| During the past decades,topological quantum phases beyond the Landau-Ginzburg-Wilson phase transition fr amework have not only fundamentally updated our knowledge of the phases of matter but also have become an important research direction in condensed mat-ter physics.Under the equilibrium state theory,one can characterize and classify the topo-logical quantum phases according to the bulk topological invariants defined on the ground state.Due to the existence of the bulk-edge correspondence,the topological phases have the edge states that are immune to local defects and disordered perturbations at the same time.In the experiment,researchers use the angle-resolved photoelectron spectroscopy,scanning transmission electron microscopy and r tansport measurement to detect the edge state,and then characterize the topological properties of the system.However,the experimental detec-tion of local edge states is not a direct measurement of non-local bulk topological invariants.Moreover,measuring the topological properties of equilibrium states requires great care to initialize the system.Considering the limitation of the equilibrium theory in the charac-terization of topological phases,based on the development of cold atom technology,the nonequilibrium dynamical characterization of topological phases has attracted considerable attention in recent years.In this thesis,we will focus on fi nding simple and efficient dy-namical characterization schemes for different topological phases based on experimentally measurable spin texture in the nonadiabatic evolution of the system.This thesis mainly includes the following contents:（1）The dynamical characterization of the Z-type topological phase and the Z2-type topological phase in slow quench are studied.We fi rst propose an exactly solvable Landau-Zener model that can be directly applied to the dynamical characterization of the topological phases.Then we present two different schemes to characterize the bulk topology of the sys-tem based on the so-called spin inversion surface.The i frst one needs only one quenching process,but requires measuring the gradients of spin texture on the spin inversion surface.The second one only needs to directly measure the value of spin texture on the spin inver-sion surface,but requires introducing other quenching processes.Moreover,high-order spin inversion surface or band inversion surface relying on the dimension reduction approach can also characterize the bulk topology of system.The highest order band inversion surface and spin inversion surface are a series of paired points,which greatly simplify the characteriza-tion strategy.Compared with the sudden quench,we demonstrate that the bulk topology can be captured not only by the band inversion surface,but also by the spin inversion surface.In particular,direct characterization of topological phases based on band inversion surface and spin inversion surface can be realized.（2）The dynamical characterization of layered systems in sudden quench is studied.After analyzing the spin textures of two different stacking models,we find that the terms in the Hamiltonian,which anticommute with other terms,can have common band inversion surface in the two stacking models.However,the terms,which do not anticommute with other terms,should be treated case by case.In addition,for the stacking model that can be block diagonalizable,the topological number can be accurately captured by the dynamical field on the band inversion surface.However,for the stacking model without block diago-nalization,the dynamical field on the band inversion surface does not reflect the exact value of the topological number,but only reflect whether the system is topological or not.Finally,the magnitude of the interlayer transition can also be given by the spin texture itself,which helps us intuitively detect the magnitude of the interlayer transition through the spin texture experimentally.（3）The dynamical characterization of different types of quenching processes in slow quench is studied.We study not only the processes between nontrivial phase and trivial phase,but also between the phases with different topological invariants.We find that the initial topological phase can also be characterized.For an arbitrary quenching process,no matter what intermediate phase the quenching process undergoes,the spin texture only records the topological information of the initial phase and final phase.Moreover,different types of quenching processes can be distinguished from the configurations of band inversion surface and spin inversion surface in the spin texture.Compared with the sudden quench,the processes in slow quench from a topological phase to a trivial phase can always be distinguished from other types of processes. |
