Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state

Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state

Diagrammatic analysis for normal state of Hubbard model proposed in our previous paper is generalized and used to investigate superconducting state of this model. We use the notion of charge quantum number to describe the irreducible Green's function of the superconducting state. As in the pr...

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Дата:2012
Автори: Moskalenko, V.A., Dohotaru, L.A., Digor, D.F., Cebotari, I.D.
Формат: Стаття
Мова:English
Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2012
Назва видання:Физика низких температур
Теми:
Свеpхпpоводимость, в том числе высокотемпеpатуpная
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/117894
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Цитувати:Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state / V.A. Moskalenko, L.A. Dohotaru, D.F. Digor, I.D. Cebotari // Физика низких температур. — 2012. — Т. 38, № 10. — С. 1167–1174. — Бібліогр.: 21 назв. — англ.

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spelling irk-123456789-1178942017-05-28T03:03:44Z Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state Moskalenko, V.A. Dohotaru, L.A. Digor, D.F. Cebotari, I.D. Свеpхпpоводимость, в том числе высокотемпеpатуpная Diagrammatic analysis for normal state of Hubbard model proposed in our previous paper is generalized and used to investigate superconducting state of this model. We use the notion of charge quantum number to describe the irreducible Green's function of the superconducting state. As in the previous paper we introduce the notion of tunneling Green's function and of its mass operator. This last quantity turns out to be equal to correlation function of the system. We proved the existence of exact relation between renormalized one-particle propagator and thermodynamic potential which includes integration over auxiliary interaction constant. The notion of skeleton diagrams of propagator and vacuum kinds were introduced. These diagrams are constructed from irreducible Green's functions and tunneling lines. Identity of this functional to the thermodynamic potential has been proved and the stationarity with respect to variation of the mass operator has been demonstrated. 2012 Article Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state / V.A. Moskalenko, L.A. Dohotaru, D.F. Digor, I.D. Cebotari // Физика низких температур. — 2012. — Т. 38, № 10. — С. 1167–1174. — Бібліогр.: 21 назв. — англ. 0132-6414 PACS: 71.27.+a, 71.10.Fd http://dspace.nbuv.gov.ua/handle/123456789/117894 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Свеpхпpоводимость, в том числе высокотемпеpатуpная
Свеpхпpоводимость, в том числе высокотемпеpатуpная
spellingShingle Свеpхпpоводимость, в том числе высокотемпеpатуpная
Свеpхпpоводимость, в том числе высокотемпеpатуpная
Moskalenko, V.A.
Dohotaru, L.A.
Digor, D.F.
Cebotari, I.D.
Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state
Физика низких температур
description Diagrammatic analysis for normal state of Hubbard model proposed in our previous paper is generalized and used to investigate superconducting state of this model. We use the notion of charge quantum number to describe the irreducible Green's function of the superconducting state. As in the previous paper we introduce the notion of tunneling Green's function and of its mass operator. This last quantity turns out to be equal to correlation function of the system. We proved the existence of exact relation between renormalized one-particle propagator and thermodynamic potential which includes integration over auxiliary interaction constant. The notion of skeleton diagrams of propagator and vacuum kinds were introduced. These diagrams are constructed from irreducible Green's functions and tunneling lines. Identity of this functional to the thermodynamic potential has been proved and the stationarity with respect to variation of the mass operator has been demonstrated.
format Article
author Moskalenko, V.A.
Dohotaru, L.A.
Digor, D.F.
Cebotari, I.D.
author_facet Moskalenko, V.A.
Dohotaru, L.A.
Digor, D.F.
Cebotari, I.D.
author_sort Moskalenko, V.A.
title Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state
title_short Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state
title_full Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state
title_fullStr Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state
title_full_unstemmed Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state
title_sort stationary property of the thermodynamic potential of the hubbard model in strong coupling diagrammatic approach for superconducting state
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2012
topic_facet Свеpхпpоводимость, в том числе высокотемпеpатуpная
url http://dspace.nbuv.gov.ua/handle/123456789/117894
citation_txt Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state / V.A. Moskalenko, L.A. Dohotaru, D.F. Digor, I.D. Cebotari // Физика низких температур. — 2012. — Т. 38, № 10. — С. 1167–1174. — Бібліогр.: 21 назв. — англ.
series Физика низких температур
work_keys_str_mv AT moskalenkova stationarypropertyofthethermodynamicpotentialofthehubbardmodelinstrongcouplingdiagrammaticapproachforsuperconductingstate
AT dohotarula stationarypropertyofthethermodynamicpotentialofthehubbardmodelinstrongcouplingdiagrammaticapproachforsuperconductingstate
AT digordf stationarypropertyofthethermodynamicpotentialofthehubbardmodelinstrongcouplingdiagrammaticapproachforsuperconductingstate
AT cebotariid stationarypropertyofthethermodynamicpotentialofthehubbardmodelinstrongcouplingdiagrammaticapproachforsuperconductingstate
first_indexed 2025-07-08T12:58:45Z
last_indexed 2025-07-08T12:58:45Z
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fulltext © V.A. Moskalenko, L.A. Dohotaru, D.F. Digor, and I.D. Cebotari, 2012 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 10, pp. 1167–1174 Stationary property of the thermodynamic potential of the Hubbard model in strong coupling diagrammatic approach for superconducting state V.A. Moskalenko1,2, L.A. Dohotaru3, D.F. Digor1, and I.D. Cebotari1 1Institute of Applied Physics, Moldova Academy of Sciences, Chisinau 2028, Moldova 2The Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia 3Technical University, Chisinau 2004, Moldova E-mail: moskalen@thsun1.jinr.ru Received February 13, 2012, revised April 17, 2012 Diagrammatic analysis for normal state of Hubbard model proposed in our previous paper is generalized and used to investigate superconducting state of this model. We use the notion of charge quantum number to describe the irreducible Green's function of the superconducting state. As in the previous paper we introduce the notion of tunneling Green's function and of its mass operator. This last quantity turns out to be equal to correlation func- tion of the system. We proved the existence of exact relation between renormalized one-particle propagator and thermodynamic potential which includes integration over auxiliary interaction constant. The notion of skeleton diagrams of propagator and vacuum kinds were introduced. These diagrams are constructed from irreducible Green's functions and tunneling lines. Identity of this functional to the thermodynamic potential has been proved and the stationarity with respect to variation of the mass operator has been demonstrated. PACS: 71.27.+a Strongly correlated electron systems; heavy fermions; 71.10.Fd Lattice fermion models. Keywords: strong correlated electron system, Dyson equation, Green's function, periodic Anderson model. 1. Introduction The present paper generalizes our previous work [1] on diagrammatic analysis of the normal state of the Hubbard model [2–4] to the superconducting state. Now we shall assume the existence of pairing of charge carriers and non-zero Bogolyubov quasi-averages [5] and, consequently, of the Gor'kov anomalous Green's functions [6]. The central idea of standard BCS theory of convention- al superconductivity is formation of Cooper pairs due to the presence of attractive interaction between electrons. Such attractive interaction can be of electron–phonon kind with mechanism based on the polarizability of ionic lattice in metal. After the discovery in 1986 of high-temperature superconductivity in cuprate compounds with layered pe- rovskite structure begins the era of unconventional super- conductivity with possible alternative mechanisms of su- perconductivity. One of such possible mechanism is spin fluctuation exchange [7] one based on the conception of spin polarization of electrons. One of the most frequently used model for unconven- tional superconductivity is the Hubbard model. We shall discuss below its properties. The main property of the Hubbard model consists in the existence of strong electron correlations and, as a result, of the new diagrammatic elements with the structure of Kubo cumulants and named by us as irreducible Green's func- tions. These functions describe the main charge, spin and pairing fluctuations of the system. The new diagram technique for such strongly correlated systems has been developed in our earlier papers [8–18]. This diagram technique uses the algebra of Fermi operators and relies on the generalized Wick theorem which con- tains, apart from usual Feynman contributions, additional irreducible structures. These structures are the main ele- ments of the diagrams. In superconducting state, unlike the normal one, the ir- reducible Green's functions can contain any even number of fermion creation and annihilation operators, whereas in normal state the number of both kinds is equal. Therefore we need an automatic mathematical mechanism which V.A. Moskalenko, L.A. Dohotaru, D.F. Digor, and I.D. Cebotari 1168 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 10 takes into account all the possibilities to consider the inter- ference of the particles and holes in the superconducting state. With this purpose we use the notion of charge quantum number, introduced by us in [8] and called α-number, which has two values = 1α ± according to the definition , = 1; = , = 1, C C C α + α⎧⎪ ⎨ α −⎪⎩ (1) were C is a fermion annihilation operator. In this new representation the tunneling part of the Hubbard Hamilto- nian can be rewritten in the form = 1,1 = ( ) 1= ( ) , 2 H t C C t C C ′σ σ ′σ −α α ′α σ σ ′α − σ +′ ′ − = ′α − ∑∑ ∑ ∑∑ x x xx x x xx x x x x (2) with the definition of the tunneling matrix elements 1 1 ( ) = ( ), ( ) = ( ), ( = 0) = 0. t t t t t − ′ ′− − ′ ′− − x x x x x x x x x (3) In this charge quantum number representation the operator H ′ has an additional multiple α for every vertex of the diagrams and additional summation over α . All the Green's functions depend of this number. In interaction representation operator H ′ has a form 1( ) = ( ) ( 0 ) ( ). 2 H t C C−α + α ′α σ σ ′ασ ′ ′τ α − τ+α τ∑∑ x x xx x x (4) The main part of the Hubbard Hamiltonian 0 0 0 0 = , = , = i i i i i i i H H H H H H C C Un n+ σ σ ↑ ↓ σ ′+ −μ + ∑ ∑ (5) contains the local part 0 ,H where μ is the chemical po- tential and U is the Coulomb repulsion of the electrons. 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 Sec. 2 we define the one-particle Matsubara Green's functions in terms of α representation and develop the diagrammatic theory in the strong coupling limit. In Sec. 3 we establish relation between the full thermo- dynamic potential and the renormalized one-particle Green's function in the presence of additional integration over auxiliary constant of interaction λ and prove the sta- tionarity theorem both for a special functional consisting of skeleton diagrams and for a renormalized thermodynamic potential shown to be its equivalent. 2. Diagrammatic theory We shall use the following definition of the Matsubara Green's functions in the interaction representation 0 ( | ) = ( ) ( ) ( ) , c G x x TC C U′ ′αα α −α ′ ′σ σ′ ′− τ τ βx x (6) where x stands for ( , , )σ τx , index c of 0... c〈 〉 means the connected part of the diagrams and 0...〈 〉 means thermal average with zero-order partition function 0 0 e / Tr e .H H−β −β 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 H ′λ instead :H ′ 0 ( ) = exp ( ( ) ),U T H d β λ ′β −λ τ τ∫ (7) 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 [13] is the following: 1,1 0 , 1, 1 0 , 1,1 0 , 1, 1 1,1 ( | ) = ( ) ( ) ( ) = ( , | , ), ( | ) = ( ) ( ) ( ) = ( , | , ), ( | ) = ( ) ( ) ( ) = ( , | , ), ( | ) = ( | ). c c c G x x TC C U G G x x TC C U F G x x TC C U F G x x G x x ′ ′σσ λλ ′σ σ − ′ ′σ σ λλ ′σ σ − ′ ′σ σ λλ ′σ σ − − λ λ ′ ′− τ τ β = λ ′ ′τ τ ′ ′− τ τ β = λ ′ ′τ τ ′ ′− τ τ β = λ ′ ′τ τ ′ ′− xx x x x x x x x x x x (8) As a result of application of the generalized Wick theorem we obtain for propagator (6) the diagrammatic contribu- tions depicted on the Fig. 1. In superconducting state, unlike the normal state, the propagator lines do not contain arrows which determine the processes of creation and annihilation of electrons be- cause indices α can take two values = 1α ± 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 contribution of the ordinary Wick theorem and give the Hubbard I ap- proximation. The contributions of the diagrams (c) and (d) of Fig. 1 are Stationary property of the thermodynamic potential of the Hubbard model Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 10 1169 ir 1 1 1 1 1 0 1 1 ir 1 2 1 2 0 (0) 1 21 2 1 21 2 1(c) : ( ) ( 0 ) ( ) ( ) 2 ( ), 1(d) : ( ) ( ) ( ) ( ) 2 ( ) ( ) ( , | , ) , TC C C C t TC C C C t t G −α α ′α + −α ′′ α −α α ′α −α ′′ α α α α ′τ τ + α τ τ × ′× α − ′τ τ τ τ × ′ ′ ′× α − α − τ τ x x11 x x21 1 1 1 1 2 2 1 2 where ir 0...〈 〉 means the irreducible two-particle Green's function [2–5] and summation or integration is understood 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, depicted as the rectangles. In higher orders of perturbation theory more complicated many- particle irreducible Green's functions (0)ir [1, 2,..., ]nG n ap- pear. These functions are local, i.e. with equal lattice site indices. Therefore the diagram (c) in Fig. 1 can be dropped since it contains a vanishing matrix element, ( ) = 0.t −x x The process of renormalization of the tunneling amplitude shown in the diagrams (c) and (d) leads to the replacement of the bare tunneling matrix element ( )tα ′α −x x in (c) by a renormalized quantity ( | )T x x′αα ′ . This process is de- termined by the equation 1 1 (0)(0) 1( | ) = ( | ) ( | )T x x T x x T x x′′ α α′ α αα α ′σ σ ′ ′σ σ σ σ′ ′ ′+ × 1 2 2 1 2 2 (0) 1 2 2( | ) ( | ),G x x T x xα α α α σ σ σ σ× (9) where (0) ( | ) = ( ) ( 0 ) ,T x x t′α α + ′ ′αα α σσ′σ σ ′ ′ ′δ α − δ τ − τ −α δx x (10) and 1 2Gα α is the full one-particle propagator. The quanti- ty T ′α α is shown in the diagrams as a double dashed line. We then introduce the notion of correlation function ( | )x x′αα ′Λ which is the infinite sum of strongly con- nected 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 the skeleton diagrams for correlation function. In such skele- ton diagrams we replace thin dashed lines by double dashed lines and obtain the definition of ( | )x x′αα ′Λ shown in the Fig. 2. There are two kinds of λ dependence in the diagrams of Fig. 2. One is conditioned by dependence of T ′αα λ and the second is determined by λ being an explicit pre-factor in the diagrams. In Hubbard I approximation only the free propagator line is taken into account. All the contributions of Fig. 2 except the last one are local and their Fourier re- presentation is independent of momentum. Only these dia- grams are taken into account in dynamical mean field theory [19]. The last diagram of Fig. 2 has the Fourier re- presentation which depends of momentum. As a result of diagrammatic analysis we can formulate the Dyson-type equation for full one-particle Green's func- tion ( = , ) :x τx 1 2 001 2 1 2 ( | ) = ( | )G x x x x d d ββ ′ ′αα αα ′ ′σσ σσ σ σ ′ ′Λ + τ τ ×∑ ∑ ∫∫ x x 1 1 2 2 1 1 2 2 (0) 1 1 2 2( | ) ( | ) ( | ).x x T x x G x x′αα α α α α ′σσ σ σ σ σ ′× Λ (11) This equation can be written in the operator form: 0 1 0 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ= (1 ) = (1 ) .G T T− −−Λ Λ Λ − Λ (12) Using Eqs. (9) and (12) we obtain the Dyson equation for the tunneling Green's function 0 0 1 0 1 0ˆ ˆ ˆ ˆ ˆ ˆ ˆ= (1 ) = (1 ) ,T T T T T− −−Λ − Λ (13) + ...+ G �� � � + + (a) λ 2 + + λ 2 2 (d) (f) λ 3 48 (b) (e) + λ 3 6 = �� x �� x�� x–� � x� � –� � x� � � –� � x� � �� x �1, 1–�1, 1� –�1, 1� (с) –� � x� � �1, 1 �1, 1 �2, 2–�1, 1� �� x –� � x� � –� � x� � –� � x� � – 2�2, � –�1, 1� �� x – 3�3, � �2, 2 – 2�2, � �3, 3 �1, 1 �3, 3 �2, 2 �1, 1 �� x – 2�2, � – 3�3, � –�1, 1� Fig. 1. The examples of the first orders perturbation theory diagrams for propagator. Solid thin lines depict zero order one-particle Green's functions and rectangles depict two- and four-particle irreducible Green's functions. Thin dashed lines correspond to tunneling matrix elements. Double solid line corresponds to renormalized propagator. V.A. Moskalenko, L.A. Dohotaru, D.F. Digor, and I.D. Cebotari 1170 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 10 where the correlation function Λ has the role of mass ope- rator for the renormalized tunneling Green's function. In Appendix A we demonstrate the equivalence of the Eq. (11) to usual [6] representation of superconducting Green's functions. 3. Thermodynamic potential The thermodynamic potential of the system is deter- mined by the connected part of the mean value of evolu- tion operator: 0 0 1( ) = ( ) ,cF F Uλλ − β β (14) with λ equal to 1. In the Fig. 3 are depicted the first order diagrams for 0( ) cUλ β . The notations in Fig. 3 are = ( , )nn τn and = ( , 0 ).n nn +′ ′ τ + αn The first three diagrams in Fig. 3 are of chain type and correspond to the Hubbard I approximation. The next diagrams contain the rectangles which represent our irre- ducible Green's functions. Indeed, some of these dia- grams are equal to zero when the dashed lines are self- closed by virtue of the relation (0) = 0.t However, when these dashed lines are replaced by renormalized quanti- ties Tλ their contributions are different from zero and should be retained. Such renormalized tunneling quanti- ties will be used in the next part of the paper. The contri- Fig. 2. The skeleton diagrams for correlation function ( | ).x x′αα ′Λ The rectangles depict the many-particles irreducible Green's func- tion. The double dashed lines depict the full tunneling Green function ( | ).T x x′αα λ ′ + � 2 ...++ � 3 6 � 2 8 + � 3 48 +−� �� � � ( | ) =x x� ��������, ,x – , ,� �x� � � T� T� T� T� T� T� T� T� T� ��������, ,x ��������, ,x ��������, ,x ��������, ,x – , ,� �x� � � – , ,� �x� � � – , ,� �x� � � – , ,� �x� � � Fig. 3. The first orders of perturbation theory contributions. The skeleton diagrams for functional ( ).Y ′ λ � U�( )� c 0 = − � 2 − � 2 4 − � 3 6 + ... �1, 1 –�2,�1, 1 �1, 1 �2, 2 �2, 2 �3, 3 − − + + �1, 1 �1, 1 �1, 1 3 4 �� 8 2 �2, 2 �2, 2 �3, 3 3 48 � 4 384 � �3, 3 �2, 2 �4, 4 �1, 1 �3, 3 �2, 2 + − − + ... − �1, 1 �3, 3 �1, 1 4 48 � 4 16 � �4, 4 �2, 2 �2, 2 �4, 4 �3, 3 Stationary property of the thermodynamic potential of the Hubbard model Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 10 1171 butions of the fifth and eighth diagrams on the right-hand side of Fig. 3 are 2 ir 31 1 1 2 3 1 1 0 1 ( ) ( ) ( ) ( 0 ) 4 TC C C Cαα −α−α + ′ ′− τ τ τ τ + α ×1 32 1 1 2 31 2 3 ( ) ( ) ( )t t tα α α′ ′ ′× α − α − α − ×1 1 2 2 3 3 2 3(0) 2 3( , | , ),G α α ′× τ τ2 3 32 4 ir 1 1 2 3 4 0 1 ( ) ( ) ( ) ( ) 48 TC C C Cα α−α −α ′ ′+ τ τ τ τ ×1 32 4 1 21 2 ( ) ( )t tα α′ ′×α − α − ×1 1 2 2 3 2 1 ir 4 4 3 2 12 0 ( ) ( ) ( ) ( )TC C C Cα −α α −α ′ ′× τ τ τ τ ×4 3 1 3 43 4 ( ) ( ) ,t tα α′ ′×α − α −3 3 4 4 respectively. Comparison of the diagrams of Fig. 1 for the ( )nG ′αα (nth order of perturbation theory for the one-particle propaga- tor) to the contributions of Fig. 2 for 1 0 ( ) cnU + λ β ((n + 1)-th order for evolution operator) allows us to establish the fol- lowing simple relation ( 1) :n ≥ 1 ( 1) 1 1 1 10 10 1 ( ) = ( – ) 2 cn d U t λ + αλ ′α σ λβ ′β − λ α × λ ∑∑∫ 11 1 1 1 1 1 1 ( ) 1( | 0 ) ,nG α α + σ λ ′× − −α1 1 and as the result we have 1 1 1 10 110 1 1( ) = ( ) 2 c d U t λ λ α ′α σ λ ′β − β λ α − × λ ∑∑∫ 11 1 1 1 1 11 1 01 1 1 10 1 ˆˆ( | 0 ) Tr ( ). 2 d G T G λ α α + λσ λ λ′× − −α = − λ λ∫1 1 (15) Taking into account Eqs. (12) and (13) we obtain 1 10 1 110 1 ˆ ˆ( ) = Tr ( ). 2 c d U T λ λ λ λ λ β − λ Λ λ∫ (16) Then from (14) and (16) it follows that 1 0 1 1 110 1 ˆ ˆ( ) = Tr ( ) 2 d F F T λ λ λ λ λ + λ Λ = β λ∫ 1 1 0 110 1 ˆ ˆ= Tr ( ) , 2 d F T λ λ λ λ + Σ β λ∫ (17) where ˆˆ =λ λΣ λΛ (18) has the role of mass operator for tunneling Green's func- tion T̂λ . For them Dyson equation exists 0 0ˆ ˆ ˆ ˆˆ= .T T T T+ Σ (19) Equation (17) can be rewritten in the form ( ) 1 ˆ ˆ= Tr ( ). 2 dF T d λ λ λ λ Σ λ β (20) The Eqs. (15) and (17) establish the relation between the thermodynamic potential and renormalized one-particle propagator Ĝλ or tunneling Green's function T̂λ . Both these quantities depend on auxiliary parameter λ which is inte- grated over. As have been proved by Luttinger and Ward [20,21], for normal state of weakly correlated systems, it is possible to obtain another expression for the thermodynamic potential without such additional integration. In our previous paper [1], for the normal state of Hub- bard model, we have obtained such an equation in the form of special functional. We now consider its generalization to the case of superconductivity. For this purpose we introduce the functional 1( ) = ( ) ( ),Y Y Y ′λ λ + λ (21) where 0 1 1 ˆ ˆ ˆ ˆ( ) = Tr{ln( 1) }, 2 Y T Tλ λ λλ − λ Λ − + λΛ (22) and ( )Y ′ λ is the functional constructed from skeleton dia- grams depicted on Fig. 4. From Figs. 2 and 4 it is possible to obtain the relation ( ) 1= ( | ). 2( | ) Y x x T x x ′α α λ′αα λ ′δ λ ′λΛ ′δ (23) Now we shall take into account the following functional derivatives based on the Eqs. (12) and (13): 0 ˆˆ ˆ ˆTr (ln( 1)) = Tr , ( | ) ( | ) T T T x x T x x′ ′αα αα λ λ ⎛ ⎞δ δΛ Λλ − − ⎜ ⎟⎜ ⎟′ ′δ δ⎝ ⎠ ˆ ˆTr ( ) = ( | ) ( | ) T x x T x x ′α α λ′αα λ δ ′Λλ λΛ + ′δ \ ˆ ˆ+ Tr . ( | ) T T x x λ λ ′αα λ ⎛ ⎞δΛ λ⎜ ⎟⎜ ⎟′δ⎝ ⎠ (24) Fig. 4. The rectangles depict the irreducible Green's functions. The double dashed lines depict the tunneling renormalized Green's functions ( | ).T x x′αα ′ Y �(λ) = {λ + − G (0) +1 2 λ 2 4 λ 3 24 + ...+ }+ λ 4 192 λ4 24 V.A. Moskalenko, L.A. Dohotaru, D.F. Digor, and I.D. Cebotari 1172 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 10 As a consequence of these equations we have 0ˆ ˆ ˆ ˆTr{ln( 1) } = ( | ), ( | ) T T x x T x x ′α α λ λ λ λ′αα λ δ ′λ Λ − + λ Λ λΛ ′δ (25) and 1( ) = ( | ). 2( | ) Y x x T x x ′α α λ′αα λ δ λ λ ′− Λ ′δ (26) With the functional derivative of ( )Y ′ λ given in (23) we obtain the stationarity property of the functional ( )Y λ : ( ) = 0. ( | ) Y T x x′αα λ δ λ ′δ (27) Using the definition (18) of the mass operator ˆ λΣ we can rewrite the functional 1( )Y λ in the form 0 1 1 ˆ ˆˆ ˆ( ) = Tr{ln ( 1) }, 2 Y T Tλ λ λλ − Σ − + Σ (28) and prove the second form of stationarity property ( ) = 0.ˆ Y λ δ λ δΣ (29) To demonstrate this equation it is sufficient to use the Dy- son equation (19) in the form 1 0 1ˆ ˆ= ,T T − − λ λ+ Σ (30) and the derivatives: 1ˆ( ) ( | ) = , ˆ( ) ( | ) xy x'y' T y y x x ′− ββ λ ′ ′αβ α β′αα λ ′δ δ δ δ δ ′δ Σ ˆ( ) ( | ) ˆ ˆ= ( ) ( | )( ) ( | ), ˆ( ) ( | ) T y y T x y T y x x x ′ββ ′ ′α β βαλ λ λ′αα λ ′δ ′ ′ ′δ Σ ˆ ˆˆTr ( ) = ( ) ( | ) ˆ( ) ( | ) T T x x x x ′α α λ λ λ′αα λ δ ′Σ + ′δ Σ ˆ ˆˆ( ) ( | ),T T x x′α α λ λ λ ′+ Σ (31) 0ˆ ˆˆTr{ln( 1)} = ( ) ( | ). ˆ( ) ( | ) T T x x x x ′α α λ λ′αα λ δ ′Σ − − ′δ Σ Therefore we have 1( ) 1 ˆ ˆˆ= ( ) ( | ), ˆ 2( ) ( | ) Y T T x x x x ′α α λ λ λ′αα λ δ λ ′− Σ ′δ Σ (32) and ˆ( ) ( | )( ) ( )= ˆˆ ˆ( ) ( | ) ( ) ( | ) ( ) ( | ) T y yY Y x x T y y x x ′ββ λ ′ ′ ′αα ββ αα λ λ λ ′′ ′ δδ λ δ λ = ′ ′ ′δ Σ δ δ Σ 1 ˆ ˆˆ= ( ) ( | ) , 2 T T x x′α α λ λ λ ′Σ (33) where the usual convention about summation over the re- peated indices has been adopted. As a result we obtain the stationarity property (29) of the functional ( )Y λ versus the change of the mass operator .λΣ This mass operator for = 1λ coincides with correla- tion function of our strongly correlated model. Now it is necessary to find a relation between the ther- modynamic potential ( )F λ and the functional ( ).Y λ Consider first the value of the derivative ( ) / .dY dλ λ The λ dependence of the functional ( )Y λ is of two kinds: through λΣ and also explicit through the factors nλ in front of the skeleton diagrams for the functional ( ).Y ′ λ Due the stationarity property (29) we obtain ( ) ( ) ( ) ( ) ( )= | | = | . ddY Y Y dY dY d d d d λ Σ Σ Σλ λ λ λ ′Σλ δ λ ∂ λ λ λ + = λ δΣ λ ∂λ λ λ (34) Here we took into account that the 1( )Y λ part of func- tional ( )Y λ (see Eqs. (21) and (28)) does not explicitly dependent on .λ By using the definitions of ( )Y ′ λ (see Fig. 4) and of λΛ (see Fig. 2) it is easy to establish the property: ( ) ( ) 1 1ˆ ˆ ˆ ˆ= | = Tr( )= Tr( ). 2 2 dY Y T T d Σ λ λ λ λλ ′λ ∂ λ λ λ λ Λ Σ λ ∂λ β β (35) From the Eqs. (20) and (35) we have ( ) 1 ( )ˆ ˆ= Tr( ) = , 2 dY dFT d dλ λ λ λ λ Σ λ λ β λ (36) and we therefore obtain 0( ) = ( ) ,F Y Fλ λ + (37) since for = 0λ the perturbation is absent ( = 0) = 0Y λ and 0( = 0) = .F Fλ Now we set = 1λ and obtain 0= (1),F F Y+ (38) with the stationarity property = 0.ˆ Fδ δΣ (39) Stationary property (39) helps to obtain such thermody- namical quantities as entropy and specific heat = / ,S dF dT− = ( / ).C T dS dT Thermodynamic potential depends of temperature in two forms: one dependence is explicit and second is through mass operator and because of / = / ( / )( / ),dF dT F T F d dT∂ ∂ + δ δΣ Σ as a conse- Stationary property of the thermodynamic potential of the Hubbard model Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 10 1173 quence of (39) the second term in the last formula can be omitted. 4. Conclusions We have further developed the diagrammatic theory pro- posed for strongly correlated systems many years ago to establish the stationarity property of the thermodynamic potential in the superconducting state of the Hubbard model. First, we have introduced the notion of charge quantum number which gives the possibility to consider the pres- ence of irreducible Green's functions with an arbitrary number of creation or annihilation Fermi operators in su- perconducting state. We have introduced the notion of tunneling Green's function and its mass operator, which turns out to be equal to the correlation function of the fermion system. We have proven the existence of the Dyson equation for this function and establish the exact relation between the thermodynamic potential and renormalized one-particle propagator. This relation contains an additional integration over the auxiliary constant of interaction .λ We have constructed a special functional based on the skeleton diagrams for the propagator and for the evolution operator which contain the irreducible Green's functions and full tunneling Green's functions. We have proven the existence of the stationarity proper- ty of this functional and establish its relation with thermo- dynamic potential. It is important to emphasize that there is a close similar- ity between our results obtained for two different models of strongly correlated systems such as periodic Anderson model (PAM) and the Hubbard model (HM). From com- parison of the results obtained for the PAM (see paper [18]) and the results of the present paper for the HM the topological coincidence of the diagrams for both models has been revealed. For example the skeleton diagrams of Fig. 3 of paper [18], obtained for Λ functional of PAM topologically coincide with the skeleton diagrams of our Fig. 2 for the same functional, but of quite a different model. In order to obtain a complete coincidence, it is necessary to replace the full Green's function ( )cG iω of conduction electrons of PAM by the renormalized tunneling Green's function ( )T iω of the HM. The same similarity exists between other functionals of these models. For example, comparison of the skeleton diagrams of Fig. 10 of paper [18] with the diagrams of Fig. 4 of the present paper reveals the full coincidence upon replacement of the Green's functions cG by T . This comparison allows us to conclude that from the thermody- namic point of view the PAM can be reduced to the HM if we replace the Green's function of the conduction electrons of PAM subsystem by tunneling Green's function of hop- ping electrons of HM. We also note that the skeleton representation of our functional allows to select the local irreducible Green's functions as can be seen from Fig. 2 of our paper and Fig. 10 of paper [18]. These quantities contain only fluctu- ations in time, unlike the nonlocal ones which include both fluctuations in time and space. The coefficients of local diagrams (see Fig. 10) vary with the order of perturbation theory as 11 / (2 !)n n− for > 1.n Only such local diagrams are relevant for DMFT, so that one can attempt to carry out the summation of this class of diagrams. Two of us (V.A.M and L.A.D) would like to thank Pro- fessor N.M. Plakida and Dr. S. Cojocaru for a very helpful discussion. Appendix A. Gor'kov–Nambu representation We consider Eq. (11) in Fourier representation. By in- serting specific values of charge quantum number = 1λ ± we obtain ( = ( , i ))nk ωk ________________________________________________________ 1 1 1 1 1,1 1,1 1,1 1,1 1, 1 1,1 1 1, ,( ) = ( ) ( )є ( ) ( ) ( )є ( ) ( ),G k k k G k k G k− − −′ ′ ′ ′σσ σσ σσ σ σ σ −σ −σ σΛ + Λ −Λk k (A.1) 1 1 1 1, 1 1, 1 1,1 1, 1 1, 1 1,1 1 1, , ,, , ,( ) = ( ) ( )є ( ) ( ) ( )є ( ) ( ),G k k k G k k G k− − − − −′ ′ ′′σ −σ σ −σ −σ −σσ σ σ −σ σ −σΛ +Λ +Λ −k k (A.2) 1 1 1 1 1,1 1,1 1,1 1,1 1,1 1,1 1 1, , , , , ,( ) = ( ) ( )є ( ) ( ) ( )є ( ) ( ),G k k k G k k G k− − − − −′ ′ ′ ′−σ σ −σ σ −σ σ σ σ −σ −σ −σ σΛ +Λ +Λ −k k (A.3) 1 1 1 1 1,1 1,1 1,1 1, 1 1,1 1,1 1 1, , , , , ,( ) = ( ) ( )є ( ) ( ) ( )є ( ) ( ).G k k k G k k G k− − −′ ′ ′ ′−σ −σ −σ −σ −σ σ σ −σ −σ −σ −σ −σ− Λ − −Λ +Λ − −k k (A.4) Here 1, 1 1,1 1 1 1є ( ) = є( ), є ( ) = є( ), є( ) = ( )e , є( ) = 0, ( ) = ( ).i xt G k G k N − − − ′ ′σσ σ σ− − −∑ ∑k x k k k k k k x k (A.5) V.A. Moskalenko, L.A. Dohotaru, D.F. Digor, and I.D. Cebotari 1174 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 10 Assuming that the system is in a paramagnetic state, that superconductivity has a singlet character and using the definitions (8) together with the additional ones: 1,1 1, 1 1,1 ( ) = ( ), ( ) = ( ), ( ) = ( ), k k k Y k k Y k − σσ σ σσσσ − σσσσ Λ Λ Λ Λ (A.6) we obtain the following results: ( )(1 є( ) ( )) є( ) ( ) ( ) ( ) = , ( ) k k Y k Y k G k d k σ σ σσ σσ σ σ Λ − − Λ − − −k k ( ) ( ) ( ) = , ( ) = , ( ) ( ) Y k Y k F k F k d k d k σσ σσ σσ σσ σ σ (A.7) ( ) = (1 є( ) ( ))(1 є( ) ( ))d k k kσ σ σ− Λ − − Λ − +k k є( )є( )Y ( ) ( ),k Y kσσ σσ+ −k k which coincide with those found in the papers [10,11]. In spinor representation the system of Eqs. (A.1)–(A.7) has the form ˆ ˆˆ ˆ ˆ= є ,G GΛ +Λ (A.8) were ( ) ( )ˆ = , ( ) ( ) G k F k G F k G k σ σσ σσ σ ⎛ ⎞ ⎜ ⎟− −⎝ ⎠ ( ) ( ) є( ) 0ˆ ˆ= , є = . ( ) ( ) 0 є( ) k Y k k Y k k k σ σσ σσ σ Λ⎛ ⎞ ⎛ ⎞ Λ ⎜ ⎟ ⎜ ⎟−Λ − − −⎝ ⎠⎝ ⎠ (A.9) By using Eq. 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