content_image/437561.jpg content_image/437559.jpg content_image/437560.jpg content_image/437562.jpg content_image/437563.jpg content_image/437557.jpg content_image/437558.jpg
This interaction is considered as a main parameter of the model and is taken into account in zero approximation of our theory. The operator H ′ describes electron hopping between lattice sites of the crystal and is considered as a perturbation. 

We shall use the grand canonical partition function in our thermodynamic perturbation theory. 

The paper is organized in the following way. In section II we determine the definition of one-particle Matsubara Green’s functions by using α representation and develop the diagrammatic theory in the strong coupling limit. 

In section III we establish relation between the full thermodynamic potential and the renormalized one- particle Green’s function in the presence of additional integration over auxiliary constant of interaction λ and prove the stationarity theorem both for a special func- tional consisting of skeleton diagrams and for a renormal- ized thermodynamic potential shown to be its equivalent. 

We shall use the following definition of the Matsubara Green’s functions in the interaction representation 

<img src='content_image/437561.jpg'>

where x stands for ( − → x , σ, τ ), index c of 〈 ... 〉 c 0 means the connected part of the diagrams and 〈 ... 〉$_{0}$ means thermal average with zero order partition function e $_{−}$βH 0 T re $_{−}$βH 0 . 

We use the series expansion for the evolution operator U ( β ) with some generalization because we introduce the auxiliary constant of interaction λ and use λH ′ instead H $^{′}$: 

with T as the chronological operator. At the last stage of calculation this constant λ will be put equal to 1. 

The correspondence between definition (6) and usual one [12] is the following: 

<img src='content_image/437559.jpg'>

As a result of application of the Generalized Wick The- orem we obtain for propagator (6) the diagrammatic contributions depicted on the Fig.1 In superconducting 

## II. DIAGRAMMATIC THEORY 

<img src='content_image/437560.jpg'>

state, unlike the normal state, the propagator lines do not contain arrows which determine the processes of cre- ation and annihilation of electrons because indices α can take two values α = ± 1 and every vertex of the diagram describes different possibilities. 

In Fig.1 the diagram a) is the zero order propagator, the diagram b) and more complicated diagrams of such kind are of chain type. They correspond to the contribu- tion of the ordinary Wick theorem and give the Hubbard I approximation. The contributions of the diagrams c) and d) of Fig.1 are 

<img src='content_image/437562.jpg'>

<img src='content_image/437563.jpg'>

where 〈 ... 〉 ir 0 means the irreducible two-particle Green’s function [2 − 5] and summation or integration is under- stood here and below when two repeated indices are present. Spin index has been omitted for simplicity. In the diagram c) the equality of lattice sites indices − → x = − → 1 ′ = − → 1 = − → x ′ is assumed and in diagram d) − → x = − → 1 ′ = − → 2 = − → x $^{′}$. The diagrams Fig.1 c), d) and e) contain irreducible two-particles Green’s functions, de- picted as the rectangles. In higher orders of perturba- tion theory more complicated many-particle irreducible Green’s functions G (0) ir n [1 , 2 , ..., n ] appear. These func- tions are local, i.e. with equal lattice site indices. There- fore the diagram c) in Fig.1 can be dropped since it con- tains a vanishing matrix element, t ( − → x − − → x ) = 0. The pro- cess of renormalization of the tunneling amplitude shown in the diagrams c) and d) leads to the replacement of the bare tunneling matrix element αt$_{α}$ ( − → x ′ − − → x ) in c) by a renormalized quantity T α $^{$^{′}$α}$( x $^{′}$| x ). This process is deter- mined by the equation 

<img src='content_image/437557.jpg'>

where 

<img src='content_image/437558.jpg'>

and G α 1 α 2 is the full one-particle propagator. The quan- tity T α $^{′}$α is shown in the diagrams as a double dashed line. 

We then introduce the notion of correlation function Λ αα $^{′}$( x | x $^{′}$) which is the infinite sum of strongly connected parts of propagator’s diagrams. If we now omit from these diagrams all those contained in the process of renor- malization of the tunneling matrix element, we obtain 

