Spin Exchange in Fermi Sea - Evaluating S_x \cdot S_{x+1}

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The discussion focuses on evaluating the expected value of the spin exchange operator \( S_x \cdot S_{x+1} \) within the context of a Fermi sea ground state described by the Hamiltonian \( H = -2 t \sum_k \cos(k) f^+_k f_k \). The challenge arises from the need to transition from the momentum basis to the position basis to compute this expectation value. Participants suggest utilizing a Fourier transform to facilitate this conversion, ensuring accurate evaluation of the spin exchange operator in the desired basis.

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

[tex]H = -2 t \sum_k \cos(k) f^+_k f_k[/tex]

(t is some parameter in units of energy).

How do I evaluate the expected value of the spin exchange operator [tex]S_x \cdot S_{x+1}[/tex] 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.
 
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Fourier transform?
 

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