Topological Quantum Phase Transitions Of Chern Insulators In Disk Geometry

In 1988 Haldane proposed a toy model in a crystal lattice and realized the topological insulator in the lattice.The topological properties of the Haldane model are usually characterized by Chern number,and we refer to the topological insulators like the Haldane model as the Chen insulators.It has been extremely hard to realize the Haldane model experimentally in ordinary condensed matter systems because of the unusual staggered magnetic flux assumed in the model.Until recently the model was experimentally implemented in shaking optical lat-tices.However an infinite system is impossible to achieve,the study of finite size is very important.This paper takes Haldane model as an example to expand in finite size.We investigate the topological quantum phase transitions of Chern insulators on disk geometry with 6-fold rotational symmetry.We first verified the high number phase diagram obtained by Doru Sticlet considering the distant hopping parameters.Considering the nearest-neighbor hopping parameters,the next-nearest-neighbor hopping parameters,even the next-next-nearest-neighbor hopping parameters and the staggered flux parameters of Haldane model,we can obtain rich topological quantum phases with large Chern numbers.These topo-logical quantum phases can be identified based on the edge states,the density of energy level and the local density of states.On the basis of the original disk geometry,we add the trap potential which only depends on the radius of the disk geometry.The phase diagram about trap potential is obtained.Furthermore,we use machine learning as an effective way to automatically identify different phases and phase diagrams for the Haldane model on the disk with the hopping and the staggered flux parameters.This article is divided into six parts,the article is arranged as follows.In Sec.1,we mainly introduce the research background of this paper,briefly state the origin and development of Haldane model,the implementation of Haldane model in experiment,and relate knowledge of machine learning.In Sec.2,we introduce the basic knowledge of Haldane model in the periodic boundary conditions and some basic concepts and characteristics of Haldane model on disk geometry with 6-fold rotational symmetry in the opening boundary conditions.We verify the phase diagrams of the Haldane model adding the long-distant hopping parame-ters,find the characteristics of edge states in periodic boundary conditions and obtain the topological quantum phase of the Haldane model with the large Chern number on disk geometry.In Sec.3,we gain the phase diagram for(?)—t2 and the large Chern number phase.diagram for(?)—t3 according to the Kubo formular for comparison.The corresponding law of the energy spectrum and the number of Chern is found.The effect of size effect on topological quantum phase is s-tudied.In Sec.4,we add the trap potentialwhich only depends on the radius of the disk geometry,like the harmonic potential,linear potential and so on.With the addition of the trap potential,the topology property of the system will be destroyed owing to the appearance of the disturbance state.We obtain the phase diagram about trap potential.In Sec.5,we use machine learning to automatically identify topological quantum phase and topological quantum phase diagrams on disk geometry according to the characteristics of the energy spectrum.We utilize softmax regression within a few lines of code,a multi-classification in supervised machine learning,to mostly recover the phase diagrams on disk geometry with 6-fold rotational symmetry.Sec.6 is the summary of master’s thesis and the prospect of the development and exploration of the subject.