Topological Characteristics Of Double-chain System And Quantum Walk

The topological characteristics of the physical system is a hot researching topic.The energy band structure of the system determines the topological properties,which are usually described by topological invariants.In this thesis,some basic concepts of the topological quantum system are firstly introduced,and the topological characteristics of the SSH model are analyzed by the method of the topological invariants and energy band theory.Then,on this basis,we research the topological characteristics of the double-chain SSH model and behaviors of single-particle quantum walk on it.The entanglement Hamiltonian of the subsystem is defined by the reduced correlation matrix of the composite system,and the entanglement Chern number is calculated by its eigenfunction.We study the topological properties of the double-chain SSH model by entanglement Chern number.The inter-chain coupling changes the topological properties of the system.In the absence of inter-chain coupling,the entanglement Chern number jumps from 0 to 1 with the variation of the lattice dimerization parameter.Meanwhile,the system changes from a non-trivial topological state to a topological state.After considering the inter-chain coupling,the entanglement Chern number jumps twice,which enriches the topology of the system.Contrasting to the energy spectrum of the system,it is found that the transition point of entanglement Chern number is corresponding with the degeneration of the energy between the two bands.The eigen states with zero energy in the topological regions are edge states.Compared with the winding number of the total system,it is found that the entanglement Chern number coming from the reduced correlation matrix also can be used to describe the topological characteristics of the double-chain SSH model.Due to SSH model has chiral invariance,the average chiral displacement is defined.It is found that its variance with dimer parameter at some moment is similar to that of winding number.So it can also be used to dynamically describe the topological properties of the system.The single-particle quantum walk on the lattice space is affected by the topology of the system.When there is zero-energy bound state in the band gap,that is to say,the system is in a topological state,the particle starting from the boundary of the finite lattice system will be bound on the boundary with certain probabilities.The bound strength is related to coupling parameters between adjacent grid points.When the system is in a non-topological state,the particle probabilities at starting boundary disappears greatly with the time evolution of the system.