FIG. 2: Diagrammatic building blocks for perturbative cal- culations: (a) the boson propagator g ( ω ; p , p $^{′}$), Eq. (3.16), (b) the interaction terms, Eqs. (3.12a-3.12c). FIG. 3: First order diagram from 〈S 2 〉 . (3.16) and Eq. (3.17) we obtain in the first order where n 0 p is the bare Fermi distribution. We can represent Eq. (3.20) in a somewhat different form using the identity Substituting Eq. (3.21) into Eq. (3.20), we see that the 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 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 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 as 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 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) 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 ΔΩ (1) = ΔΩ (1) H + ΔΩ (1) F (3.25) ΔΩ (1) = T 〈S 2 〉 + δ Ω $^{(1)}$. (3.19) δ Ω (1) = N$_{d}$ [ 2 n $^{2}$V (1) (3.18) + ( V$_{r,r}$ + 2 ¯ V ) n − 4 n $^{2}$V $^{(1)}$] ΔΩ (1) H = δ Ω (1) + δ $^{′}$Ω (1) = 2 N$_{d}$n $^{2}$V$_{q}$$_{=0}$ (3.24) ΔΩ (1) F = − N$_{d}$ ∫ V$_{q}$n 0 $_{p}$n 0 p + $_{q}$( d p d q ) (3.22) δ $^{′}$Ω (1) = N$_{d}$ ∫ [ 2 n 0 $_{p}$n 0 p + q ( V$_{r,r}$ + ¯ V ) + ( n 0 p + n 0 p + q ) ( V$_{q}$/ 2 − V$_{r,r}$ − ¯ V ) ] ( d p d q )(3.23) T ∑ ω n 0 p ′ − n 0 p iω − ε$_{p}$ + ε$_{p}$ ′ = 1 2 ( n 0 p ′ − n 0 p ) coth ε$_{p}$ ′ − ε$_{p}$ 2 T = n 0 $_{p}$n 0 p ′ − 1 2 ( n 0 p + n 0 p ′ ) . (3.21) T 〈S 2 〉 = T 2 ∑ ω ∫ 〈 Ψ$_{p}$$_{,}$$_{p}$$_{+}$$_{q}$$_{;}$$_{−}$$_{ω}$( ρ )Ψ$_{p}$ ′ + q , p ′ ; $_{ω}$( ρ$_{1}$ ) 〉 × ( − i ) u$_{1}$U$_{q}$$_{;}$$_{ω}$ ( σ, σ$_{1}$ ) θ $^{∗}$θ$_{1}$ [ n 0 p ′ + q − n 0 p ′ ] dρdρ$_{1}$ ( d p d p $^{′}$d q ) = − N$_{d}$T ∑ ω ∫ n 0 p + q − n 0 p iω − ε$_{p}$ + ε$_{p}$$_{+}$$_{q}$ [ V$_{q}$ − 2 V$_{0}$ ] ( d p d q ) (3.20) 12