Fig. 2: Results of EHM at half filling.

a–c are with on-site repulsion of U = 1.6 and d–f of U = 4, with the corresponding mini phase diagrams also depicted. a, d show the results of the Luttinger parameter K_ρ, (b, e) are the results of central charge c determined from the entanglement entropy scalings, and c, f are the spin structure factors at π. Here the K_ρ results in a, d are obtained via the second-order polynomial fitting of S_c(k) in the small-k regime, with the error bars estimated by using different k-range for fitting (Supplementary Note 3). The insets in c, f show S_m(π) vs. L at various Vs. a–c reveal the system undergoes a transition from spin density wave (SDW) to triplet superconducting (TS) phase at V_c ≃ −1 and then enters the phase separation (PS₂) regime for V < V_s ≃ −1.55. d–f show no intermediate phase but a direct first-order transition from SDW to PS₂ at V_s ≃ −2.42.