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Stability of $(N+1)$-body fermion clusters in multiband Hubbard model

We start with a variational approach and derive a set of coupled integral equations for the bound states of $N$ identical spin-$\uparrow$ fermions and a single spin-$\downarrow$ fermion in a generic multiband Hubbard Hamiltonian with an attractive onsite interaction. As an illustration we apply our integral equations to the one-dimensional sawtooth lattice up to $N \le 3$, i.e., to the $(3+1)$-body problem, and reveal not only the presence of tetramer states in this two-band model but also their quasi-flat dispersion when formed in a flat band. Furthermore, for $N = \{4, 5, \cdots, 10 \}$, our DMRG simulations and exact diagonalization suggest the presence of larger and larger multimers with lower and lower binding energies, conceivably without an upper bound on $N$. These peculiar $(N+1)$-body clusters are in sharp contrast with the exact results on the single-band linear-chain model where none of the $N \ge 2$ multimers appear. Hence their presence must be taken into account for a proper description of the many-body phenomena in flat-band systems, e.g., they may suppress superconductivity especially when there exists a large spin imbalance.