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location of the extra charge, leading to the double occu- pation of both orbitals at the same site. 11 In this case, the dominant spin arrangement of the ground state is 

Once again, a stripe-like AMF background is used, since for J ∼ 1 and close to the density ρ = 2 our numerical data show (see Fig. 4(a)) that the ( π, 0) AFM order is the dominant one, but it could have been FM as well. 

In the limit of couplings considered here, the ground state energy for the doped two-electron system is 

Note that if the two extra electrons that are in the state ( ↓↑ ↓↑ ) were to move in opposite directions (after consid- ering the presence of small but nonzero hopping terms) they would break two ferromagnetic on-site links. Such a state would have a large energy and it is therefore“forbid- den” for − U$_{eff}$ ≫ 1. However, if the two extra electrons move “together” (i.e. ↓ $_{↓}$) in the same direction, forming a spin triplet, then no other on-site FM links are broken. Thus, to minimize the energy the two-electrons added to the system must form a bound state, at least in the limit where the hoppings amplitudes are very small compared with the Hubbard and Hund couplings (the same spin dominant picture works when instead of adding electrons we remove two electrons, i.e. for the two-hole problem). This argument, known from previous investigations using chains, 11 explains the binding of electrons in the limit − U$_{eff}$ ≫ 1. More explicitly, the binding energy of the doped two-electrons/holes system in the limit considered here is given by 

Based on the discussion above we conclude that for − U$_{eff}$ ≫ 1, there is an indication of pairing, and perhaps superconductivity, in the two-orbital model. In fact, the on-site interorbital pairing state found here in this ex- treme regime results to be a spin triplet and transforms according to the irreducible representation A$_{2g}$ of the group D$_{4h}$ . 32 Interestingly, Exact Diagonalization calcu- lations in two-dimensional clusters, still in the FM state of the model but U ′ > J , found indications of a pairing state with the same characteristics and symmetry but with electrons at distance of one lattice spacing from each other. 12,13 The argument presented above for this pair- ing is actually valid in any dimension, and also it works for the two doped holes case, as already mentioned. To confirm these argumentations, Δ$_{b}$ was calculated numer- ically for large values of − U$_{eff}$ . An excellent agreement between the numerical data and the analytic expression (Eq. (7)) was found. 

Now the crucial question is whether there is binding of holes/electrons for values of − U$_{eff}$ that may be of more 

  ( ↓ ↓ ) ( ↑ ↑ ) ( ↓↑ ↓↑ ) ( ↑ ↑ ) ( ↓ ↓ ) ↓ ↓ ↑ ↑ ) ( ↓ ↓ ) ( ↑ ↑ ↑ ↑   . 

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relevance for real materials. The answer to this ques- tion appears to be positive. In fact, the binding of elec- trons/holes has been observed numerically for U$_{eff}$ ≲ 0 and several values of the coupling J , as shown in Fig. 2. 33 More specifically, Figs. 2(a)-(b) show the region in the J − (3 J − U ) plane where the binding energies of electrons 

