Magnetic phases of the two-dimensional Hubbard model at low doping

We study the equilibrium spin configuration of the two-dimensional (2D) Hubbard model at low doping, when a long-range magnetic order is still present. We use the spin-density-wave formalism and identify three different low-doping regimes depending on the value of z=4U${\mathrm{\chi }}_{2\mathrm{D}}$ where ${\mathrm{\chi }}_{2\mathrm{D}}$ is the Pauli susceptibility of holes. When z<1, the collinear antiferromagnetic state remains stable upon low doping. As candidates for the ground state for z>1 we first examine the planar spiral phases with the pitch Q either in one or in both spatial directions. Mean-field calculations favor the spiral (π,Q) phase for 1<z<2, and (Q,Q) phase for z>2. Analysis of the bosonic modes of the spiral state shows that the (Q,Q) state has a negative longitudinal stiffness and is unstable towards domain-wall formation. For the (π,Q) state, the longitudinal stiffness is positive, but to the lowest order in the hole concentration, there is a degeneracy between this state and a whole set of noncoplanar states. These noncoplanar states are characterized by two order parameters, one associated with a spiral, and the other with a commensurate antiferromagnetic ordering in the direction perpendicular to the plane of a spiral. We show that in the next order in the hole concentration this degeneracy is lifted, favoring noncoplanar states over the spiral. The equilibrium, noncoplanar configuration is found to be close to the Néel state with a small spiral component whose amplitude is proportional to the square root of the hole concentration. These findings lead to a new scenario of spin reorientation upon doping in Hubbard antiferromagnets.