Investigation On Magnetism And Localization Systems In One-Dimensional Quantum Many-Body Physics

One-dimensional quantum many-body physics plays an important role in condensed matter physics,cold atom physics,quantum information and other fields.At first,the research on one-dimensional systems mainly focused on theory,but with the progress of experimental technology,more and more experimental platforms,such as ultracold atomic optical lattices,ion traps,superconducting qubits,and other systems have all achieved the simulation of dimensionally constrained systems.These make one-dimensional quantum many-body physics also a research hotspot experimentally.Quantum many-body physics problems are generally difficult to solve analytically,so a large number of numerical simulation methods have been developed,such as Quantum Monte Carlo(QMC),Numerical Renormalization Group(NRG),density matrix renormalization group(DMRG),Matrix Product States(MPS),Time Evolving Block Decimation(TEBD)and other algorithms.These make it possible to study quantum many-body systems through numerical simulations.Recently,magnetism,dynamics,and localization,as three important properties of quantum many-body systems,have been extensively studied in theory,numerical and experiment.The combination of these properties also produces many novel physical phenomena,such as kinetic frustration,dynamic localization,many-body localized discrete time crystal and so on.Aiming at these properties,this paper mainly studies three kinds of quantum many-body systems through theoretical and numerical simulation methods,including quantum magnetism systems,periodically driven systems and many-body localization systems.Among them,the quantum magnetic system belongs to the equilibrium states system,while the periodically driven system and the many-body localization system belong to the non-equilibrium states system.We mainly discuss the phase transitions in these three types of systems.Among them,quantum magnetism systems have quantum phase transitions in the general sense,periodically driven systems have transitions controlled by the driving frequency,and many-body localization systems contain phase transitions induced by the disorder or quasi-periodicity of the systems.The main conclusions and innovations of this paper are as follows:1.The isotropic Majumdar-Ghosh(MG)model is extended to the fully anisotropic model,and the ground state phase diagram of the model is obtained,which proves the existence of an exact dimer phase under anisotropic parameters.In one-dimensional or quasi-one-dimensional magnetic materials,there are a large number of compounds whose ground states are dimer states.Most of the previous theories are based on the isotropic J1-J2 model to understand such materials,but the real materials inevitably have anisotropy.Therefore,in Chapter 3,we generalize the isotropic MG model to the fully anisotropic model.Through theoretical and numerical calculations,we find that this anisotropic model has an accurate dimer ground state,and demonstrate the strict phase boundary of the exact dimer phase.We discuss the corresponding observation results of the model in the experiment,indicating that the model is expected to be used for the theoretical description of dimer ground state materials.Due to the existence of anisotropy,this model is more consistent with the real materials.2.In the periodically driven Bose-Hubbard model,multifrequency resonance phenomenon similar to those in two-level systems are found,and precession and nutation dynamics similar to those in classical mechanics and single-particle systems are observed.Previous studies on periodically driven systems have mostly focused on the extreme cases of high frequency and low frequency.For the high-frequency case where the driving frequency is much higher than the energy scale of the system,the Floquet theory can be used to describe it.For the low-frequency case where the driving frequency is much lower than the energy level spacing of the system,it can be well understood using the adiabatic theorem.In Chapter 4,we study the intermediate frequency range of the periodically driven Bose-Hubbard model and find that when the driving frequency(ω)is a rational multiple of the energy gap(ΔE12)between the ground state and the first excited state of the system at the initial moment,that is,ω=βΔE12(β=p/q,where p and q are mutually prime and p>q),under suitable parameters,the wave function can periodically recover to the initial state,and the recovery period is related to β.In this case,allowing us to map the many-body system to the single-body system,thus confirming the existence of multifrequency resonant phases and the dynamic behaviors of the system consistent with precession and nutation dynamics.This phenomenon is expected to be experimentally realized by ultracold atoms in periodically driven optical lattices.3.Using the Bethe ansatz method,the relationship between many-body localization and Anderson localization is established,which shows that many-body localization can be regarded as infinite-dimensional Anderson localization in the virtual lattice with infinite-range correlated disorder.There are similarities between Anderson localization and many-body localization,but there are also fundamental differences.Currently,no research has shown the relationship between these two cases.In Chapter 5,we use the Bethe ansatz method to show that many-body localization can be viewed as infinite-dimensional Anderson localization in the virtual lattice with infinite-range correlated disorder.Using theoretical and numerical simulation methods,through the study of the energy levels,wave function and dynamic properties of the system,we find that the system slowly crossover from Anderson localization to many-body localization as the virtual lattice dimension increases.Thus verifying the correctness of this physical picture.Our findings provide a new perspective on the understanding of many-body localization and a theoretical basis for experimentally exploring the relationship between these two types of localization.4.Using random matrix theory,the statistical properties of long-range level spacing ratio and stochastic level spacing ratio of many-body localization systems are studied,and the corresponding analytical expressions of probability density functions are obtained.Previous studies have shown that the transition from an ergodic phase to a manybody localization phase can be characterized by the statistical property of the energy spectrum’s consecutive level-spacing ratio.But in numerical calculations or experiments,if we only know part of the discontinuous spectrum information,can we still judge the many-body localization transition of the system?To this end,we propose two new statistics in Chapter 6:the long-range level spacing ratio and the stochastic level spacing ratio.Using random matrix theory,we obtain the probability density functions of these two statistics through reasonable approximations and verify them numerically.In addition,we specifically calculate the one-dimensional spin-1/2 system with disorder,showing that the critical disorder strength and critical exponent obtained from these two statistics are consistent with the previous results given by the consecutive level-spacing ratio.Therefore,the information of partial discontinuous spectrum can still judge the transition of the system from the ergodic phase to the many-body localization phase.