  We use a stochastic series expansion quantum Monte Carlo method to study the
phase diagram of the one-dimensional extended Hubbard model at half filling for
small to intermediate values of the on-site (U) and nearest-neighbor (V)
repulsions. We confirm the existence of a novel, long-range-ordered
bond-order-wave (BOW) phase recently predicted by Nakamura (J. Phys. Soc. Jpn.
68, 3123 (1999)) in a small region of the parameter space between the familiar
charge-density-wave (CDW) state for V > U/2 and the state with dominant
spin-density-wave (SDW) fluctuations for V < U/2. We discuss the nature of the
transitions among these states and evaluate some of the critical exponents.
Further, we determine accurately the position of the multi-critical point,
(U_m,V_m)= (4.7 +/- 0.1, 2.51 +/- 0.04) (in energy units where the hopping
integral is normalized to unity), above which the two continuous SDW-BOW-CDW
transitions are replaced by one discontinuous (first-order) direct SDW-CDW
transition. We also discuss the evolution of the CDW and BOW states upon hole
doping. We find that in both cases the ground state is a Luther-Emery liquid,
i.e., the spin gap remains but the charge gap existing at half-filling is
immediately closed upon doping. The charge and bond-order correlations decay
with distance r as r^{-K_rho}, where K_rho is approximately 0.5 for the
parameters we have considered. We also discuss advantages of using parallel
tempering (or exchange Monte Carlo) -- an extended ensemble method that we here
combine with quantum Monte Carlo -- in studies of quantum phase transitions.
