Kibble-Zurek Mechanism And Defect Production In 1D Topological Model

Kibble-Zurek mechanism(KZM)describes the non-equilibrium dynamics and the formation of defects in a system which quenches across a continuous phase transition point at finite rate.This theory is widely applied in transitions of liquid He,super-conductors,ultracold atoms,etc.,even early universe.By means of adiabatic-impulse approximation,KZM provides a prediction of defect density without solving compli-cated differential equations.In this thesis,we will discuss KZM and defect production in a one-dimensional(1D)topological insulator with 3rd-nearest-neighbor hoppings.In the first and second chapter of the thesis,we reviewed backgrounds and basic concepts(edge states,winding number,etc.)of topological insulators.Then we intro-duced core concepts of KZM and some conclusions of transition dynamics.In the third chapter we discussed the bulk states evolution of the 1D topological model with 3rd-nearest-neighbor hoppings in periodic boundary conditions.The Hamil-tonian matrix is block diagonalized by Fourier transformation,and it decomposes into multiple decoupled 2-level systems in momentum space.We calculated the evolution analytically using Landau-Zener formula.Then we checked the KZ scaling of defects numerically.In the fourth chapter,we investigated the properties of our model with open bound-aries.In open boundary conditions,the system has edge states in nontrivial topological phases caused by bulk-boundary correspondence.We calculated the edge states and their evolution.Though solving dynamical equations analytically is difficult,KZM paradigm provides an easy approximation to defect production,and we obtained the adiabatic-impulse boundary(freeze-out time)in the process.While the system quenches slowly,bulk states and edge states may interfere,creating an oscillation in defect density,which can be analyzed by KZM paradigm.Bulk states and edge states have complicated couplings when the system quenches across different phase regions with open bound-aries,and the evolution behavior of bulk states with open boundaries are different from the ones with periodic boundaries.We calculated defect densities in several quench paths,open boundaries are proved to have nontrivial effect on defect production.Overall,we investigated the non-equilibrium dynamics of 1D topological system with open and periodic boundaries.This work provides some insights into KZM of topo-logical systems.Methods and conclusions in this thesis can be references in researches on quench dynamics and defect production of complicated systems.