  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.
