Generalized two-leg Hubbard ladder at half-filling: Phase diagram and   quantum criticalities
  The ground-state phase diagram of the half-filled two-leg Hubbard ladder with inter-site Coulomb repulsions and exchange coupling is studied by using the strong-coupling perturbation theory and the weak-coupling bosonization method. Considered here as possible ground states of the ladder model are four types of density-wave states with different angular momentum (s-density-wave state, p-density-wave state, d-density-wave state, and f-density-wave state) and four types of quantum disordered states, i.e., Mott insulating states (S-Mott, D-Mott, S'-Mott, and D'-Mott states, where S and D stand for s- and d-wave symmetry). The s-density-wave state, the d-density-wave state, and the D-Mott state are also known as the charge-density-wave (CDW) state, the staggered-flux (SF) state, and the rung-singlet state, respectively. Strong-coupling approach naturally leads to the Ising model in a transverse field as an effective theory for the quantum phase transitions between the SF state and the D-Mott state and between the CDW state and the S-Mott state, where the Ising ordered states correspond to doubly degenerate ground states in the staggered-flux or the charge-density-wave state. From the weak-coupling bosonization approach it is shown that there are three cases in the quantum phase transitions between a density-wave state and a Mott state: the Ising (Z_2) criticality, the SU(2)_2 criticality, and a first-order transition. The quantum phase transitions between Mott states and between density-wave states are found to be the U(1) Gaussian criticality. The ground-state phase diagram is determined by integrating perturbative renormalization-group equations. It is shown that the S-Mott state and the SF state exist in the region sandwiched by the CDW phase and the D-Mott phase. 
M. Tsuchiizu and A. Furusaki