Low-energy structure of the homometallic intertwining double-chain ferrimagnetsA3Cu3(PO4)4(A=Ca,Sr,Pb)

Abstract

Motivated by the homometallic intertwining double-chain ferrimagnets ${A}_{3}{\mathrm{Cu}}_{3}{(\mathrm{P}{\mathrm{O}}_{4})}_{4}$ $(A=\mathrm{Ca},\mathrm{Sr},\mathrm{Pb})$, we investigate the low-energy structure of their model Hamiltonian $\mathcal{H}={\ensuremath{\sum}}_{n}[{J}_{1}({\mathbit{S}}_{n:1}+{\mathbit{S}}_{n:3})+{J}_{2}({\mathbit{S}}_{n+1:1}+{\mathbit{S}}_{n\ensuremath{-}1:3})]∙{\mathbit{S}}_{n:2}$, where ${\mathbit{S}}_{n:l}$ stands for the ${\mathrm{Cu}}^{2+}$ ion spin labeled $l$ in the $n\text{th}$ trimer unit, with particular emphasis on the range of bond alternation $0l{J}_{2}∕{J}_{1}l1$. Although the spin-wave theory, whether up to $O({S}^{1})$ or up to $O({S}^{0})$, claims that there exists a flatband in the excitation spectrum regardless of bond alternation, a perturbational treatment, as well as the exact diagonalization of the Hamiltonian, reveals its weak but nonvanishing momentum dispersion unless ${J}_{2}={J}_{1}$ or ${J}_{2}=0$. Quantum Monte Carlo calculations of the static structure factor further convince us of the low-lying excitation mechanism, elucidating similarities and differences between the present system and alternating-spin linear-chain ferrimagnets.