Although assuming that doping creates holes primarily on oxygen sites, eve derive explicitly a single-band e6'ective Hamiltonian for the high-T, Cu-oxide superconductors. Cu-0 hybridization strongly binds a hole on each square of 0 atoms to the central Cu + ion to form a local singlet. This moves through the lattice in a similar~ay as a hole in the single-band effective Hamiltonian of the strongly interacting Hubbard model.

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The effective Hamiltonian of strongly correlated electrons on a square lattice is replaced by a renormalised Hamiltonian and the factors that renormalize the kinetic energy of holes and the Heisenberg spin-spin coupling are calculated using a Gutzwiller approximation scheme. The accuracy of this renormalization procedure is tested numerically and found to be qualitatively excellent. Within the scheme a resonant valence bond (RVB) wavefunction is found at half-filling to be lower in energy than the antiferromagnetic state. If the wavefunction is expressed in fermion operators, local SU(2) and U(l) invariance leads to a redundancy in the representation. The introduction of holes removes these local invariances and we find that a d-wave RVB state is lowest in energy. This state has a superconducting order parameter whose amplitude is linear in the density of holes.

We investigate the magnetic properties of the Cu-O planes in stoichiometric Sr n−1 Cu n+1 O 2n (n=3,5,7,...) which consist of CuO double chains periodically intergrown within the CuO 2 planes. The double chains break up the two-dimensional antiferromagnetic planes into Heisenberg spin ladders with n r = 1 2 (n − 1) rungs and n l = 1 2 (n + 1) legs and described by the usual antiferromagnetic coupling J inside each ladder and a weak and frustrated interladder coupling J ′ . The resulting lattice is a new two-dimensional trellis lattice. We first examine the spin excitation spectra of isolated quasi one dimensional Heisenberg ladders which exhibit a gapless spectra when n r is even and n l is odd ( corresponding to n=5,9,...) and a gapped spectra when n r is odd and n l is even (corresponding to n=3,7,...). We use the bond operator representation of quantum S = 1 2 spins in a mean field treatment with self-energy corrections and obtain a spin gap of ≈ 1 2 J for the simplest single rung ladder (n=3), in agreement with numerical estimates. We also present results of the dynamical structure factor S(q,ω). The spin gap decreases considerably on increasing the width of the ladders. For a double ladder with four legs and three rungs (n=7) we obtain a spin gap of only 0.1J. However, a frustrated coupling, such as that of a trellis lattice, introduced between the double ladders leads to an enhancement of the gap. Thus stoichiometric Sr n−1 Cu n+1 O 2n compounds with n=3,7,11,... will be frustrated quantum antiferromagnets with a quantum disordered or spin-liquid ground state.

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