# Hubbard model in magnetic field

Let's assume I have a Hubbard hamiltonian in a magnetic field.

$$H=-t\sum_{j,\sigma}c_{j,\sigma}^\dagger c_{j+1,\sigma}+c_{j+1,\sigma}^\dagger c_{j,\sigma}+U\sum_j n_j,_\uparrow n_j,_\downarrow +\sum_j (-1)^j s_j^z B$$.

We now assume a staggered confiugration, working in the mean field approximation (MFA), we have:

MFA: $$n_j,_\uparrow n_j,_\downarrow=n_j,_\uparrow \langle n_j,_\downarrow\rangle+n_j,_\downarrow \langle n_j,_\uparrow \rangle -\langle n_j,_\uparrow\rangle \langle n_j,_\downarrow \rangle$$ (1)

Staggered: $$\langle n_j, _\uparrow\rangle=n=1-\langle n_{j+1},_\uparrow\rangle$$ and similarly for down spin. (2)

Question: I now minimize the energy with respect to $$n$$, and get a value for $$n$$. What is the physical significance of this parameter and what relation does it have to the magnetic field?

My reasoning: For example let's take $$n=0.2$$, it tells me that the probabiltyo of seeing a spin down ($$\downarrow$$) on the j-th position is $$0.8$$ . But on the next position (j+1) the probabilty of seeing a up ($$\uparrow$$) spin is $$0.8$$, following (2). So in a chain of spins we get an equal number of down spins and up spins, no matter what $$n$$ we choose, which is strange to me because one "direction" should be prefered due to the magnetic field.