Edge-edge correlations without edge states: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>η</mml:mi></mml:math>-clustering state as the ground state of the extended attractive SU(3) Hubbard chain

We explore the phase diagram of the extended attractive $\mathrm{SU}(3)$ Hubbard chain with two-body hopping and nearest-neighbor attraction at half-filling. In the large on-site attraction limit, we identify three different phases: phase separation (PS); Tomonaga-Luttinger liquid (TLL); and charge density wave (CDW). Our analysis reveals that the $\ensuremath{\eta}$-clustering state, a three-component generalization of the $\ensuremath{\eta}$-pairing state, becomes the ground state at the boundary between the PS and TLL phases. On an open chain, this state exhibits an edge-edge correlation, which we call boundary off-diagonal long-range order (bODLRO). Using the density matrix renormalization group (DMRG) method, we numerically study the phase diagram of the model with large but finite on-site interactions and find that the numerical results align with those obtained in the strong coupling limit.