1. For the single-band Hubbard Model, I want to write down a mean field theory for the possibility of occurrence of ferromagnetism in the ground state. For the model given by

$$\hat{H} = -t \sum_{\langle i, j \rangle,\sigma}( \hat{c}^\dagger_{i,\sigma} \hat{c}_{j,\sigma} + \hat{c}^\dagger_{j,\sigma} \hat{c}_{i,\sigma}) + U \sum_{i=1}^N \hat{n}_{i\uparrow} \hat{n}_{i\downarrow}$$

Lets consider only the second term in the R.H.S of the Hamiltonian so I can write only that as

$$U \sum_{i=1}^N \hat{n}_{i\uparrow} \hat{n}_{i\downarrow} = \dfrac{U}{4}\Big[ \sum_{i=1}^N(\hat{n}_{i\uparrow} + \hat{n}_{i\downarrow})^2 + (\hat{n}_{i\uparrow} - \hat{n}_{i\downarrow})^2 \Big] $$

I can now interpret the first term as average number of particles and the second term as magnetization to get the mean field theory for only the second part of the R.H.S. Is this procedure correct?

  1. How do I obtain the self-consistency equation for the ferromagnetic order parameter from this.

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