We show that charge excitations within one-dimensional extended Hubbard models in the limit of large on-site Coulomb interaction can be mapped onto pairs of spin reversals from the ferromagnetic Ising-Heisenberg spin Hamiltonian. The exciton in the extended Hubbard models is equivalent to the bound two-magnon state of the spin Hamiltonian. From exact results for the spin Hamiltonian, for the intersite Coulomb interactions limited between nearest neighbors, the lowest optical exciton is separated in energy from the electron-hole continuum only for the intersite interaction larger than half the one-electron bandwidth. The commutation relationship between spin operators allows us to numerically investigate exciton binding energies and optical absorptions from the ground state in the limit of large on-site Coulomb interaction even for cases where analytic results are not available. For intersite Coulomb interaction limited between nearest neighbors, the binding energy of the lowest optical exciton changes very weakly for realistic Peierls bond dimerization, although there is a profound change in the optical absorption. While in the undimerized chain a single optical exciton can occur, two additional excitons that occur above the threshold of the electron-hole continuum become optically visible in the dimerized case. For long-range intersite Coulomb interactions, multiple excitons can occur even for the undimerized chain, and additional exciton states become optically visible upon dimerization. With the
1/r form for the intersite Coulomb interaction, our numerical calculations suggest that the formation of at least one optical exciton is unconditional, unlike the case of nearest-neighbor interaction. The binding energy of the lowest optical exciton again depends weakly on the dimerization for long-range interaction. Our theoretical results can explain the occurrence of two charge-transfer absorption bands in half-filled band segregated stack charge transfer solids below the spin-Peierls dimerization temperature. Our results also suggest that the lowest optical state of \ensuremath{\pi}-conjugated polymers, in which Coulomb interactions are long ranged, is necessarily an exciton.
The one-dimensional (1D) tight-binding model with random nearest-neighbor hopping is known to have a singularity of the density of states and of the localization length at the band center. We study numerically the effects of random long-range (power-law) hopping with an ensemble average magnitude $〈|{t}_{\mathrm{ij}}|〉\ensuremath{\propto}|i\ensuremath{-}j{|}^{\ensuremath{-}\ensuremath{\sigma}}$ in the 1D chain, while maintaining the particle-hole symmetry present in the nearest-neighbor model. We find, in agreement with results of real-space renormalization-group techniques applied to the random XY spin chain with power-law interactions, that there is a change of behavior when the power-law exponent $\ensuremath{\sigma}$ becomes smaller than 2.