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The term ˆ H (0) int can be decoupled as follows 

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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 ). (1) 

Before decoupling the term ˆ H $_{int}$, Eq. (2.8), we rewrite it in the form 

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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 

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 ), 

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where 

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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 

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with 

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and the Hamiltonian ˆ H$_{0}$ now given by 

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 

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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 

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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 

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