Interaction of holes in a Hubbard antiferromagnet and high-temperature superconductivity

The ${\mathrm{CuO}}_2$ planes in the high-temperature superconductors are described by a two-dimensional Hubbard model. The model is investigated for an infinite system with one and two electrons less than half filling. The method used is to diagonalize the Hamiltonian exactly within a retained portion of the Hilbert space. A single hole is found not to be localized by a string potential that increases linearly with distance, although it does have a large effective mass. A pair of holes, which naively should be quite mobile, is found instead to be extremely heavy due to a previously unappreciated frustration effect that impedes their motion. This lack of mobility increases the energy of the pair so that they do not bind, contrary to some recently published results using mean field theory or intuitive arguments. The energy of one- and two-hole states is calculated as a function of wavevector k. The comparative energy of different angular momentum channels and magnetic polaron effects are discussed.