  The density matrix renormalization group and quantum Monte Carlo method are
used to describe coupled trimer chains in a magnetic field h. The Hamiltonian
contains exchange terms involving the intra-trimer coupling J1 (taken as the
unit of energy) and the inter-trimer coupling J2, plus the Zeeman interaction
for a magnetic field h along the z direction. Results for the magnetization per
trimer m are calculated in regimes of positive and negative values of the ratio
J = J2 / J1, from which the rich field-induced ground state phase diagram h vs.
J is derived, with the presence of Luttinger liquid, the 1/3-plateau (m = 1/2),
and the one of fully polarized magnetization (m = 3/2). Also, the zero-field
Lanczos calculation of spin-wave dispersion from the 1/3-plateau for $S^z = 1$
is shown at the previous regimes of J values. In addition, we also report on
the decay of correlation functions of trimers along open chains, as well as the
average two-magnon distribution. The ground state is ferrimagnetic for $0<J\leq
1$, and is a singlet for $-1\leq J<0$. In the singlet phase, the spin
correlation functions along the legs present an antiferromagnetic power-law
decay, similar to the spin-1/2 linear chain, thus suggesting that the ground
state is made of three coupled antiferromagnetically oriented chains. In the
singlet phase, the dimensionless thermal magnetic susceptibility per site
normalized by 1/|J| gets closer to $1/\pi^2$ as the temperature $T\rightarrow
0$. For the ferrimagnetic phase, we fit the susceptibility to the experimental
data for the compound Pb$_3$Cu$_3$(PO$_4$)$_4$ and estimate the model exchange
couplings: J1=74.8 K and J=0.4. These values imply a range of energies for the
magnon excitations that are in accord with the data from neutron scattering
experiments on Pb$_3$Cu$_3$(PO$_4$)$_4$ for two excitation modes. The
1/3-plateau closes only at 1/|J|=0 with J<0.
