| The Frenkel-Kontorova (FK) model describes a chain of particles coupled to their neighbors in the presence of an external periodic potential which first appeared in1938. As one of the most important nonlinear models, the classical FK model demonstrates a lot of nonlinear phenomena, such as chaos, solitons, kinks and breathers. Nowadays, this model or its generalized forms have been used in a wild range of applications, such as an adsorbate layer on the surface of a crystal, charge-density-wave transport, dry friction and a chain of coupled Josephson junctions.Comparatively, the research upon quantum FK model started much later. The first paper appeared in1989. Not much progress has been made until now due to the complexities related to the quantum many-body problems. But re-cently, due to the fast development in controlling the atomic or ionic systems in optical lattice, the quantum FK model has attracted new attentions since it has great potential to be applied in describing these systems. In order to have a better understanding of the properties of quantum FK model, we have used the density-matrix renormalization group (DMRG) technique to carry out the numer-ical work. As one of the most efficient methods in dealing with strongly-coupled physical systems, DMRG has provided us the results with much higher preci-sions than those methods, such as Monte Carlo or variation methods. The work we have done are summarized as follows.1. We investigate the application of the Density-matrix renormalization group (DMRG) algorithm to a one-dimensional oscillator chain and compare the results with exact solutions, aiming to improving the algorithm efficiency. We want to find out how the algorithm can give quite accurate results. The calculation errors of ground energy and the energy gap between the ground state and the first excited state are also analyzed, which are critically depen-dent upon the size of the system, the number of retained subblock states, and the number of states targeted during the DMRG procedure.2. The incommensurate quantum one-dimensional FK model is investigate by a DMRG algorithm. We focus on the ground state properties such as the entanglement, the ground state energy, and the first energy gapA△E12(the energy separation from the lowest excitation state to the ground state). We find the quantum effect play a major role in the process of phase transition. The variation of K or h will lead a great change in the properties of the model. As we increase the quantum fluctuations, the correspond numerical results demonstrate a transition from the pinned state to a sliding one by passing through a so called instanton glass phase. However,no expected quantum critical point can be justified by our present data.3. We have investigated in details the phase transition from the pinned state to the sliding one in the commensurate quantum FK model with a DMRG al-gorithm. Similarly, the entanglement, the ground state energy, and the first energy gap△E12have been studied. The FK model will undergo a quan-tum transition as the quantum fluctuations increases gradually, but quite different from the incommensurate case, there isn’t a glass phase during the process. By investing the entanglement of the FK chain, we give the phase diagram.4. The time evolution problems have been investigated. Dealing with the time-dependent Schrodinger equation for strongly interacting many-body systems is a challenging job. By using the adaptive time-dependent DMRG (adaptive tDMRG), we try to investigate the real-time evolution of a given initial state of FK model chains. 5. Besides the FK model, we have investigated the entanglement dynamics for two atoms interacting nonsymmetrically with a single-mode l-photon coherent state based upon the Tavi-Cumming model. |
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