Ground-state phase diagram and thermodynamics of coupled trimer chains

Abstract

The density matrix renormalization group and quantum Monte Carlo methods are used to describe coupled trimer chains in a magnetic field $h$. The Hamiltonian contains exchange terms involving the intratrimer coupling ${J}_{1}$ (taken as the unit of energy) and the intertrimer coupling ${J}_{2}$, 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 $J={J}_{2}/{J}_{1}$, from which the rich field-induced ground state phase diagram $h$ versus $J$ is derived, with the presence of a Luttinger liquid, the 1/3 plateau ($m=1/2$), and the one of fully polarized magnetization ($m=3/2$). Also, zero-field Lanczos calculation of the 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 $0<J\ensuremath{\le}1$, and is a singlet for $\ensuremath{-}1\ensuremath{\le}J<0$. 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 $1/{\ensuremath{\pi}}^{2}$ as the temperature $T\ensuremath{\rightarrow}0$. For the ferrimagnetic phase, we fit the susceptibility to the experimental data for the compound ${\mathrm{Pb}}_{3}{\mathrm{Cu}}_{3}{({\mathrm{PO}}_{4})}_{4}$ and estimate the model exchange couplings: ${J}_{1}=74.8\phantom{\rule{4pt}{0ex}}\text{K}$ 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 ${\mathrm{Pb}}_{3}{\mathrm{Cu}}_{3}{({\mathrm{PO}}_{4})}_{4}$ for two excitation modes. The 1/3 plateau closes only at $1/|J|=0$ with $J<0$.