  We study a generalized Hubbard model on the two-leg ladder at zero
temperature, focusing on a parameter region with staggered flux (SF)/d-density
wave (DDW) order. To guide our numerical calculations, we first investigate the
location of a SF/DDW phase in the phase diagram of the half-filled weakly
interacting ladder using a perturbative renormalization group (RG) and
bosonization approach. For hole doping delta away from half-filling,
finite-size density-matrix renormalization-group (DMRG) calculations are used
to study ladders with up to 200 rungs for intermediate-strength interactions.
In the doped SF/DDW phase, the staggered rung current and the rung electron
density both show periodic spatial oscillations, with characteristic
wavelengths 2/delta and 1/delta, respectively, corresponding to ordering
wavevectors 2k_F and 4k_F for the currents and densities, where 2k_F =
pi(1-delta). The density minima are located at the anti-phase domain walls of
the staggered current. For sufficiently large dopings, SF/DDW order is
suppressed. The rung density modulation also exists in neighboring phases where
currents decay exponentially. We show that most of the DMRG results can be
qualitatively understood from weak-coupling RG/bosonization arguments. However,
while these arguments seem to suggest a crossover from non-decaying
correlations to power-law decay at a length scale of order 1/delta, the DMRG
results are consistent with a true long-range order scenario for the currents
and densities.
