By using mainly numerical methods, we investigate the ground-state phase diagram (GSPD) of an
S=1 ferromagnetic-antiferromagnetic bond-alternating chain with the
XXZ and the on-site anisotropies. This system can be mapped onto an anisotropic spin-2 chain when the ferromagnetic interaction is much stronger than the antiferromagnetic interaction. Since there are many quantum parameters in this system, we numerically obtained the GSPD on the plane of the magnitude of the antiferromagnetic coupling versus its
XXZ anisotropy, by use of the exact diagonalization, the level spectroscopy as well as the phenomenological renormalization group. The obtained GSPD consists of six phases. They are the $XY
1,thelarge−D$ (LD), the intermediate-
D (ID), the Haldane (H), the spin-1 singlet dimer (SD), and the Néel phases. Among them, the LD, the H, and the SD phases are the trivial phases, while the ID phase is the symmetry-protected topological phase. The former three are smoothly connected without any quantum phase transitions. It is also emphasized that the ID phase appears in a wider region compared with the case of the GSPD of the anisotropic spin-2 chain with the
XXZ and the on-site anisotropies. We also compare the obtained GSPD with the result of the perturbation theory.
We investigate the ground-state phase diagram and critical properties of the
S=1/2 two-leg Heisenberg spin ladder system with a negative four-spin interaction using a numerical exact diagonalization method. Using the perturbation theory in the strong negative rung-coupling limit, we derive an
S=1 bilinear-biquadratic chain as an effective model. We discuss the ground-state phase diagram in this limit. Next we numerically determine a phase boundary between the rung singlet phase and the columnar dimer (CD) phase by the phenomenological renormalization group method, and one between the CD phase and the Haldane phase by the twisted boundary condition method. We confirm that the phase transition between the CD phase and the Haldane phase is of second order and this universality class is described by the
k=2 SU(2) Wess-Zumino-Novikov-Witten nonlinear $\ensuremath{\sigma}$ model, calculating the central charge and scaling dimensions.
The half-filled extended Hubbard model (containing nearest-neighbor interaction) is considered at strong coupling in fourth-order perturbation theory. The ground-state energies in the spin- and charge-density-wave phases and the resulting phase diagram are calculated in various dimensions: here d=1, d=\ensuremath{\infty}, high dimensions (d\ensuremath{\gg}1), and d=2,3 are discussed separately. In d=1 it is shown that, for U\ensuremath{\gtrsim}6, fourth-order perturbation theory leads to excellent agreement with existing Monte Carlo data. Second-order perturbation theory alone is valid only at unrealistically large values of U and V. In d=\ensuremath{\infty}, one needs fourth-order perturbation theory to obtain any nontrivial contribution to the phase diagram at all. As a consequence the fourth-order corrections to the ground-state energy and to the phase diagram are large also in d=2,3. The formalism used to obtain the perturbative expansion is discussed in an appendix.