Low-energy structure of the intertwining double-chain ferrimagnets A_3_Cu_3_(PO_4_)_4_ (A=Ca,Sr,Pb)

Motivated by the homometallic intertwining double-chain ferrimagnets A_3_Cu_3_(PO_4_)4 (A=Ca,Sr,Pb), we investigate the low-energy structure of their model Hamiltonian H=\sum_n_[J_1_(S_{n :1}+S{n :3}) +J_2(S_{n+1:1}+S_{n-1:3})]\cdotS{n:2}, where S{n:l}_ stands for the Cu^{2+}^ ion spin labeled l in the nth trimer unit, with particular emphasis on the range of bond alternation 0<J_2/J_1<1. Although the spin-wave theory, whether up to O(S^1^) or up to O(S^0^), claims that there exists a flat band 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.