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Lexi Chandler

Lexi Chandler

Answered question

2022-05-10

M = 4 U 3 ϕ
where M is the ferromagnetic order parameter and ϕ is the auxiliary field from the Hubbard-Stratonovich transformation. The book argues that because the above equation is correct, the mean field theory which is derived from the Hartree-Fock approach is equivalent to the the saddle point approximation formalism for H-S transformation auxiliary field Lagrangian. But I can not understand the equation.

Answer & Explanation

nelppeazy9v3ie

nelppeazy9v3ie

Beginner2022-05-11Added 22 answers

You can always derive connection between auxiliary fields in saddle point and averages of fermion operators they are coupled to. For example, take a full decoupled action:
S = 1 2 m ϕ ( q ) ϕ ( q ) + g ϕ ( q ) M ( q ) + S f ,
where ϕ is an auxiliary field, S f denotes fermonic part of action and M ( q ) = 1 2 k α β ψ ¯ α ( k q ) ψ β ( k ) τ α β z is just a FT of spin in z direction. Now, consider a variational derivative with respect to ϕ q in a saddle point (where δ δ ϕ q S = 0 holds):
1 Z δ δ ϕ q Z = 1 m ϕ M F ( q ) + g M ( q ) M F = 1 Z D ( ψ ¯ , ψ ) e S δ δ ϕ q S = 0
Which means that:
1 g m ϕ ( q ) = M ( q ) .
I hope it answers your question.

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