Phase diagram of a coupled trimer system at half filling using the Hubbard model

Flat band systems have recently attracted significant attention due to their instability under small perturbations, which can lead to the stabilization of many exotic quantum phases. We study a trimer ladder which shows a middle flat band in the absence of onsite Coulomb interaction. We investigate the quantum phases of the Hubbard model on this geometry using exact diagonalization, density matrix renormalization group, and perturbation theory. We construct a quantum phase diagram in the plane of the next-nearest-neighbor hopping parameter ${t}_{2}$ and onsite Coulomb interaction $U$, revealing five distinct quantum phases. At low $U$ and moderate to high magnitude of ${t}_{2}$, the system exhibits metallic behavior, while at large $U$ and small magnitude of ${t}_{2}$, it transitions to a ferrimagnetic insulator phase, similar to those observed in certain trimer materials. In the small ${t}_{2}$ limit, the Fermi energy is in the flat band, leading to localization of the electrons within the trimer. At low $U$ and small magnitude of ${t}_{2}$, the flat band mechanism favors insulating ferrimagnetism, whereas at large $U$, ferrimagnetic states emerge from singlet dimer formation between neighboring sites of a trimer and an isolated corner spin, which connect ferromagnetically. The insulating cell spin density wave phase displays an up-up--down-down spin configuration due to competing nearest-neighbor hopping ${t}_{1}$. Interestingly, in moderate $U$ and $|{t}_{2}|>0.3$, the ground state behaves like metallic Tomonaga-Luttinger liquid.