Study On Spin Chern Numbers And The Topological Properties Of Two-dimensional Topological Insulator

In the past few years, inspired by the remarkable discovery of the quantum Hall effect (QHE), a number of theorists suggested that new classes of topological states of quantum matter might exist in nature, without spontaneously broken symmetry. Re-cently, the research of nontrivial topological states of matter has been highly stimu-lated by the discovery of new classes of materials called topological insulators (TIs). A key ingredient to the QSH effect is the strong spin-orbit coupling (SOC), and it plays an important role similar to the external magnetic field in the QHE. Contrary to an external magnetic field, however, the SOC does not break time-reversal sym-metry (TRS). The QHE systems are insulating in the bulk, but on the boundary they have gapless edge (in a two-dimensional case) or surface (in a three-dimensional case) states that are topologically protected and immune to impurities or geometric pertur-bations. A two-dimensional (2D) material, graphene, was first predicted to exhibit time reversal invariant topological states and quantum spin-Hall effect. Electrons in graphene have two inequivalent Dirac cones and additional twofold spin degeneracy, and obey a two-dimensional Dirac-Weyl equation for massless particles in the vicin-ity of the Dirac points. The time-reversal invariant topological insulators have been experimentally observed in HgTe/CdTe quantum wells. A3D TI has a bulk band gap and gapless surface state on the sample boundary. The strong3D TI is topologically nontrivial in any direction, and the surface states are characterized by an odd number of Dirac cones. The3DTI thin films also have unique physical properties, and provide a very promising system for studying various topological effects. There have been pre-dicted proximity-induced ferromagnetism or superconductivity in TIs, which may lead to many exotic phenomena such as the topological magnetoelectric effect and Majo-rana fermions. These systems will have important practical applications in spintronics and topological quantum computing. The existence of edge or surface states in the TIs does not depend on the geometric condition on the boundary, but comes from the topological characteristic of the bulk. The TIs can be distinguished from a trivial insu-lator by the nonzero topological invariant. The nontrival bulk band topology of a2D TI or QHS system is usually described by the Z2index or the spin Chern numbers. To deeply understand the topological properties of the TIs, the discrete formulation of the topological invariants and an efficient way to calculate them are highly desirable.The present dissertation consists of five chapters.In Chapter1, we give a brief introduction to the related theoretical and experi-mental research on the2D and3D topological insulators, including the2D Kane-Mele model for graphene, the2D HgTe/CdTe quantum well, the3D TI, the3DTI thin films, and several topological invariants. We also briefly review some theoretical methods of calculating the topological invariant, which can be used to determine the topological nature of insulating systems.In Chapter2, we study the topologically invariant isospin Chern number for thin films of3DTI, Bi2Se3, in which the structure inversion asymmetry gives rise to gapped Dirac hyperbolas with Rashba-like splitting in the energy spectrum. It is shown that the isospin Chern number can be used to describe the topologically trivial and nontrivial phases and related quantum phase transitions, in the3DTI thin film where the struc-ture inversion symmetry is broken and the isospin operator is not conserved. A phase diagram of the3DTI thin films is obtained by the calculation of the isospin Chern numbers. The calculated result indicates that a strong structure inversion asymmtry al-ways tends to destroy the QSH state, and at a critical value of the structure inversion asymmetry there is a quantum transition from a nontrivial topological phase to a trivial phase. Clearly, for the system with TRS, the isospin Chern number and the Z2index lead to equivalent identification of the quantum phase. Then proceed to the next step, we study the topological quantum phase transition in the3DTI thin film in the presence of an electrostatic potential and a spin-splitting Zeeman field, the former breaking the inversion symmetry and the latter breaking the TRS. It is shown that there exist a quan-tum pseudospin Hall phase, a quantum anomalous Hall (QAH) phase, and a normal insulator phase in this system. By tuning the electrostatic potential and Zeeman field, three quantum phase transitions between them can occur accompanied with energy-band closing, each of which is characterized by a change of the quantized pseudospin Chern numbers. Very interestingly, the quantum pseudospin Hall state still exists in a range of Zeeman field, where the TRS has been broken. We further construct a tight-binding Hamiltonian of electrons on a honeycomb lattice to calculate the edge states of an armchair ribbon of N dimer chains for different phases. The calculated results are in good agreement with the edge-state picture of various quantum topological phases.In Chapter3, we show that a thin film of3DTI with a Zeeman exchange field is a realization of the famous Haldane model for quantum Hall effect (QHE) without Landau levels. The exchange field plays the role of staggered fluxes on the honeycomb lattice, and the hybridization gap of the surface states is equivalent to alternating onsite energies on the AB sublattices. As a result, the QHE in a thin film of the3DTI is equiv-alent to that in the Haldane model. The Hall conductivity is numerically calculated by use of the standard Kubo formula. A peculiar phase diagram for the QHE is predicted in the3DTI thin films under an applied magnetic field, which is quite different from those either in traditional QHE systems or in graphene.In Chapter4, we propose a topological understanding of general characteristics of edge states in a quantum spin Hall phase. Based upon a general topological argument without relying on the TRS or other symmetries, we show that as required by the non-trivial spin Chern numbers and the gauge invariance in a QSH system, edge states must appear near the sample boundary, for which either the energy gap or the gap in the spec-trum of the projected spin operator PszP needs to close on the edges. To demonstrate such a general argument, we take the Kane-Mele model for a honeycomb lattice ribbon as an example, by taking into account a uniform Zeeman field and different confining potentials near the edges of the ribbon. It is found that a TRS-broken QSH system can have either gapless or gapped edge states, depending on whether the confining potential near the boundary is of soft-wall or hard-wall type. Most prominently, corresponding to relatively soft-wall boundary potential, the gapless edge modes can remain to be robust and immune to backscattering in spite of the breaking of the TRS. It then follows that the gapless edge states are protected by the bulk topological invariant alone, rather than any symmetry.It is known that the experimental difficulty to detect the dissipationless edge states is that the two opposite movers have identical spatial probability distributions, and so presence of small TRS-breaking perturbation may couple the two edge states, giving rise to backscattering. This makes the QSH effect fragile in realistic environments, where perturbations violating the TRS are usually unavoidable. To strengthen the ex-perimental visibility of the transport measurement, we propose in Chapter5a new route to realize a robust QSH effect by deliberately breaking the TR symmetry of the edge states via magnetic manipulation. It is shown that creating a narrow ferromagnetic (FM) region near the edge of a QSH sample can push one of the counter-propagating edge states to the inner boundary of the FM region, and leave the other at the outer boundary, without changing their spin polarizations and propagation directions. The narrow FM regions near the edges can be realized either by doping of magnetic atoms such as Mn or by utilizing the proximity effect of the FM insulator. Since the two counter-propagating edge states are spatially separated into different lanes, the QSH effect becomes robust against symmetry-breaking perturbations.Finally, Chapter6presents a summary of this dissertation, and gives some outlook for the investigation.