Numerical Study of Strongly Correlated Electron Systems

In this thesis, we present some systematic studies on low dimensional strongly correlated systems: the Heisenberg spin magnets and the t-.Jz model for the two dimensional lattices. We have studied these systems through numerical and analytical research. Various physical properties of the systems are obtained in our approach. The results obtained are believed to provide useful insights about the correlated behaviors of the model systems, which are closely related and motivated by the discoveries of the high temperature superconductors. First, we have proposed a new method concerning the spin wave theory on finite lattices. In this thesis we have developed a systematic spin wave expansion for the quantum fluctuations of a generic spin Hamiltonian on a finite lattice, where 1/ :S is the small parameter. The idea was applied to the J 1 -h Heisenberg model in the square lattice and the Heisenberg model in the triangular lattice ( AHT) in two dimensions. By comparing with the exact diagonalization results, we have shown that for large enough h, the ground state with long range antiferromagnetic order is instable. The non-Neel ordered state sets in at a smaller value of h than the previous estimates. For the triangular lattice, contrary to the previous results of finite size extrapolation, we also show that the ground state on the triangular lattice indeed has long range Neel order. The new method has also been applied to the lx-ly model. Second, we have studied an anisotropic spin magnet in two dimensions, through long wavelength field theory, numerical research, combined with the finite size spin wave theory. A possible spin liquid is suggested to exist in this two dimensional system, and a phase transition from the gapless spin liquid phase to a ordered phase occurs at the critical anisotropy llc = 0.1. In particular, we have found that the gapless spin liquid phase may be thought of as a set of decoupled spin chains, exhibiting the one dimensional feature. Third, we have introduced a new method, which we called Lanczos Spectra Decoding (LSD), to extract information about dynamical correlation functions of many-body Hamiltonians with a few Lanczos iterations and without the limitation of finite size. We apply this technique to understand the low energy properties and the dynamical spectral weight of a simple model describing the motion of a single hole in a quantum antiferrornagnet: the t-.Jz model in two spatial dimensions and for a double chain lattice. The simplicity of the model allows us a well controlled numerical solution, especially for the two chain case. We have found that the single hole ground state in the infinite system is continuously connected with the N agaoka fully polarized state. Analogously we have obtained an accurate determination of the dynamical spectral weight relevant for the photoemission experiments. The spectral weight is in qualitative agreement with the old approximate techniques: the retraceable path approximation for .fz = 0 and the string theory for .fz > 0. However, contrary to the previous approximations, the band tails for .f 2 = 0 or the asymptotic .fz -+ 0 one hole ground state again approach the N agaoka energy. We have given a simple analytical argument, supported by our numerical data, showing that the vanishing of the spectral weight close to the Nagaoka energy is faster than any power law. We have also been able to show that spin charge decoupling is an exact property in the Bethe lattice but it is not fulfilled in more realistic lattices where the hole can describe closed loop paths during its motion. Our detailed numerical results on this simple model represent a benchmark for possible developments to the more interesting t-.J model in the physical small .J limit. Finally, we have discussed possible physical applications of the results, m particular, related to the high temperature superconducting materials.