M = 4 U 3 ϕwhere 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.

Question

M  =                              4          U                3              ϕwhere 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.

nelppeazy9v3ie · Accepted Answer

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  =  0Which means that:  −      1          g      m        ϕ  (  q  )  =      ⟨    M    (    q    )    ⟩    .I hope it answers your question.

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