Flux phase, Néel antiferromagnetism, and superconductivity in thet-Jmodel

Using Hubbard operators, which allow one to separate charge and spin channels, we construct mean-field equations for a fluxlike state that coexists with Néel antiferromagnetism (AF). The formal analysis shows that the equations for the real and imaginary parts of the effective hopping amplitude are inconsistent for any θ≠0 (including θ=π/4 which gives flux 1/2). The hopping amplitude is slightly suppressed by exchange renormalization. The equations for the Néel magnetization give two solutions, a band-type solution at δ≠0 and a Heisenberg-type solution for hole density δ==0. The latter solution is absent at δ→0,δ≠0. We explain this by the observation that at Hubbard U==∞ the spectral weight of the minority-spin subband in the Néel AF cannot be moved to the high-energy region at an electron density ρ<1. This forbids the Heisenberg-type of solutions even at J≫t. Therefore, this conclusion is not restricted by the approximation used. At larger δ, band antiferromagnetism (AFM) arises. In this region the Néel magnetization decreases with increasing hole density. The region where band AFM exists decreases with increasing hopping parameter. For t/J≥(t/J${)}_{\mathit{c}}$, there are no magnetic solutions below a certain hole density. Moreover, at large values of t/J (t/J≥1.2) the region where antiferromagnetic order can exist is found to be very narrow and situated near the critical hole density δ=1/3.