Study On Topological Phase Transitions And Topological Properties In Hermitian And Non-Hermitian Su-Schrieffer-Heeger Models

A persistent theme in condensed matter physics is to discover and classify topological phases of different substances.Among them,a series of discoveries related to quantum Hall effect establish the relationship between topology and energy band structure of matter,and lay the topological phase classification paradigm based on the concept of topological order.As an emerging research direction in the field of condensed matter physics,topology has not only profound physical connotations in theory,such as bulkboundary correspondence relationship and chiral symmetry,but also contains gapless boundary states,which are immune to disorder and protected by topology.Moreover,the research of topological systems has penetrated from the closed systems described by Hermitian Hamiltonian to the open systems characterized by non-Hermitian Hamiltonian.Compared with Hermitian topological systems,non-Hermitian topological systems exhibit more abundant and novel topological properties and topological classification.On the other hand,the realizability of topological systems on many experimental platforms further promotes the progress of different types topological systems.In recent decades,a variety of simulations and researches on topological insulator experimentally make it available to observe topological phase transition in higher dimensional system.In this thesis,we firstly investigate the prominent non-Hermitian bulk-boundary correspondence and non-Hermitian skin effect induced by asymmetric coupling,and reveal the relationships between the eigenstates localization and the system size and the asymmetric coupling.Moreover,based on cavity optomechanical system with periodical modulation of cavity fields,we explore the topological phase transition between topological trivial phase and nontrivial phase and the enhanced topological effect.The specific research contents are as follows:Based on a one-dimensional generalized non-Hermitian Su-Schrieffer-Heeger(SSH)lattice,we firstly investigate the effect of asymmetric coupling on the energy spectrum of the system.By analyzing the energy spectra with and without asymmetric coupling under open boundary conditions,finding that the non-Hermitian phase transition occurs because of the existence of the asymmetric coupling.It is found that the points where zero energy eigenvalues appear under open boundary conditions are not merged with the band-gap close points under periodic boundary conditions and the traditional bulk-boundary correspondence relationship breaks down in non-Hermitian systems.In order to resolve this problem,in virtue of the non-Hermitian winding number under the generalized Brillouin zone(GBZ)theory,we obtain the bulk-boundary correspondence relationship in nonHermitian systems,and depict the non-Hermitian topological phase transition boundary with different system parameters.Meanwhile,compared with Hermitian topological systems,it is also found that the asymmetric coupling leaves the eigenstates of non-Hermitian topological system all localizing at same boundary of the system.Moreover,in virtue of the mean inversion participation ratio,we also explore the relationship between the eigenstates localization and the system size and the asymmetric coupling strength.It is found that large asymmetric coupling strength and system size leave the system in localized states.Additionally,for the asymmetric coupling strength and the system size,the eigenstates localization is much more sensitive to the asymmetric coupling strength.Finally,we reveal the relationship between next nearest-neighbor coupling strength and the eigenstates localization of the system.Based on cavity optomechanical system with periodical modulation of cavity fields,we explore the topological phase transition and enhanced topological effect under steadystate regime.Through investigating the Langevin equation,we obtain the steady-state dynamics of system,and provide the restricted conditions.Under steady-state regime and restricted conditions,we realize modulating the cavity optomechanical system to topological trivial SSH phase and nontrivial SSH phase by adjusting the decay rate of cavity fields and optomechanical couplings.Meanwhile,in virtue of the effective optomechanical couplings and the energy spectrum and probability distribution of gap states,it is found that the phase transition between topological trivial phase and nontrivial phase can be realized via adjusting optomechanical couplings and the decay rate of cavity fields.Moreover,in order to make up the unapparent localing effect of gap states under topological nontrivial phase,two feasible approaches are provided to realize the enhanced topological effect of gap states.