| Chern insulator is an electronic system characterized by Chern number.It has quantized Hall conductance and protected gapless boundary state,and breaks the time inversion symmetry.Later,when the quantum spin Hall insulator was discovered,people realized that symmetry plays an important role in the classification of topological states.According to the classification scheme of time inversion symmetry,particle hole symmetry and chiral symmetry,we judge that topological insulators and superconductors with different symmetries belong to different categories,and they are not connected in topology.Physicists think it is meaningful to explore the possible connection between superconductor and insulator in the topological structure of electron.This connection makes it possible to find the Majorana fermions in superconductors.At present,in real life,people have not found the existence of Majorana fermion,but it can be derived from Majorana fermion in other ways.In condensed matter physics,people clearly recognize the importance of matter with topological band structure,and thus found new material states,such as topological insulator.Therefore,a series of models with topological nontrivial flat energy bands are proposed.These lattice models have topological nontrivial energy band,similar to Haldane model,and their bandwidth can be adjusted much smaller than the band gap,thus forming an almost flat energy band structureIn today’s society,insulators and superconductors with nontrivial topological structure have attracted and will continue to attract people’s great interest,and also have great research value.In this paper,we study the Chern insulator checkerboard lattice model with flat band and the honeycomb lattice model with d+id wave respectively.The specific contents are as follows:in the first chapter,we mainly introduce the development process of topological insulator and topological superconductivity,and in the last section,we mainly explain the reason of this paper In chapter 2,in section 2.1,the Hamiltonian equation,energy spectrum and boundary states of the checkerboard lattice spinfull Haldane model are introduced,and obtain that the model has a flat band with topological nontrivial and Chern number is C=2 In section 2.2,the mean field method is used to calculate the Hamiltonian equation and the self-consistent equation of ferromagnetism and antiferromagnetism respectively and draw their energy spectrum.We find that the degraded flat band becomes a dispersion band under the action of the antiferromagnetic order,but splits into two flat bands under the action of the ferromagnetic order.In Section 2.3,by calculating and comparing the ground state energy of ferromagnetic and antiferromagnetic order under different doping conditions,the overall phase diagram is obtained.It is found that there are seven phases:common metal,antiferromagnetic metal,ferromagnetic metal,Chern insulator with quantum anomalous Hall effect Chern numbe C=2,antiferromagnet C=2 Chern insulator,common antiferromagnetic insulator and ferromagnetic C=1 Chern insulator;In Section 2.4,the spin density wave formulas of ferromagnetic and antiferromagnetic order are calculated by the random phase approximation method,and the case of half filling d-0,quarter filling d=0.5 and other doping are analyzed in detail.The important conclusion is that when quarter filling d=0.5,the system becomes ferromagnetic C=1 Chern insulator It is also found that the spin density wave is dispersive in both ferromagnetic and antiferromagnetic order.In antiferromagnetism,the dispersion of spin wave presents linear behavior,while in ferromagnetic order,the dispersion of spin wave presents quadratic behavior.Finally,we make a concrete analysis and summary of the conclusion.In Chapter 3,we consider a chiral singlet states d+id wave superconductor on a honeycomb lattice model with nearest neighbor,next neighbor hopping and nearest neighbor pairing terms.In Section 3.1,we calculate the energy spectrum and the boundary state of the spinless Haldane model of honeycomb lattice when there are only the nearest and the next nearest neighbors,and obtain the boundary state with topological properties;In Section 3.2,the self consistent equations of the superconducting order parameters Δ and the chemical potential μ are calculated by introducing the d+id wave superconductor pairing on the nearest neighbor;In Section 3.3,the change of superconducting order parameters and gap with the interaction is drawn,and important conclusions are obtained:when the interaction is U=5 half filling,a boundary state in K space corresponds to a pair of Majorana zero energy modes in real space;In Section 3.4,the conclusions are analyzed and summarized.In the fourth chapter,the topological insulator and the topological superconductor are summarized and prospected. |
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