content_image/423151.jpg content_image/423171.jpg content_image/423152.jpg content_image/423170.jpg content_image/423176.jpg content_image/423175.jpg content_image/423172.jpg content_image/423173.jpg content_image/423174.jpg content_image/423169.jpg
<img src='content_image/423151.jpg'>

as 

where V (1) = − ∑ r ′ ˜ V$_{r,r}$ ′ + ¯ V and n is the (unperturbed) fermion density per one spin direction at site r . The first term in Eq. (3.18) comes from the shift of the thermody- namic potential, Eq. (2.13). The second line originates from the shift of the chemical potential, Eq. (2.14), and is obtained using the standard relation ∂ Ω /∂µ = − 2 nN$_{d}$ (factor 2 is due to spin). So, the total first order contri- bution ΔΩ (1) to the thermodynamic potential takes the form 

Diagrammatically, T 〈S 2 〉 is represented in Fig. 3. The correspondence of Fig. 3 to the conventional diagram- matic technique becomes evident if we associate the double-line with two single lines representing the origi- nal fermionic propagators. Then, one can easily imagine that the diagram in Fig. 3 should yield the Fock contri- bution [Fig. 1 (a)]. Actually, this is not completely cor- rect because this diagram contains also some other terms. Calculating T 〈S 2 〉 with the help of the contraction rule 

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<img src='content_image/423152.jpg'>

<img src='content_image/423170.jpg'>

(3.16) and Eq. (3.17) we obtain in the first order 

<img src='content_image/423176.jpg'>

where n 0 p is the bare Fermi distribution. 

We can represent Eq. (3.20) in a somewhat different form using the identity 

<img src='content_image/423175.jpg'>

term with V$_{q}$ in Eq. (3.20), indeed, participates in form- ing the Fock contribution ΔΩ (1) F to the thermodynamic potential 

However, there are other contributions δ $^{′}$Ω (1) in Eq. (3.20) that should be added to that of Eq. (3.18) 

The structure of the integrand in Eq. (3.23) allows one easily integrate over p and q . Then, adding Eqs. (3.18, 3.23) to each other and using Eqs. (2.5-2.7) we obtain 

<img src='content_image/423172.jpg'>

which is the standard Hartree contribution. The artificial interaction ¯ V introduced in the model drops out from the final formulas as it should. 

Thus, we come in the first order to the standard ex- pression for the correction ΔΩ (1) to the thermodynamic potential 

where ΔΩ (1) H and ΔΩ (1) F are the Hartree, Eq. (3.24), and Fock, Eq. (3.22), contributions, respectively. 

## D. Second order of the perturbation theory 

The perturbation expansion in the conventional field theory for interacting fermions leads to the diagrams shown in Fig. 4. Only the diagrams in Fig. 4 (a) and (b) 

<img src='content_image/423173.jpg'>

<img src='content_image/423174.jpg'>

<img src='content_image/423169.jpg'>

