Excitations In Topological Insulators

An important goal of condensed matter physics is to find new phases of matter.Usually phases and phase changes can be described by the phase transition theory of Landau-Ginzburg,i.e,symmetry-breaking.However,in 1980 it was discovered by Klitzing et al.that the Hall conductance is an integer multiple of a fundamental constant of nature.In 1982,Thouless et al.found that the expression for Hall conductivity is a topological invariant,the so-called Chern numbers.This new phase of matter described by topology other than broken-symmetry is a topological order.C.L.Kane and S.-C.Zhang et al.independently developed the theory of topological insulators respectively from the viewpoints of topological band theory and topological field theory,where quantum Hall and quantum spin Hall systems are typical examples.Topological insulators give a new phase of matter,which have gap states in the bulk and gapless states on the surface.This thesis contains the following aspects:1.Basics of the theory of topological insulators.2.A brief review of the theory of gauge potential decomposition and topological currents developed by the school of Professor Yi-shi Duan.This serves as a basis for the study in the following chapters.3.Within the topological field theoretical framework of topological insulators,the inner structures of gauge potentials are obtained.4.We appeal to the 't Hooft monopole model and construct a Maxwell electromagnetic sub-field of the SU(2)gauge field.In such a model there exist two types of first Chern classes,which lead to the Berry-type vortices in two dimensions and monopole defects in three dimensions.It is addressed that the present popular theory of topological insulators gives topological numbers,like the Chern numbers,only in the integration form,and cannot link the numbers to topological defects directly.In contrast,our approach is able to go a step further and demonstrates the differential structures of the defects.5.As an example,the two-band model is considered,which serves as an typical evaluation of the unit vector of hamiltonian in the group space.Monopoles and vortices are located in this concrete model,their topological charges being recognized via illustrations.