Ground-state phase diagram and thermodynamics of coupled trimer chains

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.