Incommensurate spin density wave as a signature of spin-orbit coupling and precursor of topological superconductivity

On a square lattice, the Hubbard model at half filling reduces to the Heisenberg model and exhibits antiferromagnetism. When doped away from half filling this model gives rise to d-wave superconductivity. This behavior is reminiscent of the phenomenology of the cuprate family with their high T(c)d-wave superconductivity and their antiferromagnetic parent compound. It is therefore interesting to study an extension of the Hubbard model which includes spin orbit coupling. We have previously studied this model away from half filling [see, for example, Farrell and Pereg-Barnea, Phys. Rev. B 89, 035112 (2014)] and found that the addition of spin-orbit coupling and Zeeman field leads to topological superconductivity with d + id pairing function. In this paper we are interested in the 'parent compound' of this state. Namely, we study the half filling, strong coupling limit of the square lattice Hubbard model with spin orbit coupling and Zeeman field. The strong coupling expansion of the model is a spin model which contains compass anisotropy and Dzyaloshinsky-Moriya interaction on top of the usual Heisenberg term. We analyze this spin model classically and find an incommensurate spin density wave (ISDW) for low Zeeman fields. This ISDWhas a wave vector (Q) over right arrow which deviates from (pi, pi) by an amount which is proportional to the spin-orbit coupling and can therefore serve as a signature. We study the stability of the ISDW phase using spin wave theory and find a stable and an unstable region. At higher but moderate Zeeman fields we find a tilted antiferromagnet and a ferromagnet at high Zeeman fields.