in HgTe quantum wells.Unlike the QH systems,where time reversal(TR)symmetry must be broken,the QSH systems preserve the TR symmetry.The main ingredient is the existence of strong spin-orbit coupling,which acts as spin-dependent magnetic fluxes coupled to the electron momentum.The QSH state is characterized by a bulk band gap and gapless helical edge states on the sample boundary.The existence of the edge states is due to nontrivial topological properties of bulk energy bands.However,the bulk band topology of the QSH systems cannot be classified by the first Chern number,which always vanishes.Instead,it is classified by new topological invariants,namely,the Z2 index or the spin Chern numbers.For TR-invariant systems,both Z2 and spin Chern numbers were found to give an equivalent description.The robustness of the Z2 index relies on the presence of the TR symmetry.In contrast,the spin Chern numbers remain to be integer-quantized,independent of any symmetry,as long as both the band gap and spin spectrum gap stay open.They are also different from the first Chern number for the QH state,which is protected by the bulk energy gap alone.The spin Chern numbers have been employed to study the TR-symmetry-broken QSH effect.The QSH system is an example of the 2D topological insulators(TIs).Its gener-alization to higher dimension led to the birth of 3D TIs.A 3D TI has a bulk band gap and surface states on the sample boundary.The metallic surface states provide a unique platform for realizing some exotic physical phenomena,such as Majorana fermions and topological magnetoelectric effect.The 3D TIs have been experimentally observed in Bi1-xSbx,Bi2Te3,and Bi2Se3 materials,which greatly stimulates the research in this field.The 3D TIs with TR symmetry are usually classified by four Z2 indices,and are divided into two general classes:strong and weak TIs,depending on the sum of the four Z2 indices.In the presence of disorder,while the weak TIs are unstable,the strong TIs remain to be robust.The Z2 indices are essentially defined only on the TR-symmetric planes in the Brillouin zone,and do not provide information about the distribution of surface states in the full momentum space.When the TR symmetry is broken,the Z2 indices become invalid.Therefore,a more general characterization scheme for the bulk band topology,which does not rely on any symmetry and can provide more information about the distribution of surface states,is highly desirable.In this dissertation,we will discuss topological phase transition in the 2D and 3D TI model without TR symmetry by calculating the spin Chern number.The dissertation consists of five chapters:In chapter one,we give an introduction for the related experimental and theoretical background,theoretical methods and a brief outline of some fundamental conceptions.In chapter two,we investigate the fate of the QSH effect in the presence of the Rashba spin-orbit coupling and an exchange field,which break both inversion and TR symmetries.We calculate the spin Chern number Cs analytically and use this integer invariant to distinguish different topological phases in the model with breaking TR symmetry.It is found that the QSH state characterized by nonzero spin Chern numbers C ± = ±1 persists when the TR symmetry is broken.A topological phase transition from the TR-symmetry-broken QSH phase to a quantum anomalous Hall phase occurs at a critical exchange field,where the bulk band gap just closes.It is also shown that the transition from the TR-symmetry-broken QSH phase to an ordinary insulator state cannot happen without closing the band gap.In chapter three,topological phase transitions in a three-dimensional(3D)topo-logical insulator(TI)with an exchange field of strength g are studied by calculating spin Chern numbers C±(kz)with momentum kz as a parameter.When |g| exceeds a critical value gc,a transition of the 3D TI into a Weyl semimetal occurs,where two Weyl points appear as critical points separating kz regions with different first Chern numbers.For |g|,Cv(kz)undergo a transition from ± 1 to 0 with increasing |kz|to a critical value kzc.Correspondingly,surface states exist for |kz|<kzc,and vanish for|kz|? kzc.The transition at |kz|= kzc is accompanied by closing of bulk spin spectrum gap rather than energy gap.In chapter four,we theoretically investigate the magnetoresistance effect of an u1-trathin Bi2Se3 film sandwiched between two ferromagnetic insulators(FIs).It is found that the conductance is quantized to be e2/h and vanishing,respectively,for parallel and antiparallel magnetization configurations of the two FIs,which stems from a tran-sition of the Bi2Se3 film from the quantum anomalous Hall phase to a conventional insulator.This quantum magnetoresistance is robust against disorder scattering.The last chapter presents a summary of this dissertation,and then gives some outlook for the investigation. |
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