The term ˆ H (0) int can be decoupled as follows and φ (1) r ( τ ) is also a real periodic field, φ (1) r ( τ ) = φ (1) r ( τ + β ). The matrix ( V $^{(1)}$) − 1 r,r ′ is a matrix inverse to the matrix V (1) r,r ′ . The integral over φ (1) r ( τ ) in Eqs. (2.15, 2.16) converges because ( V $^{(1)}$) − 1 r,r ′ is positive definite. Introducing the field we write finally the partition function Z , Eq. (2.9), in the form where c ( τ ) and ¯ c ( τ ) are creation and annihilation oper- ators in the interaction representation [introduced anal- ogously to Eq. (2.10)], φ (0) r ( τ ) is a real field with the bosonic periodicity, φ (0) r ( τ ) = φ (0) r ( τ + β ). Before decoupling the term ˆ H (1) $_{int}$, Eq. (2.8), we rewrite it in the form where V (1) = ∑ r ′ V (1) r,r ′ , n = 〈 c + $_{rσ}$c$_{rσ}$ 〉 is the fermion den- sity per spin direction, and N$_{d}$ is the number of sites in the system. The second term in Eq. (2.13) renormalizes once more the chemical potential, such that now it equals and the Hamiltonian ˆ H$_{0}$ now given by with Z [ φ ] equal to with the modified chemical potential µ ′ from Eq. (2.14). We calculate the trace over the fermionic operators c, c + introducing an additional variable 0 ≤ u ≤ 1, re- placing φ$_{rσ}$ ( τ ) by uφ$_{rσ}$ ( τ ) in Eq. (2.19) and writing Z [ φ ] in the form In Eq. (2.21), Z$_{0}$ is the partition function of the ideal Fermi gas, where ϵ$_{k}$ are the eigenvalues of the Hamiltonian ˆ H$_{0}$ , Eq. (2.20), and The last term in Eq. (2.13), (2 n $^{2}$V $^{(1)}$N$_{d}$ / 2, is a trivial contribution to the thermodynamic potential Ω. The fermion density n should be calculated using the shifted chemical potential µ $^{′}$, Eq. (2.14), and, thus solv- ing a self-consistency equation. We assume that this has been done and consider the first term in ˆ H (1) int in Eq. (2.13) decoupling it by integration over another field φ (1) r ( τ ), is the electron Green function in the field φ$_{rσ}$ ( τ ). In Eq. (2.22), the operators c, ¯ c and ˆ H ( uφ ) int ( τ ) are written in the interaction representation using ˆ H$_{0}$ , Eq. (2.20), as the bare Hamiltonian and G (0) r,r ′ ,σ ( τ, τ $^{′}$) is the Green function of the ideal Fermi gas described by the Hamil- tonian ˆ H$_{0}$ . The symbol 〈 ... 〉 0 in Eq. (2.23) stands for the where G ( uφ ) r,r ′ ,σ ( τ, τ $^{′}$) = $_{−}$〈 T$_{τ}$ c$_{rσ}$ ( τ ) ¯ c$_{r}$ ′ σ ( τ $^{′}$) exp ( − ∫ β 0 ˆ H ( uφ ) int dτ )〉 0 〈 T$_{τ}$ exp ( − ∫ β 0 ˆ H ( uφ ) int dτ )〉 0 (2.23) T$_{τ}$ exp ( − ∫ β 0 ˆ H (1) int ( τ ) dτ ) (2.15) = ∫ T$_{τ}$ exp ( ∑ r,σ ∫ β 0 φ (1) r ( τ ) (¯ c$_{rσ}$ ( τ ) c$_{rσ}$ ( τ ) − n ) dτ ) × W$_{1}$ [ φ $^{(1)}$] Dφ (1) [ ∫ β 0 W$_{1}$ [ φ $^{(1)}$] Dφ (1) ] − 1 W$_{1}$ [ φ $^{(1)}$] = exp ( − 1 2 ∑ r,r ′ ∫ β 0 φ (1) r ( τ ) ( V (1) ) − 1 r,r ′ φ (1) r ′ ( τ ) dτ ) (2.16) Z$_{0}$ = ∏ k,σ ( 1 + exp ( − ϵ$_{k}$ T )) (2.22) Z [ φ ] = Z$_{0}$ exp [ ∑ r,σ ∫ β 0 ∫ 1 0 σφ$_{rσ}$ ( τ ) (2.21) × ( G ( uφ ) r,r,σ ( τ, τ + 0) − G (0) r,r,σ ( τ, τ + 0) ) dudτ ] µ ′ = µ − 1 2 ( V$_{0}$ + ¯ V ) + 2 nV $^{(1)}$. (2.14) ˆ H (1) int = − 1 2 ∑ r,r ′ ,σ,σ ′ V (1) r,r ′ ( c + $_{rσ}$c$_{rσ}$ − n ) ( c + r ′ σ ′ c$_{r}$ ′ σ ′ − n ) − 2 n ∑ r V $^{(1)}$c+ $_{rσ}$c$_{rσ}$ + N$_{d}$ 2 V $^{(1)}$(2 n ) $^{2}$, (2.13) ˆ H$_{0}$ = ∑ r,r ′ ,σ,σ ′ ( − t$_{r,r}$ ′ − δ$_{r,r}$ ′ µ $^{′}$) c + $_{rσ}$c$_{r}$ ′ $_{σ}$, (2.20) Z [ φ ] = Tr$_{¯}$ c,c [ exp ( − β ˆ H$_{0}$ ) T$_{τ}$ exp ( − ∫ β 0 ˆ H ( φ ) int ( τ ) dτ )] , ˆ H ( φ ) int ( τ ) = ∑ r,σ σφ$_{rσ}$ ( τ ) [¯ c$_{rσ}$ ( τ ) c$_{rσ}$ τ ) − n ] . (2.19) Z = ∫ Z [ φ ] W [ φ ] Dφ ∫ W [ φ ] Dφ , W [ φ ] = W$_{0}$ [ φ $^{(0)}$] W$_{1}$ [ φ $^{(1)}$] (2.18) × W$_{0}$ [ φ $^{(0)}$] Dφ (0) ( ∫ 0 W$_{0}$ [ φ $^{(0)}$] Dφ (0) ) − $^{1}$, W$_{0}$ [ φ $^{(0)}$] = exp ( − 1 2 V$_{0}$ ∑ r ∫ β 0 ( φ (0) r ( τ ) ) 2 dτ ) (2.12) T$_{τ}$ exp ( − ∫ β 0 ˆ H (0) int ( τ ) dτ ) (2.11) = ∫ T$_{τ}$ exp ( ∑ r,σ ∫ β 0 σφ (0) r ( τ ) ¯ c$_{rσ}$ ( τ ) c$_{rσ}$ ( τ ) dτ ) β − φ$_{rσ}$ ( τ ) = φ (0) r ( τ ) + σφ (1) r ( τ ) (2.17) 4