4 U$_{eff}$ > 0. Neutron scattering measurements for pnic- tides also show a similar spin order. 31 We have observed that the stripe-like AFM structure appears only when the plaquette-diagonal hopping amplitudes ( t$_{3}$ and t$_{4}$ ) are of value similar as those of the nearest-neighbor hop- ping amplitudes. If we force t$_{3}$ = t$_{4}$ =0 then the peak in the spin structure factor S ( q ) appears at wavevector q =( π, π ). Note also that for the two-leg geometry, the ( π, 0) AFM state is not, naturally, degenerate with the (0 , π ) AFM state. Due to this fact, a study in a ladder geometry could make a better connection with the two- dimensional results of the pnictide materials where ( π, 0) is favored over (0 , π ) by a lattice distortion. For quasi-one-dimensional systems, a true long-range magnetic order is replaced by a power-law decay of the spin-spin correlations. Thus, in order to analyze the range of the magnetic order, we have also investigated the spin-spin correlation function along one of the legs (say, leg 1) defined as C$_{spin}$ ( l ) = 1 M ∑ | i − j | = $_{l}$〈 S z i, $_{1}$S z j, 1 〉 , (4) where M is the number of site pairs ( i, j ) satisfying l = | i − j | . In practice, we have averaged over all pairs of sites separated by distance l , in order to minimize boundary effects (a few sites at the edges were also discarded while implementing this averaging procedure). In Fig. 1(c), the spin correlation function C$_{spin}$ ( j ) is shown for the 2 × 24 cluster and using U$_{eff}$ = − 0 . 5. The dashed line is a fit of the numerical data with the function ˜ C$_{spin}$ = a cos( πx ) x 1 / 3 . (5) Similar results were found for the 2 × 16 cluster. The ob- served power-law decay suggests that a two-dimensional system with the same model and parameters would de- velop long-range magnetic order at zero temperature. B. Doping with two holes or electrons Let us now consider the effect of doping with charges this ladder system. If tendencies toward the pairing of the extra charges are unveiled, they would be an indicator that this model could become superconducting in a two dimensional geometry, for the couplings here considered. Let us start with the calculation of the binding energy of two doped holes/electrons. This binding energy is de- fined as Δ$_{b}$ = E (2) + E (0) − 2 E (1), where E ( n ) is the ground state energy with (4 L + n ) holes/electrons (4 L is the number of electrons corresponding to the “undoped” limit where there is an electron per orbital and per site). On a finite system the binding energy Δ$_{b}$ > 0 is posi- tive if the electrons/holes do not form a bound state, 5 while in the thermodynamic limit Δ$_{b}$ should vanish in the absence of pairing. On the other hand, if the extra holes/electrons form a bound state, then Δ$_{b}$ < 0 even on Below, the dominant spin arrangement in the ground state at density ρ = 2 is shown to guide the discussion For completeness, let us address this limit in detail, although it is clear that large J compared with U ′ leads to an effective attractive interaction. 11 Let us denote the states of the one-site problem via the symbol ( s$_{y}$ s$_{x}$ ) , where the “arrows” s$_{x}$ ( s$_{y}$ ) represent the electrons (with their spins projections) at the orbital x ( y ). For this one-site problem with two electrons and the limit considered here where J is large, the ground state energy is degener- ate. Its value is e$_{2}$ = U$_{eff}$ , and the three corresponding eigenstates are ( ↓ ↓ ) , ( ↑ ↑ ) , and [ ( ↑ ↓ ) + ( ↓ ↑ ) $_{]/}$√$_{2.}$ Since − U$_{eff}$ ≫ 1, we can approximate the Hamiltonian as H = H$_{kinetic}$ + H$_{int}$ ∼ H$_{int}$ , and the (highly degener- ate) ground state energy of the two-leg model becomes E (0) = 2 L × e$_{2}$ = 2 L × U$_{eff}$ . Before presenting our numerical results for the binding energies, let us first consider the following strong coupling regime defined by − U$_{eff}$ = 3 J − U = J − U ′ ≫ 1. As it will be argued below, and as it was discussed in pre- vious investigations for one-dimensional chains, 11 in this regime the binding of hole/electrons is clearly present. a finite cluster, and this would be indicative that effective attractive forces are present in the system.   ( ↓ ↓ ) ( ↑ ↑ ) ( ↓ ↓ ) ( ↑ ↑ ) ( ↓ ↓ ) ↓ ↓ ↑ ↑ ↓ ↓ ↑ ↑ ↑ ↑   , using only the up and down projections of the spin one states at every site for simplicity. Such a state is to be expected since for J ≫ 1 the alignment of the two spins at the two orbitals in the same site will occur, and as U ∼ 2 J ≫ 1 then having two electrons in the same or- bital is not allowed. Of course, any other configuration (such as a fully ferromagnetic state) is also equally likely as the one shown in the figure for − U$_{eff}$ ≫ 1. However, the hopping terms will lift the large degeneracy, and our numerical results for the spin-spin correlations presented before indicate that hopping terms favor the ( π, 0) AFM configuration, for a wide range of couplings in the un- doped limit. For this reason, here we have chosen to present the dominant spin arrangement of the ground state as a stripe-like AFM state, but it could be fully FM as well. Regardless of this detail, the arguments we will use below to obtain the energies E ( n ) will hold true both for a stripe-like AFM state, as well as for a FM state. Let us now add two extra electrons to the undoped system. In the limit being considered here, where the hopping amplitudes are negligible, the best way for the system to minimize its energy is to have the two electrons located on the same site, since in this way less on-site ferromagnetic links are broken. The on-site Hubbard U energy penalization is the same whether the doubly oc- cupied orbitals are at the same site or not, and since U ′ is negligible with respect to J in the limit considered, then the (effectively attractive) Hund coupling determines the