  We have investigated quantum critical behavior in the doped two-leg extended
Hubbard ladder, by using a weak-coupling bosonization method. In the ground
state, the dominant fluctuation changes from the conventional d-wave-like
superconducting (SCd) state into density-wave states, with increasing
nearest-neighbor repulsions and/or decreasing doping rate. The competition
between the SCd state and the charge-density-wave state coexisting with the
p-density-wave state becomes noticeable on the critical point, at which the gap
for magnetic excitations vanishes. Based on the Majorana-fermion description of
the effective theory, we calculate the temperature dependence of the magnetic
response such as the spin susceptibility and the NMR relaxation rate, which
exhibit unusual properties due to two kinds of spin excitation modes. On the
quantum critical point, the spin susceptibility shows paramagnetic behavior
with logarithmic corrections and the NMR relaxation rate also exhibits
anomalous power-law behavior. We discuss the commensurability effect due to the
umklapp scattering and relevance to the two-leg ladder compounds
Sr_{14-x}Ca_xCu_{24}O_{41}.
