Research Of Low Dimensional Topological Insulation Phase And Phase Transition

In 1973,J.Michael Kosterlitz and David J.Thouless first proposed the concept of topological phase and topological phase transition of matter in a two-dimensional condensed matter system.Different from the Landau phase transition theory of common materials,the topological phase of topological materials was characterized by the topological invariants of the system's energy band characteristics.Topology,as a mathematical concept,it mainly presented a property that the geometric type of object keeps constant in the continuous deformation process.In condensed matter physics,the topological properties of matter made some related states stable under environmental disturbances.For example,unlike the mediocre insulators,the most intuitive representation of topological insulators was the conductive surface states caused by topological properties.The topological order was characterized by measuring the degeneracy of the ground state.The topological quantum states would have some kind of quantum entanglement,etc.The method used to characterize the internal characteristics of the system is the focus of our research.In this paper,the theoretical calculation was used to derive the eigenvalues of the momentum space,and the energy band diagram was drawn.The accurate numerical simulation method was used to calculate the quasi-particle spectrum,edge state,entanglement spectrum,von Neumann entropy,excitation energy gap,etc.The physical quantity was a comprehensive analysis of the specific conditions of these criteria.Firstly,we used the method of density matrix product state to studied the topological quantum phase of the Fermi model with the open-boundary condition in the optical lattice in the next-dimensional chain.The study found that the entanglement spectrum was in the system topology state.In the process of double degeneracy,the entanglement spectrum was destroyed and the entanglement spectrum was destroyed.At the same time,the Von Neumann entropy had a sharp peak.We could infer that the point was a phase transition point.In order to verified the correctness of the conclusion more accurately,a quasi-particle energy spectrum was performed in the topological phase interval,and there were two degenerate modulo localities on the boundary of the chain.At the same time,the excitation gap also had a gap closure point at the phase transition point.Combining all physical quantities,this point was the phase transition point of the transition of the system topology to the non-topological phase.Secondly,we studied the topological properties of the Bose-Hubbard system in the half-filled state of the one-dimensional sawtooth lattice.We still used several methods to determine the topological phase transition of the system,and proved that the system had a topological non-mean Moment insulation state.It showed that degeneracy of the entanglement spectrum was not a sign of the judgment topology.Finally,we used the Fermion lattice model driven by spin-orbit coupling in a one-dimensional sawtooth lattice to conduct in-depth research on other physical quantities,and analyzed the quasi-particle spectrum at a fill factor of 1/4(or 3/4),although the system exists.Edge state,but calculations showed that the system did not have the characteristics of a topological state.Proved that the quasi-particle spectrum is not necessarily a sign of judging the topology.At the same time,in order to verify the influence of the sawtooth lattice on the model,we had calculated that the Fermi-Hubbard model had no topological phase transition in the one-dimensional chain lattice for the simplest Fermi-Hubbard model.By placing the model in a sawtooth lattice,the quasi-particle energy spectrum also exhibited an energy gap and a boundary state,indicated that the energy gap was caused by a sawtooth lattice.