Finite hole-doping of the $t-J$ and related models within the non-crossing approximation

The spin and hole dynamics in a 2D hole-doped quantum antiferromagnet is studied for small but finite hole-doping fractions within the two-dimensional $t-J$ and the related $t-t^\prime-t^{\prime\prime}-J$ model. The non-crossing approximation is used to sum up the self-energy diagrams and the Dyson's equations for both the hole and magnon Green's functions are solved self-consistently. The evolution with doping of the hole and magnon spectra with doping is determined, the Fermi surface topology is studied and the doping-dependent staggered magnetization of the system is computed. The latter determines the doping fraction up to which the spin wave theory remains a reasonable approximation to describe the doped antiferromagnet.