Figure 2

(a) Diamond chain with four plaquettes. (b) One-particle localized state of a diamond plaquette. This state result from the combination of two one-plaquette states with opposite momentum. (c) Network associated with the TB Hamiltonian for a diamond chain with four plaquettes, with one spin up and one spin down. The state shown corresponds to a combination of all sixteen CL states, with the sites at red(blue) having finite(zero) particle wave-function amplitude. If Hubbard interacting terms are added and projected onto the subspace of extended states, the previous network is modified in the anti-diagonal squared bubble structures (marked by green circles). These modifications are illustrated in detail (at the right) for the top right squared bubble structure. The nodes that correspond to electronic configuration with the spins occupying sites B and C (sites 2 and 3 in this example) are linked to each other, and these links correspond to long range hoppings, t′, pair tunneling terms, Δ, and spin exchange terms, J. The projection of the Hubbard terms also generate long range spin interactions, V, and the lowering of the Hubbard interaction in the red nodes, U′<U. Figure generated in Mathematica 8, https://www.wolfram.com/.