# Spin & the Fermi Sea

## Main Question or Discussion Point

I've found the simplest hopping Hamiltonian for fermions (diagonal in momentum space) has a so-called Fermi sea ground state.

$$H = -2 t \sum_k \cos(k) f^+_k f_k$$

(t is some parameter in units of energy).

How do I evaluate the expected value of the spin exchange operator $$S_x \cdot S_{x+1}$$ in this state? I am having trouble because the ground state obvious in the momentum basis, and it is not written in the position basis.