We investigate the charge and spin gaps, and the spin structure in half-filled one-dimensional Hubbard superlattices with one repulsive site and L0 free sites per unit cell. For odd L0, it is correlated metal at the particle–hole symmetric point, and then turns into band insulator beyond this point. For even L0, the system has a Mott insulator phase around the particle–hole symmetric point and undergoes a metal–insulator transition with on-site repulsion U increasing. For large U,there exists a multiperiodic spin structure, which results from the ferromagnetic（antiferromagnetic） correlation between the nearest neighboring repulsive sites for odd（even） L0.
We propose a theoretical model to detect the quantum phase transition in a triangular quantum dot molecule with frustration. The boundaries of the phase diagram are accurately determined by the transmission. For small frustration t, as the interdot Coulomb repulsion V increases, the system undergoes a Kosterlitz–Thouless（KT） transition from the Kondo resonance state with a transmission peak at zero energy to the Coulomb blocked state with zero transmission, which is followed by a first transition to the V-induced resonance（VIR） state with unitary transmission. For large frustration t, as V increases, the orbital spin singlet without transmission transits to the VIR state through a KT transition.
We study the charge oscillation in the triangular quantum dots symmetrically coupled to the leads. A strong charge oscillation is observed even for a very small level difference. We attribute this oscillation behaviour to the many- body effect in the strongly correlated system instead of the physical scenarios based on the mean-field approach in the previous works for the two-level dot. The level difference induces the difference of the occupations between different dots, while the symmetry of the many-body states favours the homogeneous distribution of the charge density on the three dots. The interplay of these two factors results in the charge oscillation.
