The ground state of a mobile vacancy in a square-lattice spin-1/2 Heisenberg antiferromagnet described by the t-J Hamiltonian is investigated numerically for an 18-site cluster. The ground state has spin 1/2, and the bottom of the vacancy band is at momentum k=(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2), for t/J4, which is consistent with the spin-wave perturbation theory. The structure of the spin configuration in the vicinity of the vacancy is examined using a number of correlation functions. The latter reveal the existence of a dipolar distortion of the staggered magnetization, as suggested by the semiclassical theory.
The possibility of a perturbation treatment for quantum crystals is investigated. It is found that the usual form of the unperturbed Hamiltonian, being a single-particle operator, is not suitable for this situation, since it does not produce correlations necessary in a theory of quantum crystals. These correlations prevent the unperturbed ground state or low excited states from containing multiply occupied lattice sites. An unperturbed Hamiltonian is proposed which generates these correlations but is no longer a single-particle operator. The rules for a diagrammatic representation of a perturbation expansion starting from this unperturbed Hamiltonian are given. They differ from the usual rules for Goldstone diagrams. In particular, the linked-cluster theorem is not valid. Adding and subtracting pairs of mutually cancelling diagrams, in a manner similar to that in which Pauli-principle-violating diagrams are treated in the standard perturbation theory for fermions, we find new rules which, at least at low temperatures, are identical to those for spinless fermions. This result is essentially independent of the statistics of the actual lattice particles. In the case where the lattice particles are actually fermions with spin \textonehalf{}, e.g.,
3He, we find an expansion formally identical to the problem of spinless fermions interacting among themselves and with a set of spins \textonehalf{} localized at lattice sites.
The ground-state wave function and energy are calculated for the one-band Hubbard Hamiltonian on a 4\ifmmode\times\else\texttimes\fi{}4 cluster of the square lattice up to first and second order of the on-site Coulomb repulsion U, respectively. Changes induced by a deviation of the number of electrons from half-filling (one electron per lattice site) are closely followed. Charge-charge and spin-spin correlation functions are directly obtained from the ground-state wave function and show the existence of antiferromagnetic correlations that extend over the whole cluster at half-filling. Correlations among sites separated by more than a lattice site diminish very rapidly as holes are introduced into the cluster. The analysis is extended to the negative-U case. Ground-state properties change qualitatively.