We study the elementary excitations and magnetization of a one-dimensional spin-1/2 XX chain comprising trimer-dimer units ($J_1-J_1-J_2-J_3-J_2$ topology) under a transverse magnetic field ($h$). Using Green's function theory and the Jordan-Wigner transformation, we map the system to spinless fermions and focus on antiferromagnetic (AFM) interactions. At zero temperature, distinct 1/5 and 3/5 magnetization plateaus emerge, determined by the overall periodicity $Q=5$, with the number of plateaus matching the number of excitation gaps above the Fermi level of spinless fermions. The magnetic phase diagram in the (h-J) plane features a Luttinger liquid (LL) state, a gapless AFM state, two magnetization plateau states, and a fully polarized gapped magnetic state. The widths of the LL and gapless AFM phases are found to be proportional to the bandwidths [$\gamma \propto \left|E\left(k=0\right)-E\left(k=\pi \right)\right|$] of the respective elementary excitations, whereas the widths of the magnetization plateau states are found to be linked to the gaps between excitations. Our study opens a path for exploring interacting trimer-dimer spin chains in quantum magnetism using experimental techniques, such as neutron scattering and/or through other theoretical or numerical approaches, including quantum Monte Carlo and density matrix renormalization group methods. Furthermore, our study extends the Oshikawa-Yamanaka-Affleck condition to generalized cluster chains, showing that the allowed magnetization plateau states are governed by the global periodicity of the chain (e.g., $Q=5$ for a trimer-dimer chain), rather than the local periodicity of individual units ($Q=3$ for a trimer or $Q=2$ for a dimer).

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• Magnetism
• Density matrix renormalization group
• Quantum Monte Carlo
• Spin chains