Fig. 1: Extended Hubbard model and phase diagrams.

a illustrates the BSCO compound and corresponding extended t-U-V Hubbard model with NN hopping t, on-site repulsive U, and NN attractive V terms. b, c show the quantum phase diagrams of the EHM at half and quarter fillings, respectively. There are triplet superconducting (TS), spin density wave (SDW), and Tomonaga-Luttinger liquid (TLL) phases. There are also three phase separation phases with singly (PS₁) doubly (PS₂) or fractionally (PS_x) occupied clusters. The solid black lines represent the asymptotic phase boundary \(V=-\frac{U}{2}-\frac{8\ln 2}{3U}\) in the strong coupling limit, and the dashed line for V = −U/2 in the weak coupling limit^30,54. The blue circle in c represents the parameters U = 8 and V = − 1 of the doped 1D cuprate chain BSCO⁴⁶.