Theoretical Study On The Properties Of Zero Modes In The Non-hermitian Lattices

Recently,non-Hermitian systems have been greatly studied both in experimental and theoretical fields.It has been revealed that non-Hermititian can greatly alter the topological behaviors that were established in the Hermitian cases,such as the edge modes influenced by gain and loss distributions.On the other hand,topological insulator exhibits an insulating bulk and gapless edge states under open boundary conditions(OBC),which can be characterized by a topological invariant,such as the Su-Schrieffer-Heeger(SSH)model.The chiral symmetry of the SSH model leads to nontrivial topology which can be probed by the winding number and the presence or absence of two-fold degenerate zero modes in the thermodynamic limit under OBC.In fact,the experimental realization of the topological systems is usually of finite size.Due to the boundary coupling in a finite system,the zero modes will deviate from exact-zero energy.Therefore,in this thesis,we mainly study the recovery of topological zero modes in a finite systems.We first introduce the P operator,the T operator,the PT combination operator and the properties of each operator.Then we introduced the traditional SSH model,the one-dimensional SSH model may be the simplest two-band topological system initially introduced to study the polyacetylene.Due to the boundary coupling in a finite system,the zero modes will deviate from exact-zero energy.Song et al.suggested that one can reduce the coupling of the boundary modes and recover the zero modes in a finite system by increasing the alternating gain or loss strength in a one-dimensional parity-time(PT)symmetric SSH model.Therefore,in this article,we mainly study whether the topological zero modes can always be recovered by the PT-symmetric potential in a small system case.we study the edge modes of the SSH model with a pair of balanced gain and loss at two end sites in both large and small size cases.In the cases,we find the zero modes can be recovered by modulating the gain or loss strength.However,the failure of the recovery of the edge modes for the SSH model with another types of PT-symmetric potentials in the small size limit.Our results indicate that the energies of the edge modes in the infinite size case decide whether or not the success of the recovery of the zero modes by tuning the strength of PT-symmetric potential in a finite system.If the energies of the edge modes amount to zero in the thermodynamic limit under an open boundary condition(OBC),the recovery of the zero modes will break down by increasing the gain or loss strength for a finite system.Finally,we discuss the edge modes of a spin-orbit coupled SSH chain with PT-symmetry and get the same conclusion.Our results can be easily simulated in a silicon waveguide platform with controlled gain or loss.