Magnetic properties of the Hubbard sigma model with three-dimensionality.

Abstract

It has been broadly accepted that the magnetism may play an important role in the high-${\mathit{T}}_{\mathit{c}}$ superconductivity in the lamellar ${\mathrm{CuO}}_{2}$ materials. In this paper we give a quantitative study of the magnetic properties in and around the N\'eel ordered state of three-dimensional quantum antiferromagnets such as ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$ with and without small hole doping. The study is based on the model we call the Hubbard \ensuremath{\sigma} model. It is a (3+1)-dimensional ${\mathrm{CP}}^{1}$ or ${\mathit{S}}^{2}$ nonlinear \ensuremath{\sigma} model derived as an effective field theory describing the low-energy spin dynamics of the three-dimensional Hubbard t-J model with a weak interlayer coupling. The effect of hole dynamics is taken into account in the leading approximation by substituting the ${\mathrm{CP}}^{1}$ coupling and the spin-wave velocity with ``effective'' ones including the tree and one-loop corrections by hole fermions. Stationary-phase equations for the one-loop effective potential of the ${\mathit{S}}^{2}$ model are analyzed. Based on them, various magnetic properties of the system, such as N\'eel temperature, spin-correlation length, staggered magnetization, specific heat, and susceptibility as functions of anisotropic parameters, temperature, etc., are investigated in detail. The results show that our anisotropic field-theory model with certain values of parameters gives a good description of the magnetic properties indicated by experiments on ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$ in both the ordered and the disordered phases. These results are supported in part by the renormalization-group analysis. In the doped case it is observed that the hopping effect of holes is very large and reduces the N\'eel temperature significantly.