Theoretical Studies On Topological Phase Transitions In One-dimensional Quantum Systems

The study of topological phase is the most important frontier hotspot in the field of condensed matter in recent years.There are two core issues,the first is to find and construct the non-trivial topological phase of the system,and the second is to explore the bulk-boundary correspondence.Among all kinds of quasiparticles,excitons are fundamental composite quasiparticles,among which indirect excitons have high cooling rate and long lifetime and are most likely to achieve exciton condensation.In addition,the control of exciton motion by periodic potential is experimentally realized,which provides the possibility to realize topological phase.Considering the complex interactions of excitons systems,it is necessary to theoretically explore whether indirect excitons have nontrivial topological phases.Moreover,the study of the topological phase of the indirect excitons system can provide theoretical guidance for the experiment.In addition,previous studies of topological phases have mainly focused on Hermitian systems,however many physical systems have non-Hermitian properties and can be described by non-Hermitian Hamiltonians.Non-Hermitian systems show many special behaviors that are different from Hermitian systems,especially the bulkboundary correspondence is broken due to the non-Hermitian skin effect.The topological invariants are currently redefined in the generalized Brillouin zone,but the solution in the generalized Brillouin zone requires a lot of numerical computation.Therefore,it is very important to explore analytical methods for realizing bulk-boundary correspondence in nonHermitian systems.First of all,an experimental method for the realization of topological phase by indirect excitons in semiconductor coupled quantum wells is proposed.The research focuses on the spin-orbit coupling of electrons and holes and the modulation effect of Zeeman field on topological phase.It is found that there is a pair of zero-energy boundary states can survive due to the protection of time-reversal,particle-hole,chiral,and inversion symmetries.The topological invariant of the system can be characterized by the winding number,which can be used to distinguish the different topological regions of a band.It was also found that a uniform(staggered)perpendicular Zeeman field can turn trivial regions into nontrivial through a topological phase transition.On the basis of this research,the polarization of the excitons in the topological phase is analyzed.Secondly,it is proposed that any non-Hermitian model has a topological counterpart.By equating the non-Hermitian skin effect to the AB effect,the imaginary magnetic flux generated under the periodic boundary condition will disappear under the open boundary condition,thereby eliminating the non-Hermitian skin effect,and a topological counterpart is constructed.And we use the SSH model to illustrate.The topological counterpart can be discussed in the Brillouin zone,and the analytical form of the winding number is obtained.A non-Hermitian model with domain structure composed of two chains is discussed in this way.And the analytical solution agrees with the numerical solution.This method provides a new way to study the topological properties of non-Hermitian systems.Finally,the topological properties of the non-Hermitian SSH model with spin-orbit coupling are investigated.In the model with Dresselhaus spin-orbit coupling,a topological counterpart is constructed taking into account the relationship between the non-Hermitian skin effect and the non-Hermitian AB effect.And the analytical form of the phase transition point of the topological counterpart is obtained.In the model with Rashba spin-orbit coupling,the rigorous solution to the phase transition point is obtained by solving the eigenequation in the generalized Brillouin zone.