Based on a symmetry argument, we investigate the ground-state properties of newly synthesized metal-halide ladder compounds (C8H6N4)[Pt(C2H8N2)X]2(ClO4)4·2H2O (X = Cl, Br, I). Employing a fully dressed two-band Peierls-Hubbard model, we systematically reveal possible charge-or spin-ordered states. Numerical phase diagrams demonstrate a variety of competing Peierls and Mott insulators with particular emphasis on the transition between two types of mixed-valent state of Pt II and Pt IV driven by varying interchain hopping integrals and Coulomb interactions.

In terms of spinless fermions and spin waves, we describe magnetic properties of a spin-1 2 ferromagnetic-antiferromagnetic bond-alternating chain which behaves as a Haldane-gap antiferromagnet. On one hand, we employ the Jordan-Wigner transformation and treat the fermionic Hamiltonian within the Hartree-Fock approximation. On the other hand, we employ the Holstein-Primakoff transformation and modify the conventional spin-wave theory so as to restore the sublattice symmetry. We calculate the excitation gap, the specific heat, the magnetic susceptibility, magnetization curves, and the nuclear spin-lattice relaxation rate with varying bond alternation. These schemes are further applied to a bond-alternating tetramerized chain which behaves as a ferrimagnet. The fermionic language is particularly stressed as a useful tool to investigate onedimensional spin-gapped antiferromagnets, while the bosonic one works better for ferrimagnets.

We make a matrix-product variational approach to spin-1 2 ferromagnetic-antiferromagnetic bondalternating chains with anisotropy on their ferromagnetic bonds, especially under the open boundary condition. The rich phase diagram containing the Haldane, large-D, and two types of Néel phases is well reproduced with only two variational parameters. The on-bond anisotropy has a significant effect on the ferromagnetic coupling between neighboring spins and induces novel edge states peculiar to spin-1 2 chains.