Low Dimensional Topological States And Its Dynamics Far From Equilibrium

In this dissertation I have studied the representation of multi-band topological states by trajectories of Majorana stars;quench dynamics in topological band systems and in incommensurate lattice;and topological states in dissipative systems.In chapter 1,I first give a brief introduction of the discovery of topological states of matter,and show the conception of classification and the topological states and the calculation of topological invariants.Then I introduce the theories and experiments of dynamical quantum phase transition,and make a comparison of dynamical quantum phase transition and phase transition in equilibrium.At last,I show the method called third quantization,and show how to use it in solving the non-interacting Lindblad master equation.In chapter 2,by using the Majorana stellar representation,we study the multi-band topological states.As a major example,we study the topological states protected by time reversal symmetry and inversion symmetry.We show these symmetries will restrict the distribution of Majorana stars,then the Berry phase of band equals to the summary of Berry phase of each Majorana star.If the band is nontrivial,the trajectories of Majorana stars on Bloch will cover the equator.On the contrary,if the band is trivial,the trajectories of Majorana stars cancel out.Because the topological property of the band can be represented by the high symmetric points,we also show the distribution of Majorana stars at high symmetric points,and prove that it definitely determined the topology of the band.In chapter 3,we first study the quench dynamics in topological band systems.By studying the one dimensional two band system,we show there exists some fixed points in momentum space,which will lead to nontrivial topological structure of the composed momentum-time manifold.The topology of momentum-time manifold will reduce to a series of two dimensional spheres,and on each sphere,we can define a dynamical Chern number.Furthermore,we demonstrate the relation of dynamical Chern number and the topological invariant of initial and final Hamiltonian.In class AIII and BDI,the Hamiltonian preserve chorial symmetry,the topological invariant is winding number,we prove that the difference of winding number of initial and final Hamiltonian is the lower bound of the number of nontrivial dynamical Chern number.In class D,the Hamiltonian preserve particle hole symmetry,the topological invariant is Pfaffian,we prove the difference of the Pfaffian of initial and final Hamiltonian equals to the first dynamical Chern number.In this chapter,we also study the quench dynamics of AA model.First if the amplitude of incommensurate potential quenchs process is between zero and infinity,we give the exact solution of Loschmidt echo,which is the zero order Bessel function.Because the Bessel function have a series of zeros,then the Loschmidt echo have exact zeros,which exhibits the behavior of dynamical quantum phase transition,the dynamical free energy density divergence at critical time in the logarithm formula.If the initial and final incommensurate potential take arbitrary values,we numerical calculated Loschmidt echo,we show either the initial and final incommensurate potential locate in the localized and extended phases,Loschmidt echo will approach zero at some time interval.However,if both initial and final incommensurate potential locate in localized phase or extended phase,Loschmidt echo will oscillate around a finite number and never reach zero.In chapter 4,we study the dissipative Kitaev model.By adding the gain and loss at each site,we solve the Lindblad master equation by third quantization formula.The exact rapidities is given and the edge modes would exist in some region.By calculating the dynamics of correlation matrix,we show the existence of edge modes affect the dynamics of particle numbers at the boundary.However,the the edge modes have no influence on the non-equilibrium steady state.