Magnetic excitations of a doped two-dimensional antiferromagnet

Magnetic excitations of the two-dimensional (2D) t-J model are considered in the presence of a small concentration of holes c. The spin-wave approximation used implies long-range antiferromagnetic ordering from the beginning. Migdal’s theorem is shown to be valid for the model considered. The energy spectrum of the magnons is determined with the help of the one-pole approximation for the hole Green’s function. If the concentration of mobile holes is larger than a critical value an additional branch of overdamped magnons arises near the Γ and M points of the Brillouin zone. This is connected with the generation of electron-hole pairs (the Stoner excitations) by magnons. The appearance of such excitations means the destruction of the long-range antiferromagnetic order. For parameters presumably realized in cuprate perovskites this happens for several percent of holes per site. The relation between the critical concentration and the hole concentration destroying the 3D long-range ordering in ${\mathrm{La}}_{2\mathrm{-}\mathit{x}}$ ${\mathrm{Sr}}_{\mathit{x}}$ ${\mathrm{CuO}}_4$ is discussed. The arising short-range order is characterized by the instantaneous spin correlation length ξ∼${\mathit{c}}^{\mathrm{-}1/2}$, in coincidence with the experimentally observed dependence in this crystal.