Numerical Study Of One Dimensional Strongly Correlated Lattice Models

Because of its rich underlying physics, strong correlated system is one of the most important issues in both theoretical and experimental aspects in condensed matter physics. In strong correlated systems, the interactions between particles can not be neglected or averaged, and the band theory does not work. Generally, since most of these problems can not be analytically solved, numerical methods are very important in the strong correlated systems. Comparing with the case in higher dimensions, one dimensional system can be handled with density matrix renormalization group method with much higher accuracy. The results in one dimension are also helpful for understanding the physics in two- or three dimen-sions.In chapter 1, we firstly introduce basic properties of the ground state of several one dimensional strong correlated systems. Generally, one dimensional Fermi system can be described by Luttinger liquid theory, which is the simplest example of non-Fermi liquid. In Luttinger liquid, the elementary excitations are the charge density wave and the spin density wave; the decay rate of correlation functions is determined by the Luttinger parameter; and the system shows the behavior of spin-charge separation. For one dimensional Boson systems, there exist the superfluid-Mott insulator phase transitions. For one dimensional quasi-periodic spin systems, there are plateaus at certain positions corresponding to the period of the ground state in the zero-temperature magnetization curves. On the other hand, we briefly introduced three popular numerical methods in strong correlated systems: the exact diagonalization method, the quantum Monte Carlo method and the density matrix renormalization group method.In the first section of chapter 2, we further introduce the basic theory and techniques of the exact diagonalization method. Next we discuss the key aspects of the density matrix renormalization group method, together with its basic process. We have also discussed the efficiency and accuracy of the method. Further more, we have improved the real space parallel scheme of the method. At last, we proposed that the lattice in higher dimension can be projected onto one dimension using some skills, so that it can be handled using DMRG.In chapter 3, we study systematically the effect of long-range dipolar interac-tions on the ground-state phase diagram of one-dimensional t - J model by means of density matrix renormalization group. While the basic phases described by the Luttinger parameter Kp, namely, the repulsive Luttinger liquid (metallic phase, Kp< 1), attractive Luttinger liquid (superconducting phase, Kp> 1), and the phase separation (Kp → ∞) are similar to those of the conventional t - J model, the presence of the long-range dipolar interactions leads to significant differences. (ⅰ) At high density regime, the phase boundaries of these three phases are pushed to even large-J region; At low density regime, (ⅱ) these phase boundaries shift to-ward small-J region and most importantly (ⅲ) the spin-gap region spreads across the boundary of Kp = 1, suggesting an exotic metallic phase with spin gap, which is absent in the conventional t - J model. The result indicates that the long-range dipolar interactions have a significant influence on the ground-state properties of the one-dimensional t -J model. Its implication on the pseudogap phenomenon of the hole-doped cuprates is briefly discussed.In chapter 4, we investigate the ground state phase diagram of hard-core boson system with repulsive two-body and attractive three-body interactions in one-dimensional optical lattice. When these two interactions are comparable and increasing the hopping rate, physically intuitive analysis indicates that there ex-ists an exotic phase separation regime between the solid phase with charge density wave order and superfluid phase. We identify these phases and phase transitions by numerically analyzing the density distribution, structure factor of density-density correlation function, three-body correlation function and von Neumann entropy estimator obtained by density matrix renormalization group method. These exotic phases and phase transitions are expected to be observed in the ultra-cold polar molecule experiments by properly tuning interaction parameters as suggested in Methods by Buchler, Michell and Zoller [Nature Physics 3,726 (2007)], which is constructive to understand the physics of ubiquitous insulating-superconducting phase transitions in condensed matter systems.In chapter 5, we unveil nontrivial topological properties of zero-temperature magnetization plateau states in periodically modulated quantum spin chains un- der a uniform magnetic field. As positions of plateaus are uniquely determined by the modulation period of exchange couplings, we find that the topologically non-trivial plateau state can be characterized by a nonzero integer Chern number and has nontrivial edge excitations. Our study clarifies the topological origin of the well-known phenomena of quantized magnetization plateaus in one-dimensional quantum spin systems and relates the plateau state to the correlated topological insulator. Furthermore, we propose that there might be fractional topological magnetization plateaus in the presence of long-range interactions.Chapter 6 summarizes the presented work and gives a outlook of our future work.