Studies On Mobility Edge, Topology And Dynamics Of Several Quasiperiodic And Flat Band Models

The discovery and study of novel quantum phases and quantum phase transitions is one of the central topics in condensed matter physics.In recent years,due to showing their rich quantum phenomena,quantum disordered systems,topological physics and quantum quench dynamics have attracted extensive research interest.In the theoretical research of these subjects,physical systems are generally abstracted as tight-binding lattice models.By adjusting the parameters of models,such as the potential strength and the parameters of quantum quenching,the physical laws contained in these systems can be deduced.In this thesis,we mainly study quantum phase transitions in several quasiperiodic and flat band lattice models,and the concrete contents are outlined as follows: Firstly,two off-diagonal slowly varying quasiperiodic models are studied,and the analytical expressions of energy-dependent mobility edges are given.Secondly,topological phase transitions of one-dimensional topological superconductor and one-dimensional topological insulator driven by quasiperiodic modulations are studied,and the numerical results of topological phase transition points are given.Finally,the quench dynamics in one-dimensional flat band model is studied,the analytical expression of the critical time for the occurrence of dynamical quantum phase transition is given,and a non-trivial example of Loschmidt echo always being 1 is given.In Chapter 1,we mainly introduce the physical background and knowledge of quantum disordered systems and flat band networks.It mainly includes: the concept of Anderson localization,scaling theory,one-dimensional quasiperiodic system(Aubry-André model),two precisely solvable quasiperiodic models with mobility edges and flat band networks.In Chapter 2,we study the effect of a commensurate off-diagonal modulation on the mobility edge in a slowly varying quasiperiodic model.Using the semi-classical WKB technique,we analytically give the expression of mobility edges.When the incommensurate potential strength is less than a critical value,there are four mobility edges in the energy spectrum of the system.We also calculated the inverse participation ratio(IPR),density of states and Lyapunov exponent of the model.By comparison,we find that the analytical results are in good agreement with the numerical results.In Chapter 3,we study the mobility edges in incommensurate off-diagonal models.When only the off-diagonal term of the system is the incommensurate modulation and the diagonal term is constant,mobility edges always exist in the energy spectrum of the system,no matter how strong the on-site potential is.When the off-diagonal and diagonal terms of the system are both incommensurate modulations,the singularity appears in the change of the position of the mobility edge in the energy spectrum.These interesting phenomena are different from those widely studied models driven by the potential(diagonal)disorder.In Chapter 4,we mainly study the topological phase transition of one-dimensional topological superconductors(one-dimensional p wave superconductor chain)driven by the on-site potential disorder.We consider the slowly varying quasiperiodic disorder in which mobility edges generally exist.We find that with the increase of the disorder strength,the system transforms from a topological superconducting phase to a topologically nontrivial insulating phase,and finally to a topologically trivial insulating phase.Then,we use the transfer matrix method to examine the topological transition point of the system numerically.In addition,we find that when the disorder strength is less than a critical threshold,the energy spectrum of the system has four mobility edges,and when the disorder strength is greater than the critical threshold,all eigenstates of the system become localized states.In Chapter 5,we mainly study the topological phase transition of one-dimensional topological insulators(Su-Schriffer-Heeger model)driven by the hopping disorder.We consider two kinds of disordered configurations,one is the Aubry-André quasiperiodic disorder,the other is the slowly varying quasiperiodic disorder.We use the transfer matrix method to calculate the topological phase transition point of the system numerically.It is found that under the driving of these two disorders,the behaviors of the system near the topological phase transition points are different.The phase transition point of the former can be well determined under the finite size,while that of the latter can not be well determined when the size is finite,it is necessary to implement finite size analysis and numerical extrapolation to determine the phase transition point.In Chapter 6,we mainly study the time-dependent evolution of the initial state after a quantum quench in the cross-stitch flat band networks.We find that under a certain quenching scheme,the system can also undergo dynamical quantum phase transition,even if the quantum quench does not cross the quantum critical point.However,in previous studies the quench of dynamical quantum phase transition is mostly found to cross the quantum critical point.We give an analytical expression of the critical time for the onset of dynamical quantum phase transition,and numerically calculate the time-dependent evolution of the system.Our analytical results are in good agreement with the numerical results.We also studied the inverse quench scheme and found an interesting phenomenon: even if the initial state before the quench is not the eigenstate of the Hamiltonian after the quench,the probability of the time-dependent evolution state returning to the initial state(Loschmidt echo)is always 1,which is a new phenomenon that has not been reported before.In summary,the research in this thesis gives some further understandings of the novel quantum phases and phase transitions in low-dimensional quasiperiodic and flat band lattice models.It is a valuable reference for the scientific development of quantum disordered systems,topological physics and quantum quench dynamics.