Study On Topological Phase And Domain Fractional Charge Of One - Dimensional Topological Insulator - SSH Model

The discovery of topological insulator motivates the enthusiasm of people. It is a great victory of Condensed Matter Physics. It is a cross section of conductor and insulator. Its bulk state is insulating state and can’t conduct, but its surface state or boundary state is metallic state which can conduct. People did the research about domain wall of one dimensional topological insulator before, but they didn’t consider the effect of spin-orbit coupling and magnetic. We use theory deduction and numerical calculation to research the property of one dimensional topological insulator. In the thesis, it consists these five parts:(1) One-dimensional SSH model of commensurability 2, it is a one dimensional topological insulator. We conclude that SSH model has two topological phases, when the system is in different phases, the winding number values of system are different. When winding number equals-1, boundary state appears, the system is in topological nontrivial phase. When winding number equals 0, boundary state will disappear, the system is in topological trivial phase. We can distinguish whether the system has boundary state according to the value of winding number, judge whether the system is in topological trivial phase or nontrivial phase.(2) One-dimensional SSH model of commensurability 2 with no spin has two different phases:topological trivial phase and topological nontrivial phase. When one half of a chain is in one phase, another half is in the other phase, there is a domain wall between them, a localized state will appear around domain wall. Now domain wall has a charge of 1/2, it is a fractional charge. When considering spin, its charge is 1.(3) For One-dimensional SSH model of commensurability 2, if we consider spin orbit couping and marnetic field, comparing with only spin, the charge of domain wall comes back to 1/2.(4) One-dimensional SSH model of commensurability 3 with no spin has three different phases. When a half of a chain is in one phase of 3, the other half is in another phase of 3, we discover that the charge of domain wall is 1/3. When considering spin, the charge of domain wall is 2/3, comparing 1/3, it is twice.(5) For One-dimensional SSH model of commensurability 3, if we take spin orbit coupling and magnetic field into consideration, these gaps will change with different coefficients of spin orbit coupling and magnetic field, they will result in the changes of gaps, the gaps have a process from opening to closing, from closing to opening, it means that quantum phase transition will appear. What’s more, with the transition of phase, the charge of domain wall will change too. When fermi surface is in some gaps, the charge will transform between 1/3 and 2/3. When fermi surface is in some other gaps, the gaps just change from opening to closing, but no closing to opening, at the moment, the charge of domain wall is 1/3. When fermi surface is in different gaps, the topological properties are different, the charges of domain wall are different too.