Dynamical Bulk-edge Correspondence For Discrete Quantum Systems

Discrete quantum systems have been the cornerstone of quantum theory,due to their analytical and numerical tractability comparing with the continuum quantum systems.Science and technology progress increasingly permitted the experimental investigation on the regime of discrete quantum systems.Through investigation on discrete quantum systems people are able to capture counterintuitive physical phenomena more easily,such as Bloch oscillation,bound atom pairs in repulsive potentials,etc.In recent years,optical lattice,photonic crystal,coupled cavity array are hot topics.To ensure the reality of the spectrum and the unitary time evolution,traditional quantum mechanics requires the system to be Hermitian.In the 90's of 19 th century,Bender and his collaborators found that non-Hermitian Hamiltonian with parity timereversal(PT)symmetry can have an entirely real quantum mechanical energy spectrum and also the unitary time evolution,which attracts extensive attention the scholars all of the world.This thesis aims to introduce some advanced concepts in condensed matter physics and quantum physics.In Hermitian systems,we establish the relationship of topological invariants and dynamical bulk-edge correspondence,which also show the way of finding it experimentally.However,we are not content with these.We devote ourselves to generalize the ideal of dynamical bulk-edge correspondence in non-Hermitian systems.We believe that there are some findings can contribute to the mansion of physics.