We study numerically the hole pairing induced by spin-wave exchange. The contact hole-hole interaction is taken into account as well. It is assumed that antiferromagnetic order is preserved at all scales relevant to pairing. The strongest pairing is obtained for the d-wave symmetry of the gap. Dependence of the value of the gap on hole concentration and temperature is presented. For the critical temperature we obtain Tc ∼ 100 K at the hole concentration δ ∼0.2-0.3.

Using the J 1 -J 2 model, we present a description of quantum phase transition from Néel ordered to the spin-liquid state based on the modified spin wave theory. The general expression for the gap in the spectrum in the spin-liquid phase is given.

Using an analytical variational approach we calculate the hole-hole contact interaction on the Néel background. Solution of the Bethe-Salpeter equation with this interaction gives bound states in d-and p-waves with binding energies close to those obtained by numerical methods. At t/J ≥ 2 − 3 the bound state disappears. In conclusion we discuss the relation between short range and long range interactions and analogy with the problem of pion condensation in nuclear matter.

A bi-layer quantum frustrated antiferromagnet is studied using an effective action approach. The action derived from the microscopical Hamiltonian has the form of the O(3) non-linear sigma model. It is solved in the mean field approximation with the ultraviolet cut off chosen to fit numerical results. The obtained phase diagram displays a decrease in the critical value of interlayer coupling with increase of in-plane frustration. The critical point for a single-layer frustrated antiferromagnet (the J 1 -J 2 model) is estimated to be J 2c = 0.19J 1 .Typeset using REVT E X where g c = 4π/Λ is the critical coupling (Λ is the cut off). The system is ordered at T = 0 when g < g c . Knowing the critical point at J 2 = J 3 = 0 allows to eliminate uncertainty from the cut off. Using J ⊥c = 2.5J, the cut off is found to be Λ = 1.1π. It is easy to see that the critical line g = g c has the form