6 1 1.5 2 2.5 3 0 5 10 15 J 2x2 2x3 2x4 2x6 2x8 2x12 FM FM U$_{eff}$=-0.5 binding binding Figure 3: (Color online). (a) Phase diagram of the two-leg lad- der model Hamiltonian defined in Eq. 1. The region above the symbols is a fully-saturated ferromagnetic phase (see text). The blue lines correspond to regions where Δ b < − 0 . 1 for the cluster 2 × 3 with two doped electrons/holes, as presented in Figs. 2(a)-(b). C. Phase Diagram In Fig. 3 the phase diagram ( J vs. density) of the two- leg ladder for U$_{eff}$ = − 0 . 5 and several system sizes (see legend) is presented. For ρ = 2, it was observed that the total spin is zero for L even, and that it can be 0, 1, or 2 for L odd, depending of the values of J . Note that for L odd and ρ = 2, the ( π, 0) AFM configuration (+-+-+-) does not have the same number of + and - spins. For 2 < ρ < 2 . 5 (1 < ρ < 2), a ferromagnetic phase was found (the region above the symbols) with magnetic moment per site given by m = 2 − ρ/ 2 ( m = ρ/ 2). The symbols indicate the value of J$_{c}$ where the total spin sat- urates. The critical value J$_{c}$ was determined by the level crossing of the energies in the sector with S z = S max total and S z = S max total − 1. Using this procedure, we were able to obtain J$_{c}$ for large systems. For densities in the ranges ρ ≲ 1 . 25 and ρ ≳ 2 . 5, we have not found any trace of ferromagnetism. Below the FM region, it is very hard numerically to determine the total spin with good accu- racy for large systems. However, our results for the 2 × 2 and 2 × 3 clusters with two doped electrons/holes suggest that the total spin changes continuously, from maximum value at J$_{c}$ to zero at J =0. The total spin can be ex- tracted from the spin structure factor at q = (0 , 0). For a few sets of couplings we also observed, through the value of S ( q = (0 , 0)), that in fact the total spin of the ground state changes continuously for the 2 × 8 cluster as well. These results suggest that the total spin varies continu- ously below the FM region present in Fig. 3 for any clus- ter sizes. Overall, our results are qualitatively compati- ble with those found in one-dimensional systems. 11 Note also that tendencies to FM states at robust J were also reported via Exact Diagonalization methods on small clusters. 12,13 (Fig. 2(a)) and holes (Fig. 2(b)) are less than -0.1, for the case of a 2 × 3 cluster. The region where Δ$_{b}$ < − 0 . 1 was chosen to be represented, as opposed to Δ$_{b}$=0, since pre- vious experience in the context of the cuprates 5 suggests that this procedure effectively takes into account size ef- fects better. In practice, other values for this “cutoff” do not alter our qualitative conclusions. Similar results were found also for the 2 × 2 cluster. Thus far, only small clusters have been considered be- cause the numerical analysis of large clusters would be too time-consuming, particularly with regards to calcu- lating the hundreds of points that are required to extract comprehensive phase diagrams. However, larger system sizes were considered for a few selected sets of couplings, as shown in Fig. 2(c). In this figure, Δ$_{b}$ vs. 1 /L for some couplings is presented. Here, it is clearly observed that in the bulk limit the binding energies of added elec- trons/holes converge to nonzero values for some coupling sets. Close to ρ = 2, these results strongly indicate that there are pairing tendencies for U$_{eff}$ ≲ 0 . Thus, to the extend that future investigations show that J comparable to U ′ is a realistic regime for effective two-orbital mod- els, this provides a possible mechanism for pairing in real materials. In Figs. 2(a)-(b), the magnetic phase diagram for the case of two doped electrons/holes is also presented. The region above the red (bold) line is a ferromagnetic phase with the maximum total spin S$_{total}$ = 2 L − 1. Below this line, we have observed that the total spin changes con- tinuously from zero, at J = 0, up to its maximum value at J$_{c}$ . As observed in these figures and for a large region of couplings, pairing (and presumably superconductivity) co-exists with ferromagnetic tendencies. We believe the FM phase is stabilized by a mechanism that has the same characteristics as the Double Exchange (DE) mechanism. 34 In the original DE scenario, there are mobile and localized degrees of freedom. In the DE mechanism these degrees of freedom are separated and well defined. Although we do not have localized degrees of freedom in our model, from the perspective of one electron at a given orbital an electron at the same site but the other orbital behaves in some respects as a localized spin. For doped systems, when an electron moves from one site to the other, in order to minimize the kinetic energy and the energy related with the Hund coupling, all spins have to be aligned. The blue lines in Fig. 3 are the same that were pre- sented in Figs. 2(a)-(b). We expect that the region of binding extends beyond these lines up to the density ρ = 2, forming regions in parameter space where super- conductivity exists inside the phase diagram. We have also measured the spin structure factor S ( q ) away from the undoped density ρ = 2. In Fig. 4, S ( q ) is presented for some particular densities for the two-leg ladder model with size L = 16, J = 1 . 5, and U$_{eff}$ = − 0 . 5. As can be observed in Fig. 4(a), there is still a peak at