Interaction of strongly correlated electrons and acoustical phonons

We investigate the interaction of correlated electrons with acoustical phonons using the extended Hubbard–Holstein model. The Lang–Firsov canonical transformation allows one to obtain mobile polarons for which a new diagram technique and generalized Wick’s theorem is used. The physics of emission...

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Veröffentlicht in:	Физика низких температур
Datum:	2006
Hauptverfasser:	Moskalenko, V.A., Entel, P., Digor, D.F.
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Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2006
Schlagworte:	Strong Correlations
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spelling	Moskalenko, V.A. Entel, P. Digor, D.F. 2017-06-11T12:15:34Z 2017-06-11T12:15:34Z 2006 Interaction of strongly correlated electrons and acoustical phonons / V.A. Moskalenko, P. Entel, D.F. Digor // Физика низких температур. — 2006. — Т. 32, № 4-5. — С. 609–633. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 78.30.Am, 74.72.Dn, 75.30.Gw, 75.50.Ee https://nasplib.isofts.kiev.ua/handle/123456789/120196 We investigate the interaction of correlated electrons with acoustical phonons using the extended Hubbard–Holstein model. The Lang–Firsov canonical transformation allows one to obtain mobile polarons for which a new diagram technique and generalized Wick’s theorem is used. The physics of emission and absorption of the collective phonon-field mode by the polarons is discussed in detail. In the strong-coupling limit of the electron-phonon interaction, and in the normal as well as in the superconducting phase, chronological thermodynamical averages of products of acoustical phonon-cloud operators can be expressed by the products of one-cloud operator averages. While the normal one-cloud propagator has the form of a Lorentzian, the anomalous one is of Gaussian form and considerably smaller. We have established the Mott–Hubbard and superconducting phase transitions in this model. This work was supported by the Heisenberg—Landau Program. It is a pleasure to acknowledge discussions with Prof. N. Plakida and Dr. S. Cojocaru and to thank Vadim Shulezhko for asistance in diagram drawing. V.A.M. would like to thank the University of Duisburg-Essen for financial support. P.E. thanks the Bogoliubov Laboratory of Theoretical Physics, JINR, for the hospitality he received during his stay in Dubna. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Strong Correlations Interaction of strongly correlated electrons and acoustical phonons Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Interaction of strongly correlated electrons and acoustical phonons
spellingShingle	Interaction of strongly correlated electrons and acoustical phonons Moskalenko, V.A. Entel, P. Digor, D.F. Strong Correlations
title_short	Interaction of strongly correlated electrons and acoustical phonons
title_full	Interaction of strongly correlated electrons and acoustical phonons
title_fullStr	Interaction of strongly correlated electrons and acoustical phonons
title_full_unstemmed	Interaction of strongly correlated electrons and acoustical phonons
title_sort	interaction of strongly correlated electrons and acoustical phonons
author	Moskalenko, V.A. Entel, P. Digor, D.F.
author_facet	Moskalenko, V.A. Entel, P. Digor, D.F.
topic	Strong Correlations
topic_facet	Strong Correlations
publishDate	2006
language	English
container_title	Физика низких температур
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
format	Article
description	We investigate the interaction of correlated electrons with acoustical phonons using the extended Hubbard–Holstein model. The Lang–Firsov canonical transformation allows one to obtain mobile polarons for which a new diagram technique and generalized Wick’s theorem is used. The physics of emission and absorption of the collective phonon-field mode by the polarons is discussed in detail. In the strong-coupling limit of the electron-phonon interaction, and in the normal as well as in the superconducting phase, chronological thermodynamical averages of products of acoustical phonon-cloud operators can be expressed by the products of one-cloud operator averages. While the normal one-cloud propagator has the form of a Lorentzian, the anomalous one is of Gaussian form and considerably smaller. We have established the Mott–Hubbard and superconducting phase transitions in this model.
issn	0132-6414
url	https://nasplib.isofts.kiev.ua/handle/123456789/120196
citation_txt	Interaction of strongly correlated electrons and acoustical phonons / V.A. Moskalenko, P. Entel, D.F. Digor // Физика низких температур. — 2006. — Т. 32, № 4-5. — С. 609–633. — Бібліогр.: 34 назв. — англ.
work_keys_str_mv	AT moskalenkova interactionofstronglycorrelatedelectronsandacousticalphonons AT entelp interactionofstronglycorrelatedelectronsandacousticalphonons AT digordf interactionofstronglycorrelatedelectronsandacousticalphonons
first_indexed	2025-11-26T23:38:33Z
last_indexed	2025-11-26T23:38:33Z
_version_	1850781655964319744
fulltext	Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5, p. 609–633 Interaction of strongly correlated electrons and acoustical phonons V.A. Moskalenko1,2, P. Entel3, and D.F. Digor1 1Institute of Applied Physics, Moldova Academy of Sciences, Chisinau 2028, Moldova 2BLTP, Joint Institute for Nuclear Research, 141980 Dubna, Russia E-mail: moskalen@thsun1.jinr.ru 3University of Duisburg-Essen, 47048 Duisburg, Germany Received Jnne 29, 2005 We investigate the interaction of correlated electrons with acoustical phonons using the ex- tended Hubbard–Holstein model. The Lang–Firsov canonical transformation allows one to obtain mobile polarons for which a new diagram technique and generalized Wick’s theorem is used. The physics of emission and absorption of the collective phonon-field mode by the polarons is discussed in detail. In the strong-coupling limit of the electron-phonon interaction, and in the normal as well as in the superconducting phase, chronological thermodynamical averages of products of acoustical phonon-cloud operators can be expressed by the products of one-cloud operator aver- ages. While the normal one-cloud propagator has the form of a Lorentzian, the anomalous one is of Gaussian form and considerably smaller. We have established the Mott–Hubbard and supercon- ducting phase transitions in this model. PACS: 78.30.Am, 74.72.Dn, 75.30.Gw, 75.50.Ee Keywords: correlated electrons, acoustical phonons, superconducting phase transition. 1. Introduction Since the discovery of high-temperature supercon- ductivity by Bednorz and M�ller [1] the Hubbard model and related models such as RVB and t J� have widely been used to discuss the physical properties of the normal and superconducting states [2–6]. How- ever, a unanimous explanation of the origin of the con- densate in high-temperature superconductors has not emerged so far. One of the unsolved questions is in how far can phonons be involved in the formation of the superconducting state. The aim of the present pa- per is to gain further insight into the mutual influence of strong on-site Coulomb repulsion using the sin- gle-band Hubbard–Holstein model [7,8] and a recently developed diagram technique [9–13]. We consider now the most interesting case, namely super- conductivity of correlated electrons coupled to dispersive acoustical phonons. This investigation dif- fers from our previous studies [4–19] of electrons cou- pled to dispersionless optical phonons, which was addressed by most other authors [20–23]. Because the interaction between electrons and phonons is strong, we include the Coulomb repulsion in the zero-order Hamiltonian and apply the canonical transformation of Lang and Firsov [24] to eliminate the linear electron-phonon interaction. In the strong electron-phonon coupling limit, the resulting Hamil- tonian of hopping polarons (i.e., hopping electrons surrounded by phonon-clouds) can lead to an attrac- tive interaction among electrons meditated by the phonons. In this limit, the chemical potential, the on-site and inter-site Coulomb energies as well as the frequency of the collective phonon-cloud mode (which is much larger than the bare acoustical phonon fre- quencies) are strongly renormalized [17–19]. This af- fects the dynamical properties of the polarons and the character of the superconducting transition. We sug- gest that the resulting superconducting state with polaronic Cooper pairs is mediated by the exchange of © V.A. Moskalenko, P. Entel, and D.F. Digor, 2006 phonon-clouds and their collective mode during the hopping of the polarons. 2. Theoretical approach 2.1. The Lang–Firsov transformation of the Hubbard–Holstein model The initial Hamiltonian of correlated electrons cou- pled to longitudinal acoustical phonons (the polariza- tion index is omitted) has the form H H H He e� � � �ph ph 0 , (1) where H t j i a ae ij ij j i� � � ��[ ( ) ] , † � � �� 0 � �� U n n V n ni i i i j c i j ij 0 1 2, , , ' , (2) H b bph 0 1 2 � �� k k k k † , (3) H g i j q ne i j ij � � ��ph ( ) , (4) n n n a ai i i i i� �� , † . Here and in the next part of the paper the Planck constant � is considered equal to one; a ai i� �( )† are an- nihilation (creation) operators of electrons at lattice site i with spin ; bk ( )†bk are phonon operators with wave vector k; q pi i( ) is the phonon coordinate (mo- mentum) at site i, which is related to the phonon oper- ators by q b b p i b bi i i i i i� � � � � 1 2 2 ( ), ( )† † . The Fourier representation of these quantities have the form b N b b N bi i i ii i� �� 1 1 k kR k k kR k e e, ,† † q N q p N pi i i ii i� �� 1 1 k kR k k kR k e e, , q b b p i b bk k k k k k� � � �� 1 2 2 ( ), ( )† † . (5) In this Hamiltonian U0 and Vij c are the on-site and inter-site Coulomb interactions, t i j( )� is the nearest neigbor two-center transfer integral (which may be extended to include also next-nearest neighbor hop- ping of electrons), g i j( )� is the matrix element of the electron-phonon interaction, � � �0 0 0� � , where �0 is the local electron energy and �0 is the chemical potential of the system. The Fourier representation of t j i( )� is related to the tight-binding dispersion �(k) of the bare electrons with band width W, t N i( ) (R k kR k � ��1 � )e , with R as nearest neighbor distance. Apparently the energy scale of the model Hamiltonian is fixed by the parameters W U g, ,0 and �k . The band filling n is an additional parameter. After applying the displace- ment transformation of Lang–Firsov [24], H H c a c ap S S S i S i S i S� � �� e e e e e e, , ,† † i� � � � (6) with S i N S p ni i i� � � ( )k k kR k, e i , (7) S g g( ) ( ) ( ),k k k k � � � we obtain the polaron Hamiltonian in the form: H H H Hp p� � �0 0 ph int , (8) H H H n Un np ip i ip i i i 0 0 0� � �� , � � � , (9) H t j i c c V n nj i ij ij c ij i jint † , ( ) '� � �� 1 2 , (10) where c a c ai i i i i ii i � � � � � �† † , ,� ��e e (11) � i i j j i j N g p p gi� � �� 1 ( ) ( ),k R Rk kR k e (12) � � � � �� 0 0 0 1 2 , , ,V U U Vph ph (13) and V i j N g g i jph e( ) ( ) ( ) . ( ) � � �� 1 k k kk k R R � (14) Hence, the effective intersite interaction isVij � � �V Vij c ij ph with Vi j� � 0. The frequency �k of acou- stical phonons is linear in k for sufficiently small wave vectors. In order to have a reasonable expression for the parameter S( )k of the canonical transformation, it is necessary that the condition g( )k � �0 0 is fulfilled. 610 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 V.A. Moskalenko, P. Entel, and D.F. Digor This condition means that the movement of phonons with infinite wavelength, which is equivalent to the macroscopic displacement of the system, cannot influ- ence its properties and must be omitted. Therefore, the Fourier representation of the direct attraction medi- ated by phonons must also vanish in this limit: V ph 0( )k � � 0. It is important to note that the Fou- rier representation of the Coulomb part of the inter-site interaction must also vanish for vanishing wave vector: V c( )k � �0 0 as a consequence of re- quired charge neutrality of the system. Therefore, the resulting direct interaction between electrons, V V Vc( ) ( ) ( )R R R� � ph , fulfills V( )k � �0 0. This will be used when analyzing the corresponding dia- grammatic contribution. When deriving the polaron Hamiltonian, it was necessary to include the shift of the polaron coordi- nate qk by e e eS S i i i q q N g n i k k kRk� � � �1 ( ) , which helps to eliminate the linear electron-phonon interaction. The polaron Hamiltonian is by nature a pola- ron-phonon operator because the new creation and an- nihilation operators ci� † and ci� entering Hp must be interpreted as operators of polarons, i.e., electrons dressed with displacements of ions that couple dynam- ically to the momentum of acoustical phonons. In the zero-order approximation (omitting Hint ) polarons are localized and phonons are free with a strongly renormalized chemical potential � and on-site interac- tion U. This last quantity can become negative if the phonon mediated attraction V ph is strong enough to overcome the direct Coulomb repulsion. The first term of the perturbation operator Hint describes tunneling of polarons between lattice sites, i.e., tunneling of electrons surrounded by clouds of phonons. The sec- ond term of this operator describes the renormalized polaron-polaron inter-site interactions. 2.2. Averages of electron and phonon operators One problem is to deal properly with the impact of electronic correlations on the polaron formation in- volving operators like (11) for the electron and phonon subsystems. This can be done best by using Green’s functions provided one finds a way to deal with the spin and charge degrees of freedom. In order to achieve this in the limit of large U, the Hubbard term can be included in the zero-order Hamiltonian. As a consequence, conventional perturbation theory of quantum statical mechanics is not an adequate tool be- cause it relies on the expansion of the partition func- tion around the noninteracting state using Wick’s theorem and conventional Feynman diagrams. In order to have a systematic description of corre- lated electrons, Hubbard [8] proposed a graphical ex- pansion around the atomic limit in powers of hopping integrals. This diagrammatic approach was reformu- lated by Slobodyan and Stasyuk [25] for the single-band Hubbard model and independently by Zaitsev [26] and further developed by Izyumov [27]. In these approaches, the complicated algebraic struc- ture of the projection or Hubbard operators was used. We have found an alternative way in the sense that our diagram technique involves simpler creation and annihilation operators for electrons at all intermediate stages of the theory and Hubbard operators only when evaluating final expressions [9–13]. In this approach, averages of chronological products of interactions are reduced to n-particle Matsubara Green’s functions of the atomic system. These functions can be factorized into independent local averages using a generalization of Wick’s theorem (GWT) which takes strong local correlators into account, see Refs. 9, 10, and 17 for de- tails. Application of the GWT yields new irreducible on-site many-particle Green’s functions or Kubo cumulants. These new functions contain all local spin and charge fluctuations. A similar linked-cluster ex- pansion for the Hubbard model around the atomic limit was recently formulated by Metzner [28]. But in the latter work the Dyson equation for the renormalized one-particle Green’s function was not derived, nor the correlation function which appears as main element of this equation. It is the purpose of this paper to check in how far we can use the GWT for the extended Hubbard–Holstein model given by Eq. (1). With respect to the transformed Hubbard–Holstein model, phonon operators are averaged using ordinary Wick’s theorem by taking into account the facto- rization of the phonon partition function in k space of phonon wave vectors. We define the temperature Green’s function for the polarons in the interaction rep- resentation by G c c Up c( , , \| , , ) ( ) ( ) ( ) ,x x x x � � � � �� T 0 (15) with c c c cH H H H x x x x� � � � � � � �� ( ) , ( )� �� e e e e 0 0 0 0 , � � �� x x( ) � �e eH H0 0 , for the polaron and phonon operators, respectively, with H H Hp 0 0 0� � ph. Instead of i j, we now use x x, � as site indices; � �, � are imaginary time variables with 0 � �� ; T is the time ordering operator and� is the in- verse temperature. The evolution operator is given by Interaction of strongly correlated electrons and acoustical phonons Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 611 U d H( ) exp ( )int� � � � � � � � �� T 0 . (16) The statistical averages � �� 0 c are evaluated with re- spect to the zero-order density matrix of the grand ca- nonical ensemble of localized polarons and free acous- tical phonons, e Tr e e Tr e e Tr e � � � � � � �� H H H H i b b b ip ip 0 0 0 0 k k k k k † †bkk � . (17) The subscript c in Eq. (15) indicates that only the connected diagrams have to be taken into account. The polaron part of the density matrix (17) is factor- ized with respect to the lattice sites. The on-site polaron Hamiltonian contains the polaron-polaron in- teraction which is proportional to the renormalized parameter U. Therefore, this Hamiltonian can be diagonalized only by using Hubbard operators [8]. At this stage no special assumption is made about the value of the quantityU and its sign. So we can set up the equations of motion for the dynamical quantities in this general case. Wick’s theorem of weakly cou- pled quantum field theory can be used to evaluate statistical averages of phonon operators, including, the propagator of phonon-clouds. 2.3. Phonon-cloud propagators The zero-order one-phonon Matsubara Green’s function has the form � � � � � �( , ) ( \| ) ( ) ( )x x� � � � � � � � � � ��x x x xT 0 � � � � � ��1 2 2 2 2 N g / / \| ( )\| cos ) cosh ( \| \| ) sinh k k(x x k k k � � � � � � , (18) with � � �x R R( ) ( ) ( )� ��p gj j j x . Here x is again the position and � is the imaginary time while x in Eq. (18) stands for ( , )x � . This function makes an essential contribution for small values of distances \| \|x x� � and \| \|� �� close to zero or �. For x x� � the minimum value of this func- tion is obtained for \| \|� � �� /2. Since all approxima- tions in this paper concern the strong-coupling limit of the electron-phonon interaction, we will use the series expansion of ( , )x x� near � � 0 and � �� : � � � � � � � � � � ( \| ) ( \| ) , ( \| ) ( ), 0 00 0 00 � � � � � �� c c (19) with � �c N g� �1 2 2\| ( )\|k k k (20) as collective phonon-cloud frequency [7,18]. Besides the one-phonon propagator we have also many-phonon-cloud propagators. There are two kind of one-cloud propagators, of which �( \| )x x� is the nor- mal-state one and �( \| )x x� the anomalous one of the superconducting state, given by � � � �( \| ) ( \| )x x� � � � � � �x x � � � � � ��T exp[ ( ) ( )]i i� � � �x x 0 � � � � � �� exp [ ( ) ( )] 1 2 2 0T � � � �x x � � � � � � �exp [ ( \| ) ( \| )], � �00 x x (21) � � � �( \| ) ( \| )x x� � � � � ��x x � � � � � ��T exp [ ( ) ( )]i i� � � �x x 0 � � � � � �� exp [ ( ) ( )] 1 2 2 0T � � � �x x � � � � � � �exp[ ( \| ) ( \| )] � �00 x x . (22) For the first function the maximum value of the one-phonon propagator �( \| )x is favored, while for the second one the corresponding minimum value is preferred. The Fourier representations in � space have the form � � � ��( \| ) ~( ),0 1 � ��e i n n i� � � (23a) � � � ��( \| ) ~( ),0 1 � ��e i n n i� � � (23b) where ~( ) ,( \| ) ( \| )� � � � � � i dn i� �� e e 00 0 0 (24a) ~( ) .( \| ) ( \| )� � � � � � i dn i� �� e e 00 0 0 (24b) Here � n is the even Matsubara frequency� n � � 2� �n/ . In order to find the Fourier representations of these functions we have used the peculiarities of the propagator in the strong-coupling limit of the elec- tron-phonon interaction. As proven in Appendix A, the first propagator can be written as 612 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 V.A. Moskalenko, P. Entel, and D.F. Digor � � � � � � �( \| ) ( ) ( ), ( ) ,x x x x� �� 0 with ~( ) ( ) , ~( )� � � �i i n c n c � � � � � 2 1 2 2 q , (25) � �c N g� �1 2 2\| ( )\|k k k . A more realistic value for ~( )� q is obtained by using the dependence of �( \| )x on small values of x. In this more precise approximation we find ~( ) , ( ) ,( )� � �� q xq x� � � �� 2 1 3 2 2 22 1 1 2 / / /e e� (26) where � �1 2 21 6 1 2 � �N g\| ( )\| cothk k k k . (27) This result has been obtained by an expansion of cos kx in terms of x. We also assume that g( )k depends on k only through its absolute value \| \|k . Then the Fou- rier representation of the normal phonon-cloud propa- gator is a Lorentzian and therefore the time dependence of this phonon-cloud corresponds to that of an oscilla- tor with the large collective frequency �c. For the anomalous one-cloud propagator �( \| )x x� we obtain in this approximation a Gaussian representation, see Ap- pendix A: ~( ) exp[� � �i / in n� �� 2 1 22 � � � � ( \| ) ( \| ) ( )]00 0 2 22 2/ /n� , (28) where �2 0 2� ��( \| )./ (29) The space dependence of �( \| )x i n� is more compli- cated compared to the space dependence of �( \| )x i n� because we cannot restrict the discussion to small val- ues of \| \|x . In the following we will discuss many-cloud propagators, both in the normal and superconducting states. We start with the two-cloud propagators [as before, x � ( , )x � ]: � � � � � � � �2 1 2 3 4 1 2 31 2 3 ( , \| , ) exp ( ) ( ) ( )[ ( )x x x x i� � � � �T x x x x4 4 0( )]� � � � �� exp [ ( ) ( ) ( ) ( )] ) 1 2 1 2 3 41 2 3 4 2 0T � � � � � � � �x x x x exp( ( , ; , \| , ; , )) x x x x1 1 2 2 3 3 4 4� � � � , (30) � � � � � � � �2 1 2 3 4 1 2 31 2 3 ( , , \| ) exp ( )) ( ) ( )[ (x x x x i� � � � �T x x x x4 4 2 0( )]� � � � �� exp [ ( ) ( ) ( ) ( )] ) 1 2 1 2 3 41 2 3 4 2 0T � � � � � � � �x x x x exp( ( , ; , , , \| , )) x x x x1 1 2 2 3 3 4 4� � � � , (31) where ( , ; , \| , ; , ) ( \| \| \| )x x x x x x1 1 2 2 3 3 4 4 1 3 1 3� � � � � � � � � � ( \| \| \| )x x1 4 1 4� � �� ( \| \| \| ) ( \| \| \| ) ( \| \| \|x x x x x x2 4 2 4 2 3 2 3 1 2 1 2 ) ( \| \| \| ) ( \| )� � � � � � x x3 4 3 4 2 00 , (32) ( , ; , ; , \| , ) ( \| \| \| ) (x x x x x x x x1 1 2 2 3 3 4 4 1 4 1 4 2� � � � � � � � � � � 4 2 4\| \| \| )� �� ( \| \| \| ) ( \| \| \| ) ( \| \| \|x x x x x x3 4 3 4 1 2 1 2 1 3 1 3 ) ( \| \| \| ) ( \| )� � � � � � x x2 3 2 3 2 00 . (33) The following relations exist between two- and one-cloud Green’s functions: � � � 2 1 2 3 4 1 3 2 4 1 4 2( , \| , ) ( \| ) ( \| ) exp [ ( \| ) ( \|x x x x x x x x x x x� � x x x x x3 1 2 3 4) ( \| ) ( \| )]� � � � � � �� ( \| ) ( \| ) exp [ ( \| ) ( \| ) ( \| ) (x x x x x x x x x x1 4 2 3 1 3 2 4 1 2 x x3 4\| )] , (34) � � � 2 1 2 3 4 1 2 3 4 1 4 2( , , \| ) ( \| ) ( \| ) exp [ ( \| ) ( \|x x x x x x x x x x x� � x x x x x4 1 3 2 3) ( \| ) ( \| )]� � � � � � �� ( \| ) ( \| ) exp [ ( \| ) ( \| ) ( \| ) (x x x x x x x x x x1 3 2 4 1 4 3 4 1 2 x x2 3\| )] � � � � �� ( \| ) ( \| ) exp [ ( \| ) ( \| ) ( \| ) (x x x x x x x x x x2 3 1 4 2 4 3 4 1 2 x x1 3\| )] . (35) Many-cloud phonon propagators will be present in all diagrams of the thermodynamic perturbation theory to be formulated here. As above equations show, all sites of the diagrams are joint and appear to be con- nected in the presence of acoustical phonons. In order to classify the diagrams as connected and disconnected ones, it is necessary to have the analogy of Wick’s theo- rem for many-cloud propagators similar to the theorem we had formulated for correlated electrons [9,10,17]. In the absence of such a theorem we cannot prove the existence of a linked-cluster theorem for the thermody- namic potential and for other extensive quantities. Interaction of strongly correlated electrons and acoustical phonons Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 613 This problem has been discussed in detail in Ref. 30, however, only now we are able to present a solu- tion. In order to obtain this solution, we observe that the two-cloud functions determined by Eqs. (34) and (35) have their maximum values when the arguments of the normal one-cloud functions �( \| )x x� coincide (x x� �) and the corresponding exponential factors close to these arguments approach one. There are sev- eral possibilities to achieve this and all of them have to be taken into account. We assume that as main ap- proximation the following expressions for the two-cloud propagators will result: � � �2 1 2 3 4 1 3 2 4( , \| , ) ( \| ) ( \| )x x x x x x x x� � � �� ( \| ) ( \| ) ( , \| , ),x x x x x x x x1 4 2 2 1 2 3 43 ir (36) � � �2 1 2 3 4 1 2 3 4( , , \| ) ( \| ) ( \| )x x x x x x x x� � � � �� ( \| ) ( \| ) ( \| ) ( \| )x x x x x x x x1 3 2 4 2 3 1 4 � �2 1 2 3 4 ir ( , , \| )x x x x . (37) These last equations also define the irreducible parts of the two-cloud propagators or phonon-cloud cumulants. In the strong-coupling limit the irreduc- ible functions are small and can be omitted as shown below. The validity of this statement is discussed in Appendix A, in which the Fourier representation of the normal two-cloud propagator, �2 1 1 2 2 3 3 4 4( , ; , \| , ; , )x x xi i i x i� � � � � � �� ... ... expd d i i i i� � � � � � 1 4 00 1 1 2 2 3 3 4 4� � � � !� � � � �2 1 1 2 2 3 3 4 4( , ; , \| , ; , ),x x x x (38) has been calculated in the strong-coupling limit lead- ing to �2 1 1 2 2 3 3 4 4( , ; , \| , ; , )x x xi i i x i� � � � � � � � � �( \| ) ( \| )x x x x1 3 1 2 4 21 3 2 4 � � �i i� �� ( \| ) ( \| )x x x x1 4 1 2 3 21 4 2 3 i i� �� . (39) The last equation shows that in this limit the irreduc- ible function is not relevant and Wick’s theorem has a simple form, which does no contain significant irre- ducible contributions. Similarly we obtain for �2 a form without irreducible contributions, �2 1 1 2 2 3 3 4 4( , ; , ; , \| , )x x xi i i x i� � � � � � !�� ... ( )d d i i i i� � � � � � 1 4 00 1 1 2 2 3 3 4 4� e � � � � ! � � � � �2 1 1 2 2 3 3 4 4( , ; , ; , \| , )x x x x � (40) � � � � �( \| ) ( \| ), ,x x x x1 2 1 3 4 32 1 3 4 � � ��i i� �� ( \| ) ( \| ), ,x x x x1 3 1 2 4 23 1 2 4 i i� �� ( \| ) ( \| ), ,x x x x2 3 2 1 4 43 2 1 4 i i� �� . (41) These results correspond to our preliminary esti- mates that the irreducible parts in Eqs. (36) and (37) can be omitted because they are not important in the strong-coupling limit, see Appendix A. Hence, with- out the irreducible parts the equations assume a form corresponding to Wick’s theorem applied to two- cloud propagators. This can easily be generalized to the case of a larger number of clouds. Thus, there is an analogy of having a generalized Wicks’s theorem for the case of correlated electrons [9,10] and a corre- sponding theorem for correlated phonon-clouds. This allows us now to develop a thermodynamic pertur- bation theory for correlated electrons interacting strongly with phonons. As is shown below the tunneling of polarons between lattice sites can be accompanied by either preserving or by exchanging phonon-clouds. In the strong-coupling limit these clouds are heavy, and therefore, in the case of preserving the cloud, the effective transfer matrix ele- ment is considerably diminished, leading to band nar- rowing effects. In the other case, when clouds are ex- changed, the transfer matrix element and the electronic band width remain unchanged. 3. Polaron and electron Green’s functions 3.1. Local approximation The zero-order one-polaron Green’s function is given by G x x c cp 0 0( , ) ( ) ( )� � � � � �� T x x� �� T a ax x x x� �� ( ) ( ) ( , \| , )0 � � � � �� x x, , ( ) ( )G0 , (42) where x stands now for x � ( , , )x � . In order to dis- cuss the influence of the collective mode on G x xp 0 ( , )� , we write down its Fourier transformation by making use of Eq. (25) (see Ref. 19): 614 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 V.A. Moskalenko, P. Entel, and D.F. Digor ~ ( ) ( ),G i d Gp n i p n � � �� 0 0 0� � e (43) ~ ( )G ip n� �0 � � � � � � � " # $ $ � � � �1 0 0 0 0 Z N i E E E c E E n c e e e � � � � �( )( ) � � � � � � � � � �e e e � � �� E c E E n c N i E E ( )( )0 0 � � � � � � � � � � � �e e e � � �� E c E E n c N i E E ( )( )2 2 � � � � � � % & ' ' � � � � e e e � � � � � E c E E n c N i E E 2 2 2 ( )( ) , (44) where �n is the odd Matsubara frequency and Z E E E 0 1 2� � � �� e e e � � , (45a) E E E U0 20 2� � � � , ,� � � , (45b) N c c( ) ( )� � � �e 1 1 . (45c) Equation (44) shows that the on-site transition en- ergies of polarons are changed by the energy �c of the collective mode. The delocalization of polarons due to hopping and intersite Coulomb interaction leads to broadening of the polaronic energy levels. The polaron propagator has the following antisymmetry property: ~ ( ; ; ) ( ; ; )G i G i Up n c p n c� �� 0 0� � � � � . (46) 3.2. Expansion around the atomic limit We will now investigate electron delocalization un- der the influence of Hint in Eq. (10) by making use of thermodynamic perturbation theory in the interaction representation. The averages of chronological prod- ucts of interactions are reduced to n-particle Green’s functions of the atomic system, which can be factor- ized into independent local averages of electron opera- tors and chronological products of phonon operators. The procedure relies on a generalized Wick’s theorem for electron operators, which takes into account the strong local electronic correlations, and Wick’s theo- rem for phonon-cloud operators. In addition to the normal one-polaron propagator in Eq. (15), we have also to investigate the anomalous propagators defined by F x x c c Up x x c( \| ) ( ) ,� � �� T � 0 (47a) F x x c c Up x x c( \| ) ( )� � �� T � 0 . (47b) As before, x stands for ( , , )x � . The easiest way to es- tablish (47) is to make use of a local source term of Cooper pairs, H a a a a i i i i i � (0 � � � � � ��( )† † , which is added to the local Hamiltonian (2) and switched off at the end of the calculation. In the following we shall consider the propagators F x xe ( \| )� and F x xe ( \| )� defined in terms of the electron operators a� �( ) and a� �( ) and not in terms of the polarons operators c( )� and c� �( ). In addition we have to discuss the normal electron propagator G x xe ( \| )� . These functions are defined by G x x a a Ue c( \| ) ( ) ( ) ( )� � � � � �� T x x� �� 0 , (48a) F x x a a Ue c( \| ) ( ) ( ) ( )� � � � � �� T x x� �� 0 , (48b) F x x a a Ue c( \| ) ( ) ( ) ( )� � � � � �� T x x� �� 0 . (48c) Diagrammatic contributions to the first two func- tions are shown in Figs. 1 and 2, respectively. The dia- grammatic elements are self-explanatory (see caption of Fig. 1). The diagrams (e) of Figs. 1 and 2 take into account the effective intersite interaction Vij in the second order perturbation theory. In these diagrams the rectangles represent correspondigly the non-full cumulants )3 0 and I 3 0 . The circle represents the 2-or- der Kubo cumulant for the number operator �n. The Kubo cumulant n c2 and )3 0 has the form n n nc2 2� � � � � �( � � ) , (49) ) )3 1 2 3 1 21 2 ( , \| , ) ( ; \| , )), , ,x x i i i i� � � �� x x x x , (50a) )3 1 2 1 2 0( ; \| , ) ( ) ( ) ( ) ( ) � � � � � � � �� T a a n n �� Ta a n n� �� ( ) ( ) ( ) ( )1 0 2 0 � � � � � � ��Ta a n n� �� ( ) ( ) ( ) ( )2 0 1 0 � � � � � � ��a a n n� �� ( ) ( ) ( ) ( )0 1 2 0 � � � � � � � ��2 0 1 0 2 0a a n n� �� ( ) ( ) ( ) ( ) . (50b) We find weakly connected diagrams which can be divided into two parts by cutting one electron line like c1, c3, c5, and c7 in Fig. 1. All other diagrams are strongly connected. Furthermore, we introduce nor- mal, ( \| )x x� , and anomalous, ( \| )x x� and ( \| )x x� , mass operators, of which the simplest contributions are shown in Fig. 3. For example, diagram a1 is the renormalized tunneling matrix element, whereas (b) Interaction of strongly correlated electrons and acoustical phonons Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 615 and (c) are the simplest contributions to the anoma- lous mass operators. In analogy to the rectangles rep- resenting irreducible Green’s functions, Gn 0 ir , we use also rectangles representing non-full cumulants )n 0, I n 0 , and I n 0 in Figs. 1 and 2. We introduce here the correlation functions Z x xe ( \| )� for the normal state and Y x xe ( \| )� and Y x xe ( \| )� for the superconducting state. For example, to (d) and (e) in Fig. 1 corre- sponding diagrams contribute here to Z x xe ( \| )� , while to (d) and (e) in Fig. 2 corresponding diagrams con- tribute to the correlation function Y x xe ( \| )� , which leads to + e ex x G x x Z x x( \| ) ( \| ) ( \| ),� � � � �0 (51a) � e ex x F x x Y x x( \| ) ( \| ) ( \| ),� � � � �0 (51b) � e ex x F x x Y x x( \| ) ( \| ) ( \| )� � � � �0 . (51c) Thus we have introduced the main dynamical quan- tities which determine the to Figs. 1 and 2 correspond- ing diagrammatical structure. This allows us to derive Dyson equations for the electron Green’s function sys- tem. The full electron Green’s functions can be ex- pressed as 616 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 V.A. Moskalenko, P. Entel, and D.F. Digor x x x x x x (a) 2(c ) 3(c ) 1(c ) 4(c ) 5 (c ) 6(c ) 7(c ) 8(c ) + ++ – – – – – – – = x x x x x x (b )1 2(b ) x)\|(xGe x x x x x x x x x x x x x x xx x x x 1(d ) 6(d )5 (d ) 4(d )3(d )2(d ) – x x x (e) x(x, \|i , i )1 23 i1 i2 x xxx 1 2 –1 2 –– 1 2 – 1 2 –1 2 – Fig. 1. Diagrammatic contributions to the normal one-electron propagator in the presence of phonon-clouds. Solid lines with arrows in same direction represent normal (G0) and lines with arrows in opposite directions anomalous ( , )F F0 0 prop- agators, respectively. Short-dashed lines are for the hopping matrix elements t i j( )� , long-dashed lines represent the direct polaron-polaron interactions V i j( )� , the wiggly lines stand for the normal phonon (cloud) propagators �( \| )x x� . G x x x x x x x x G x xe e e e e( \| ) ( \| ) ( \| ) ( \| ) ( \| )� � � � � �+ + 1 1 2 2 � � �+ e e ex x x x F x x( \| ) ( \| ) ( \| )1 2 2 � � �� e e ex x x x F x x( \| ) ( \| ) ( \| )1 2 1 2 � �� e e ex x x x G x x( \| ) ( \| ) ( \| )1 1 2 2 , (52a) F x x x x x x x x G x xe e e e e( \| ) ( \| ) ( \| ) ( \| ) ( \| )� � � � � �� 1 2 1 2 � � �� e e ex x x x F x x( \| ) ( \| ) ( \| )1 1 2 2 � � �+ e e ex x x x F x x( \| ) ( \| ) ( \| )1 1 2 2 � �+ e e ex x x x G x x( \| ) ( \| ) ( \| )1 1 2 2 , (52b) F x x x x x x x x G x xe e e e e( \| ) ( \| ) ( \| ) ( \| ) ( \| )� � � � � �� 1 1 2 2 � � �� e e ex x x x F x x( \| ) ( \| ) ( \| )1 1 2 2 � � �+ e e ex x x x F x x( \| ) ( \| ) ( \| )1 2 1 2 � �+ e e ex x x x G x x( \| ) ( \| ) ( \| )1 1 2 2 . (52c) Here x stands for ( , , )x � . Double repeated indices imply summation over x and and integration over �. All quantities in these equations are renormalized functions containing all diagrammatical contributions to the normal and anomalous one-electron Green’s functions. The solutions of these equations in the Fou- rier representation [k i n� k, � ] are Interaction of strongly correlated electrons and acoustical phonons Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 617 + x x + x + + + + x x + + (a) = x x x x x x x x – – (b )1 2(b ) )\|F (xe x x 2(c ) 3(c ) 1(c ) 4(c ) x x x x x x x x x x 5 (c ) 6(c ) 7(c ) 8(c ) x x x x 1(d ) 2(d ) 6(d )5 (d ) 1 2 – 1 2 – 1 2 – 1 2 – xx x 4(d ) xx 3(d ) xxx i1 (e) x i2 1 2 – x x(x, \|i , i )1 23 – – Fig. 2. Diagrammatic contributions to the anomalous one-electron propagator in the presence of phonon-clouds. G k d k ke e � � �( ) ( ) { ( )� � � � 1 + � � � + + � �� e e e e ek k k k k( )[ ( ) ( ) ( ) ( )]}, (53a) F k d k ke e �� ( ) ( ) { ( )� � 1 � � � �* + + � �� e e e e ek k k k k( )[ ( ) ( ) ( ) ( )]}, (53b) F k d k ke e �� ( ) ( ) { ( )� � 1 � � � �* + + � �� e e e e ek k k k k( )[ ( ) ( ) ( ) ( )]}, (53c) where d k k k k ke e e e � � � � �( ) ( ) ( ) ( ) ( )� � � � � �1 + + � � �� * � �� e e e ek k k k( ) ( ) ( ) ( ) � � � ![ ( ) ( ) ( ) ( )]+ + � �� e e e ek k k k ! � �[ ( ) ( ) ( ) ( )] � � �� e e e ek k k k . (54) These equations for the renormalized electron Green’s functions are exact. Since they do not contain the expo- nentially small anomalous phonon-cloud propagator �( \| )x x� , superconducting pairing is easier to achieve by electrons without phonon-clouds but moving in the en- vironment of the clouds belonging to other polarons, than by polarons moving in the same environment. We can now switch off the superconducting source term, which means that F0 and F 0 are identically zero. How- ever, the functions � �� e and � �� e survive in this limit and are equal to the order parameters of the supercon- ducting state, Y e �� and Y e ��, respectively. 4. Solvable limits The three correlation functions Ze�,Ye�� andYe�� are the infinite sums of diagrams which contain both partially and completely irreducible many-particle Green’s functions. In order to obtain a closed set of equations which can be solved (at least numerically), we restrict ourselves to a class of rather simple dia- grams which, however, contain the most important spin, charge and pairing correlations. One way to do this, is to check how the individual diagrams are influenced by the phonon fields, for example by distinguishing the case of moderate cou- pling, when Eq. (26) can be used, from the strong-coupling case where Eq. (25) holds. This helps to eliminate from the diagrams the less important ones. For example, in the strong-coupling limit � �( \| ) ,x x� , �x x holds, which allows to discard all renormalized tunneling matrix elements of the form t( ) ( \| )� � � �x x x x� 0 . In this limiting case the narrow- ing of the electronic energy band is maximum, i.e., its width is equal to zero. Since this extreme case is a bit unrealistic, we will in the following consider the case of moderate electron-phonon coupling when also the band narrowing is moderate and Eq. (26) must be used. After summing the infinite series of the most im- portant contributions we obtain the result which is shown graphically in Fig. 4. It is evident that in this approximation for the cor- relation functions no closed set of equations is ob- tained because of the complicated nature of mass oper- ators. So we have to simplify the latter quantities of which the simplest diagrams are depicted in Fig. 3. For the normal mass operator we will use the contri- bution (a1) in Fig. 3 which is given by e x x t( \| ) ( ) ( \| ) ( )� � � � � � �� x x x x� � � �0 � t /( ) exp( ( ) ) ( )x x x x� � � � � � � � � �1 2 2 . For simplicity we replace in the exponential func- tion the distance \| \|x x� � by the lattice constant a being a characteristic length over which the electrons tunnel: e x x t( \| ) ~( ) ( )� � � � � � �x x � � � � � � � � �t Wp( ) exp( ) ( ),x x � � � (55a) W ap � 1 2 1 2 . (55b) This result means that tunneling of phonon-fields leads to electronic band-narrowing effects by which the bare energy �( )k is replaced by ~( ) ( )� �k k� � e Wp . For moderate electron-phonon interaction the quantity Wp is about unity. With respect to the anomalous mass op- erators, e and * e , we observe that they are smaller than the normal one and can therefore safely be ne- glected. This will be used when expanding the equations close to the superconducting transition temperature. 618 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 V.A. Moskalenko, P. Entel, and D.F. Digor = . . .+ e� (x\|x ) (a )1 x = (x\|x )e = . . .++x xe(x\| ) . . .+ (b) x i1 i2 x (c) x xj1 j2 j1(a )2 i1 xx Fig. 3. The simplest diagrams contributing to the normal, e x x( \| )� , and anomalous, e x x( \| )� and e x x( \| )� , mass oper- ators. Another approximation is related to a simplifica- tion of the exact Dyson Eqs. (53) by omitting all anomalous mass operators, which yields [k i� ( , )]k � G k i D k k ke e e e � � � ��( ) ( ) { ( )[ ~( ) ( )]� � � � �+ +1 k � � �~( ) ( ) ( )},� �� k Y k Y ke e (56a) F k Y k D k e e e�� ( ) ( ) ( ) ,� (56b) D k k ke e e � � �� ( ) [ ~( ) ( )][ ~( ) ( )]� � � � � �1 1k k+ + � � �~ ( ~( ( ) ( ),� � �� k k) ) Y k Y ke e (56c) +� � � e ek G k Z k( ) ( ) ( ).� �0 (56d) These equations are identical in form with the Dyson equations for polaron superconductivity me- diated by optical phonons [17]. The difference to the previous work is related to the appearance of the renormalized energy ~( )� k and new correlation func- tions shown in Fig. 4. These irreducible functions depicted by rectangles are on-site quantities with equal site indices. Hence, all right-hand parts in Fig. 4 are proportional to �x x, � meaning that Ze�� and Ye�� are also local functions and corresponding Fourier representations to be independent of the polaron momentum k. In the diagrams x and i1 stand for ( , , )x � and ( , . )i1 1 1 � , respectively, whereby summation over i j i j1 1 2 2, , , and 1 2, and integra- tion over �1 and �2 is assumed. These quantities have the following analytical structure: Z Ve j ( , , \| , , ) ( ),x x x jx x � � �� 1 2 2 d d n c� � � � � � 1 2 0 3 1 2 2� � � �) ( , ; , \| , ) � � �� - � � � � x x, , [ , , \| , ; ,d d G ij 1 2 0 2 0 1 1 2 2 1 2 ir � ] ~( )~( ) ( , , \| , , )t t G i jej x x i� � � � �2 2 1 1 Interaction of strongly correlated electrons and acoustical phonons Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 619 x x x Ge2 j i2 en2 x 1j 1 j 1i j2 x Fe x x 1 2 – x )j2,j1,G2 0 ir (x \|x x x x 1i 1i j 2 1i i2 1j j 2 j2 )j 2 ,j1G 2 0 ir \|(x, x 2 1–) =x\| x(�Z �e 1 2 –) =( x\|xY �e � 1i Fe – 1 j Gp )xj 2 G 2 0 ir (x, i1\| , i21 j )xj 2G 2 0 ir (x, i1\| , – x� (x, \|i , i )1 23 0 2i 2i Fig. 4. Schematical representation of the renormalization function Ze��, and the superconducting order parameter Ye��. Rectanges with four inner points depict the two-particle completely irreducible Green’s functions ~ G2 0 ir or partial cumulant �3 0. Double lines depict the renormalized one-particle Green’s functions of electron and polaron kind, short-dashed lines stand for the hopping matrix elements and long-dashed lines represent the direct electron-electron interaction V(i – j). � � �� x x, , [ , ; , \| , ; ,d d G ij 1 2 0 2 0 1 1 2 2 1 2 ir � ] t t G i jp( ) ( ) ( \| ) ( , , \| , , ),j x x i� � �� 0 1 2 2 2 1 1 (57) Y d de ( , , \| , , ) , ,, x x x x i i � � � � � � � � � � � � � � ��1 2 1 2 01 21 2 G t t2 0 1 1 2 2 1 2 ir [ , ; , \| , ; , ]~( )~( ) � � � �� !x i x i ! � � �F i i d de ( , , \| , , ) ,1 1 1 2 2 2 1 2 0 1 2 � � � � � x x G t i i 2 0 1 1 2 2 1 1 21 2 ir ,, [ , ; , \| , ; , ]~( )�� ! � � � � � � x i ! � �~( ) ( , , \| , , ) ( \| )t F i iex i2 1 1 1 2 2 2 1 20 � � � � � � � �( \| )0 2 1� (58) and corresponding equation for Y��. Since Eqs. (57) and (58) are the result of summing an infinite series of di- agrams, the thin lined representing one-particle propagators are replaced by full normal electron (Ge) and polaron (Gp) and anomalous (Fe) functions. Fourier transformation of these quantities leads in case of spin-sin- glet channel of superconductivity to Z i V n i Ne c � �� ( ) ( ) [~( )] , , � � � �1 2 1 2 2 2 1 1 ) k k G i G i i i ie� � � � � � � 1 1 1 2 0 1 1 1 1( ) ~ [ , ; , \| , ; , ]k\| ir � � ��1 2 2 1 1 1 1 1 1 1N G i ip � � � � � �( ) ( ( )) ( ) , , k k\| k � � � ~ [ , ; , \| , ; , ],G i i i i2 0 1 1 1 1 ir � � � � (59) Y i N F ie e� � � � �� , , , ,( ) ~( )~ ( ) ( )� �� 1 2 1 1 1 1 1k k k\| k ~ [ , ; , \| , ; , ]G i i i i2 0 1 1 1 1 ir � � � �� 1 2 1 3 1 1 1 2 N � � � � , , , ~( )~ ( ) k k k � � F i i ie� � � � � 1 1 1 1 2 1 2, ( ( )) ( ) ( )� � � !k\| � � � � ! � � � �~ [ , ; , \| , ; , ],G i i i i2 0 1 1 1 1 ir � � � � (60) where V N V2 21 � � \| ( )\| , k k (61a) ) )� � � � � � � � � �, ( ) ( , ; , \| , ),� � � � � ��d d1 2 0 3 1 2 (61b) ~ [ , ; , \| , ; , ]G i i i i2 0 1 1 2 2 3 3 4 4 ir � � � � �� ( )1 2 3 4� � � ! ! � �~ [ , ; , \| , ; , ( )].G i i i i2 0 1 1 2 2 3 3 4 1 2 3 ir � � � � � � (61c) From Eq. (59) for Z ie ( )� we obtain the following expression for + e i( )� : + )e ci G i V n i N� � �� ( ) ( ) ( ) [~( )] , , � � � �0 2 2 21 2 1 1 1 k k G i G i i i ie� � � � � �. �( ) ~ [ , ; , \| , ; ,k\| 1 2 0 1 1 1 1 ir � �� 1 1 2 2 1 1 1 1 1 1 N G i ip � � � � � � k k k\| , , ( ) ( ( )) ( ) � � � ~ [ , ; , \| , ; , ]G i i i i2 0 1 1 1 1 ir � � � � . (62) Here, the renormalized and unrenormalized tunnel- ing matrix elements accompany the electron and polaron propagation, respectively. Equations (60) and (62) together with the expressions for the one-particle Green’s functions and the definitions of the irreduc- ible Green’s functions in Refs. 9, 10 and Kubo cumu- lants in (49) determine completely the properties of the superconducting phase and allow one to discuss the influence of strong electron-phonon interaction. The three irreducible two-particle Green’s functions already calculated (see Eqs. (5)–(7) in Ref. 13) are given by 620 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 V.A. Moskalenko, P. Entel, and D.F. Digor ~ [ , ; , \| , ; , ] [ ]( )(, G i i i i U 2 0 1 1 2 1 1 1ir e � � � � � � � � � � e e � � / � / � / � / � � �( )) ( ) ( ) ( ) ( ) 2 0 2 1 1 U Z i i i i , (63a) ~ [ , ; , \| , ; , ] ( ) G i i i i U Z U Z U 2 0 1 1 0 2 0 1ir e e � � � � � / � � �� ( ) ( ) ( ) ( ) ) ( ) ( ) ( , i i i i U i i i� / � / � / � � � /0 � / � / � / � 1 1 1 1� e i�1) � � 1 �1 � � � � � �( )[ ( ) ( )] [ ( ) ( ) ( ( )e � / � / � / � /0 � 2./ � / 2 1 1 2 2 1U i i i i i i i i i i� / � / � / � / � � 1 2 1 2 1 1 1 1 1 ) ( ) ( ) ( ) ( ) ( ) � � " # $ � � � � � �e � � � � � � � � � � � 1 1 1 2 1 1 2 2/ � / � / � / � / � / � /( ) ( ) ( ) ( ) ( ) ( )i i i i i i ( ) ( ) , i i� / �1 1 % & ' ' 3 4 1 51 (63b) ~ [ , ; , \| , ; , ] [ ( ) ][( G i i i i U Z U i 2 0 1 1 0 2 2 ir � � � � � � � � � � � � � � � � 1 �1 ! U i) ( ) ]2 2� ! � � " # $ $ % & ' ' �� 1 10 0 2 0 1, , ( )e e e Z U 1 2 1 2 2 2 1 2 � � � � �� U U i i� � � � � �e [ ( ) ][ ( ) ] � � � � � �� 1 2 2 2 2 2 U U U i U U � � � � � �e e ( ) [( ) ( ) ][( ) � 3 4 1 51( ) ]i�1 2 , (63c) where � , 1 is the Kronecker symbol for Matsubara frequencies and Z U 0 21 2� � � �e e � �( ), / � � � / � � � ( ) , ( ) ,i i i i U� � � � � � � . (64) For the present study � andU in Eqs. (63) and (64) are the renormalized quantities of Eq. (13). The fore-standing equations are generalized Eliashberg equations of strong-coupling superconductivity for the case that strong electron correlations have been taken into account in a self-consistent way. In spite of the ap- proximations involved the equations are rather compli- cated. In order to gain further insight into the physics behind Eqs. (59) and (62), we will linearize the equa- tions in terms of the order parameter Y�� , but not in terms of +�. Then the critical temperature Tc of the superconducting transition can be obtained from Y i N Y i G i i e e �� ( ) ~( )~( ) ( ) ~ [ , ; , \| � � �1 1 2 0k k ir � � � � � �� , ; , ] [ ~( ) ( )][ ~( ) ( )] i i i ie e 1 1 1 11 1 � � � � �k k k + + , 1 � � � � � �1 1 2 1 1 2 1 2 2 � � � � � � �� N Y i i i i i Ge ~( ) ~( ) ( ) ( ) ( ) ~k k � � � � 0 1 1 1 11 ir [ , ; , \| , ; , ] [ ~( ) ( � � � � � �� i i i i i i ie � � � � �k + � �2 1 1 21 1 21 )][ ~( ) ( )] , � � � � �� k k, + � � � � e i i i , (65) + ) + e c e i G i V n i N i � � �� ( ) ( ) ( ) [~( )] ( � � �0 2 2 2 11 2 1 1 k ) ~ [ , ; , \| , ; , ] ~( ) ( ) G i i i i ie 2 0 1 1 1 1 1 1 ir � � � � � �� k k, + 1 1,� � � � � � 1 12 2 1 1 11 11 � � � � � � � � N i i i i p p ( ) ( ) ( ) ~( ) ( ) , , k k k + � � + �1 1 2 0 1 1 1 1 , ~ [ , ; , \| , ; , ] � � � � �� G i i i iir . (66) In order to determine Tc it is necessary to solve Eq. (66) for + e i� �( ) and to insert it into Eq. (65). The next our approximation is the omitting in the right-hand part of Eq. (66) of the polaron function +p�. The reason for such approximation is the presence in this term of the phonon-cloud propagator �( )i�1 which makes the ap- pearance in the denominator of large quantity �c. Since �c is larger than other typical energies involved, the term under discussion is at moderate coupling strength smaller than all other terms in Eq. (66) and can there- fore be omitted, which leaves Interaction of strongly correlated electrons and acoustical phonons Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 621 + e Vi G i� �� ( ) ( )� � � 1 1 2 1 2 0 1 1 � � � � � � �� N i G i i i ie[~( )] ( ) ~ [ , ; , \| , ; , ]k + ir � �� ~( ) ( ) , � �� k k + e i 1 1 � 1 1 2 1 2 0 1 1 � � � � � � �� N i G i i i ie[~( )] ( ) ~ [ , ; , \| , ; , ]k + ir �� ~( ) ( ) , � �� k k + e i 1 1 (67) where G i G i V n iV c � � �� ( ) ( ) ( ).� �0 2 21 2 ) (68) Equation (67) is identical with the corresponding equation for the single band Hubbard model without phonons [10] if we replace in Eq. (67) the renor- malized quantities �,U, ~( )� k and G iV � �( ) by the ini- tial quantities �0, U0 and �( )k . This means that we have reduced the investigation of superconductivity in frame of the Hubbard—Holstein model to the analogi- cal problem with respect to the single band Hubbard model. It is instructive to analyze the contributions from the two spin channels by considering the quantities 6 � � � � � � � � � � � !��( ) [~( )] ( ) ~( ) ( ) i N i i e e 1 1 2 1 1 1 k k k + + ! ~ [ , ; , \| , ; , ],G i i i i2 0 1 1 ir � � � � (69a) 6 � � � � � � � � � � � !��( ) [~( )] ( ) ~( ) ( ) i N i i e e 1 1 2 1 1 1 k k k + + ! ~ [ , ; , \| , ; ,G i i i i2 0 1 1 ir � � � �. , (69b) and using the notation � � � �2 � �� e e e i N i i ( ) [~( )] ( ~( ) ( ) � � ��1 1 2k k k + + � ��1 1N ie ~( ) ~( ) ( ) � � �� k k k + . (70) Here it is assumed that � �( ) (k k� � ) holds with 7 � k k�( ) .0 We replace sums by integrals, 1 0N d k � �� ~ (~),� 8 � (71) 8 � � � � � 0 24 1 2 1 2 0 2 (~) ~ ( ~ ~ ) , \| ~\| ~ , \| ~\| ~ . � � ! 9 � � � �W /W W/ W/ (72) ~W is the renormalized band width ~W W Wp� � e and 80 is the semielliptic model density of states. Since we do not consider magnetic solutions, the spin index can be omitted. Making use of (63a), (63b) for the ir- reducible functions we obtain 6 � � / � / � � � � � � � � � !( ) ( )( ) ( ) ( ) ( ) i U Z i i U2 2 0 2 1 e e e ! � " # $ $ % & ' ' � � � �/ � / � � e i i i ( ) ( ) ( ) , (73a) 6 � � � � � � / � � � � � � � � � � � ( ) ( )( ) ( ) ( )[ ( ) ( ) i U Z U i U i U e 0 22 1 2 e / �( )]i 2 � � 1 �1 � � � � �U i U i i ee e � � �� / � / � ( ) ( ) [ ( ) ( )] 1 2 � � � � � " # $ $ % & ' ' ! �� U Z Ue e e 2 0 1 1 1 2 ( ) ( )( ) ! � � 3 4 1 51 �1 12 0 2/ � / � : � / � � ( ) ( ) ( ) ( ) ( ) , ( ) i i i i Ue (73b) where � � � � / � / � � � �1 1 1 1 1 e i i i ( ) ( ) ( ) , (74a) � � � � / � / � � � � � � � �� 1 1 1 1 1 1 2 1 e i i i ( ) ( ) ( ) � �U i i i e2 1 1 1 2 1 � � � / � / � � ( ) [ ( ) ( )] , (74b) : � � / � / � / � / � ; � � � 0 1 1 0 11 1 1( ) ( ) ( ) ( ) ( ) (, i i i i i ie � � �� ) ( )/ �2 1i , (74c) and ; � � �� 1 10 01, , . In case of half-filling when � �U/2, / � � �( )i i� � , / � � �( )i i� � , we have the antisymmetry property, � � � �e ei i( ) ( )� � � , and there- fore � �� 1 0. Furthermore, we find at half-filling, 6 � � � � � � � � � � ( ) ( ) [( ) ] ,i i i e2 2 2 2 (75a) 6 � � � � � � 6 �� ( ) ( ) [( ) ] ( ),i i i i e2 2 2 2 2 2 (75b) and therefore Eq. (67) is equal to 622 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 V.A. Moskalenko, P. Entel, and D.F. Digor 6 � 6 � 6 �� ( ) ( ) ( )i i i3 . (76) Away from half-filling we will for simplicity omit the contribution of the anti-parallel spin channel in Eq. (67) and introduce instead a correction factor fs , which is three at half-filling and different from three at general filling. The final equation to be investi- gated is then + + +e V s e e i G i f N i i� � � � � � � � � � � � ( ) ( ) [~( )] ( ) ~ ( ) ( � � � k k 2 1 1 ) ,k 1 � ! ! ~ [ , ; , \| , ; , ]G i i i i2 0 1 1 ir � � � � . (77) Further simplifications can be made regarding Eq. (65) for the critical temperature. We note that the main contribution to the last term results from the minimum values of frequency difference � �1 2� . When � �1 2� we get after summing over �1: 1 1 2 1 2 22 1 2 2 1 � � �� [ ( )] coth sinh ( ) , � �� i /c c c (78) so that f / c c c c � � �1 1 2 1 2 22�� coth sinh ( ) (79) can be used as a common factor in the remaining func- tion for the superconducting order parameter, which is Y ie�� ( ) � f N Y i G i i ic e � � � � � � � �� ~( )~( ) ( ) ~ [ , ; , \| , ; ,k k� �1 2 0 1 ir � � � � �� i i ie e � � � � �� 1] [ ~( ) ( )][ ~( ) ( )]1 11 1 1 k k k + + . (80) In order to solve Eq. (80) for Tc, we introduce a new function � � � � � � �� sc e e i N i ( ) ~( )~( ) [ ~( ) ( )][ ~( ) ( � � � � � 1 1 1 k k k k+ + � � � � � �� i i i i i e e e e� � � � � � �� )] ( ) ( ) ( ) ( )+ + k , (81) which allows to rewrite Eq. (80) in the form Y i f i Y i G i ie c sc e�� ( ) ( ) ( ) ~ [ , ; ,� � �� 1 1 2 0 1 ir \| , ; , ]. � �i i1 1� (82) Using furthermore ; � � � � � � �� 1 2 2 1 � � � sc ei Y i i ( ) ( ) ( ) , (83a) ; � � � � 0� � 2 � �� 2 2 2 1 � � � sc ei Y i U i ( ) ( ) ( ) , (83b) < � � � � �� 1 �1 3 4 1 51 �U f Z Z c U2 0 2 0 1 e e e( ) , (83c) Q i i i U i sc ( ) ( ) [ ( ) ][( ) ) ] � <� � � � � � � � � � � 1 2 2 2 2 , (83d) allows to obtain the solution: Y i Uf Z U/ U Q i i e c �� ; � � � ( ) [ ( )]( ) ( ) [ ( ) � � � � � �0 1 2 2 1 2 1 e ] � � � � � � � �Uf Z U/ U Q i U i c U 0 2 2 2 1 2[ ( )]( ) ( ) [( ) ( ( )� ; � � � � �e e ) ] , 2 (84) where symmetry properties of Y ie�� ( ) and � �sc i( ), � � � � � �� sc sci i Y i Y i( ) ( ), ( ) ( )� � � � (85) has been used. The two constants ;1 and ;2 are de- termined from the following system of equations, ; = � � 1 11 0 1 1 2 1� � � � � �� " # $ % & ' � f Z U U c ( )e � � � � � �� ; = � � � 2 12 0 21 2 0 f Z U U c U( ) ,( )e e ; = � � � 2 22 0 21 1 2 � � � � � �� " # $ % & ' � �f Z U U c U( )( )e e � � � � � �� ; = � � 1 12 0 1 2 1 0 f Z U U c ( )e , (86) Interaction of strongly correlated electrons and acoustical phonons Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 623 where =ij are given by the summations of = � � � � � � 11 2 2 2 � � �U i Q i i sc( ) ( ) [ ( ) ] , = � � � � � � � � 12 2 2 2 2 � � � � �U i Q i i U i sc( ) ( ) [ ( ) ][( ) ( ) ] , = � � � � � � 22 2 2 2 � � � �U i Q i U i sc( ) ( ) [( ) ( ) ] . (87) The critical temperature is then obtained by setting T Tc� in the equation: 1 1 2 111 0 � � � � � �� " # $ % & ' ! = � �f Z U U c ( )e ! � � � � � �� " # $ % & ' � �1 1 2 22 0 2= � � �f Z U U c U( )( )e e � � � � � � � � � � � � � � � � �= � � �12 2 0 2 2 2 21 2 f Z U U c U ( ) ( ( )e e ) ! ! � �( )1 0e � . (88) Equation (88) is invariant under the particle-hole transformation, n n U U U� � � � � �> � > � � � > � �1 2 2, , ( ), Z Z U0 0 2( ) ( ) exp[ ( )]� � � �> � . (89) 5. Metallic, insulating and superconducting phases In order to obtain a better understanding of the properties of the main equations, we will first check whether the normal state determined by Eq. (77) is metallic or dielectric. This can be done by analyzing the renormalized density of states, 8( )E , which is given by 8 � ( ) ( ),E g E i� � � �1 0Im (90a) g i N i in n n ( ) ( ) ~( ) ( ) ,� � � � � ��1 1 + +k k (90b) where +( )i n� has to be calculated from Eq. (77). The integration over k is again done by using the semiel- liptical form of model density of states in Eq. (72). This gives for the quantities (70) and (90b) the fol- lowing result [27]: � � � �e ei W i( ) ~ ( ),� 1 2 (91a) � � � � e i i i ( ) ( ~ ( ) ) ~ ( ) � � �1 1 2 2 3 + + , (91b) g i W i i ( ) ~ ( ~ ( ) ) ~( ) ,� � � � � �4 1 1 2+ + (92a) ~( ) ~ ( ) ,+ +i W i� �� 1 2 (92b) yielding for the renormalized density of states (DOS) in Eq. (90a) 8 � � ( ) ( ) ~ ~ ~ ( ) ~( ) ,E r E W W E E � � � � �4 4 1 1 2 Im + + (93) where ~( )+ E is the analytical continuation of ~( )+ i n� . We now address Eq. (77), which determines the normal state of the system. Using (73a) and (74), we rewrite it in the form: ~( ) ~ ( ) ~ ( ),+ i WG i f W iV s� � 6 �� 1 2 1 2 (94) with 6 �� ( )i given by Eq. (73a). In the limit E > 0 Eq. (94) becomes ~ ~ ( ~ )+ + +1 1 1 2 4 2 2� � � " # $ $ % & ' ' � a b , (95) where ~ ~( )+ +� �E i� for E � 0. The two parameters a and b and the function GV ( )0 are given by a f W U Z U s U 2 2 2 2 0 2 2 2 1 4 1 � � � � �~ ( )( ) ( ) , ( )e e e � � � � � (96a) b WGV� � 1 2 0~ ( ) � � � � �1 2 12 2 0 2 f WU U Z s U� � � � � � �~ ( )( ) ( ) ( )e e e , (96b) G n n U V ( )0 1 � � � � �� n V n c2 2 2 1 1 1 � � �� ~ ( ), e (96c) n n� 2 �, (96d) 624 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 V.A. Moskalenko, P. Entel, and D.F. Digor with � given by Eq. (74). The first term in Eq. (96c) is the value of the local one-particle Green’s function of the Hubbard model for E � 0, while the second term is the corresponding contribution from the intersite Coulomb interaction. Equation (95) has been discussed for the simpler case of half-filling when b � 0 in Refs. 17 and 27. Away from half-filling we have n ? 1, � ?U/2 and b ? 0. As Eq. (96a), (96b) show, the parameters a and b depend on chemical potential �, strong-coupling pa- rameterU and mean electron number per lattice site n. Moreover, b depends on the intersite Coulomb repul- sion. Although parameters a and b are not independent of each other, we first try to get information in case that b � 0, i.e., for � @� �GV ( ) 0, which holds for + +( ) ( )� � �i i� � . In the half-filled band case when � �U/2 and n � 1, the quantity a is equal to a f W/U fs s� �~ , 3 . (97) The equality b � 0 allows to determine + in Eq. (95) from a simpler relation, 1 1 1 1 1 12 2 2 2 � � �� " # $ $ % & ' ' � � �� a a ~ ~ ~ ~ + + + + " # $ $ % & ' ' � 0. (98) For a � 1 the first factor gives ~ ( )+2 2� �a a and hence, ~ ( ), ,+ � A � �a a a2 2 (99a) ~ ( ),+ � A � �i a a a2 2 . (99b) By inserting these solutions in Eq. (92) for the renormalized DOS we obtain results different from zero only for the expression with lower sign in Eq. (99b), i.e., for a � 2. Hence we obtain a metallic state at half-filling only if the Coulomb interaction is less than a critical value [8], U U U Wc c� �, ~1 2 3 . (100) In this case there is no gap at the Fermi level and the renormalized DOS becomes ( )a � 2 8 �( ) ( ~ ) ( ), ( ) ( ) .0 4 0 0 2� � �/ W r r a /a (101) The insulating (dielectric) phase exists if the inverse condition holds, a � 2,U Uc� , leading to the opening of an energy gap at the Fermi level. Away from half-filling, for a � 2 and b sufficiently small, the solution of Eq. (95) can be obtained from a series expansion in powers of b, ~ ~ ( ) ( ) ( ) ~( )( ) + + + b a b a a a b a a � � � � � � � � � � 1 2 2 5 3 8 1 2 2 2 2 � , (102) where ~+ is given by Eq. (99) in zero-order approxima- tion. This leads to r a a a a a b a a a/ / ( ) ( ) ( ) ( ) 0 2 2 8 12 5 8 1 2 3 2 2 3 2 2 5 2 � � � � � � � � � (103) This result shows that not b but b/ a( )� 2 should be taken as expansion parameter. Therefore the ex- pansion is not correct very close to a � 2. Another pe- culiarity of Eq. (103) is its even character in b, which follows from the fact that in the solution of Eq. (95), ~+, changes sign when the sign of b is changed. There- fore, if we have the solutions + +1 2A i of Eq. (95), then correspondingly �+ +1 2� i are solutions for b � 0. The different signs of the real parts do not mat- ter, since Eq. (93) depends only on the absolute value of +1. The numerical investigation shows that for a � 2 the role of b is not decisive, i.e., the system remains metallic. However, in the range 0 2� �a , for which the system is insulating when b � 0, the influ- ence of b is decisive because there exists a critical value, b a a/c( ) � �1 2, such that the system is metal- lic for \| \|b bc� and insulating for \| \|b bc� . Note that the parameters a and b are not independent each other and therefore such extremal parameter values as a small and b large or vice versa are not admitted by their definition (Eq. (96)). The metallic state exists for low and high band fill- ing (small and large values of �) and near half-filling ( )� � U/2 , provided thatU Uc� . The physical role of b is to enhance in each case the tendency towards metalicity. There is also the case of reappearance of the metallic phase for a � 2 and \| \| ( )b b ac� . The varia- tion of b ac( ) is shown in Fig. 5. We discuss now the superconducting phase transition with Tc obtained from Eq. (88). In order to gain some insight we need to simplify � �sc i( ) from Eq. (81). In the asymptotic limit \| \|�> B this quantity is equal to � �sc i W/( ) ( ~ )> 1 4 2 2 , (104) whereas in the low-energy limit E > 0 required here, we obtain from Eqs. (73a) and (94) the value � � �sc scE i /a W/( ) ( ) ( ~ )� > � 1 22 2 . (105) The two limits coincide for a � 2. Really we are inter- ested mainly in the low-energy limit (Eq. (105)). Be- cause of the fast convergence of the sums in Eq. (87) which contain � �sc i( ) and which determine the pa- rameters =ij of Eq. (88), we can rewrite =ij as Interaction of strongly correlated electrons and acoustical phonons Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 625 = � � 11 2 2 1 2 2 � � � ��U d z Q z c z sc ( )( ) � � �U I I c dsc� [ ( )],1 2 2 2 (106a) = � � 22 2 2 1 2 2 � � � ��U c z Q z d z sc ( )( ) � � �U I I c dsc� [ ( )]1 3 2 2 , (106b) = � � � 12 1 1 1 � ��U Q z U I sc sc ( ) , (106c) where I Q z1 1 1 1 � �� ( ) , (107a) I Q z z c 2 1 2 2 1 1 � � �� ( )( ) . (107b) I Q z z d 3 1 2 2 1 1 � � �� ( )( ) (107c) Q z z c z d sc 1 2 2 2 2( ) ( )( ) ,� � � � <� (107d) and <� � � � � � � sc c U s f U Z f � � � � � � �2 2 0 2 1 1 ( ) [ ( )] ( )( e e e e e e �( )) , 2 �U c d U z i n 2 2 2 2� � � �� , ( ) , . (108) In case that U � 0 the quantity <� :sc � 4 is posi- tive. In the other case of very strong electron-phonon interaction we haveU � 0. In this case we will use the notation <� Csc � � 4 (with negative <). Although U � 0 seems not to be the generic case, we believe that in the metallic phase in case of strong electron-phonon interaction an effective attractive interaction is possi- ble, because the local and intersite Coulomb interac- tion can be largely screened by the electron-ion interaction. However, calculation of self-consistent screening in not the aim of this paper. We just require charge neutrality and takeU as a parameter, which al- lows us to discuss the case of effective repulsive as well as attractive interactions. We first discuss the caseU � 0 and thenU � 0. The sums in In have been evaluated by contour in- tegration with the help of Poisson’s formula, see Eq. (B2). For the case : 4 2 2 2 4D �( )c d / we obtain the following equation which determines Tc: 0 1 2 2 4 1 1 4 4 2 2 4 2 2 2 4 2 2 2 � � � � � � � � �f U c d c d c d Uc sc� : : : : ( ) ( ) ( e � � E �E E �E( )) ( ) sinh sin co 2 0 2 1 1 2 2 � � " # $ $ % & ' ' � � 1 �1 �U Z U sh cos�E �E1 2� � � � � �" # $ $ % & ' ' �� U U Z UU( )( ) [ sinh sin( )2 1 2 0 1 1 2� E �E E �E �e 2 1 2 ] cosh cos�E �E� � � � � ��tanh( ) ( )( ) tanh( ) (( )� � � � � �d/ dZ U c/ cZ U2 2 1 0 2 0 e e � " # $ % & ' � 3 4 5 �e � :) 2 4 2 2U c d � � � � � � f U U U Z c c sc U 2 2 2 2 8 0 2 4 2 1� � � : : � � �( )( ) ( )( )( )e e e d c d c d c d2 4 2 2 2 4 2 2 2 2 2 2 4 1: : � � � � � � � 1 �1 ! ( ) ( ) ( ) 626 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 V.A. Moskalenko, P. Entel, and D.F. Digor 0 0.5 1.0 1.5 2.0 Parameter a 0.2 0.4 0.6 0.8 1.0 C ri tic al p ar am e te r b c Fig. 5. The critical parameter b ac( ) in the range 0 2� �a . For \|b\| > bc is the system metallic while it is insulating for \| \|b bc� . For a 2 the system is metallic regardless of the value of b. ! � � � �� tanh( ) tanh( ) sinh sin� � E �E E �Ed/ d d/ c 2 2 2 1 1 2 cosh cos sinh sin [cosh cos ]�E �E �E �E �E �E1 2 2 1 2 2 1 2 2� � � � � � � � �: � � � �4 2 2 2 2 2 c d d/ d d/ c d/ d dtanh( ) tanh( ) tanh( ) tanh( / c 2 1 1 2 2 1 2 ) sinh sin cosh cos � � �� 3 4 5 E �E E �E �E �E , (109) where the parameters E1 and E2 are given by Eq. (B3). For the special case of half-filling when � � �U/2 0 and c d U/ sc2 2 2 2 22 1 3 � � � �� ( ) , , : � � � 4 4 1 3 1 0� � � �f / c ( ) , e e (110) Eq. (109) is of simpler form, 0 1 3 1 1 5 4 4 4 2 2 2 1 1� � � � � � � � � �fc� : : � : � �� E �E E �e sinh sin�E �E �E E �E E �E �E �E 2 1 2 1 1 2 2 1 4 cosh cos sinh sin cosh cos� � � � 2 � � 1 �1 � � � 3 4 5 � � �4 2 4 9 4 4 2 10 8 4 4 2 1 2 � : � �� : : � �E tanh( ) sinh sin / fc �E �E �E : � � �� 2 1 2 2 4 4 2 2 2 [cosh cos ] tanh ( ) � � �� 1 �1 / � � � �� : � � �� E �E E �E � 2 2 1 1 2 2 2d d /tanh( ) sinh sin cosh E �E � �� E �E E �E �E1 2 1 1 2 2 1 2 2 � � � �cos tanh( ) sinh sin cosh co / s�E2 3 4 5 , (111) where E : � � 2 1 1 2 4 4 2 1 2 � A �� / . (112) If we consider at half-filling in addition �� 1 and ��c �� 1 we get : �4 4� and fc � 1 and instead of Eq. (112) we obtain E � 2 1 2 2 1 1 2� A( ) / , (113) and Eq. (111) for T Tc� becomes �� e � � � � � � � �1 2 2 1 1 2 2 2 2 1 0( ) , (114) which can not be fulfilled. This shows that for the half-filled band case and positive renormalized Cou- lomb interaction s-wave superconductivity is not fa- vored. This may change for d-wave superconductivity, which would be a technically more demanding inves- tigation in view of our expansion around the atomic limit, because of the explicit k dependence of the or- der parameter, which has to be retained in case of d-wave superconductivity. In the following we will discuss the opposite case, i.e., <� sc c d /9 �( )2 2 2 4, which includes the possi- bility to consider the negative values of < and hence <� Csc � � �4 0. This case corresponds to U � 0. The corresponding equation for Tc is then 0 1 4 2 4 1 4 2 2 2 4 2 2 2 4 2 � � � � � � � ��f U c d c d Uc sc U� C C C �( ) ( ) ( ( )e 1 2 2 2 0 1 1 2 2 ) ( ) tanh( ) tanh( ) Z U z / z z / z� � � � � � � � � � � � �� 1 �1 � � � � �� U U U Z z / z U( )( ) tanh( ) tanh( )2 1 22 0 1 1 � � �e ( )�z / z 2 2 2� � �� 4 2 2 12 0 U U d/ d Z c/ c U ( ) tanh( ) ( ) tanh( ) (( ) � � � � � �e e �� 3 4 1 51 e � ) Z0 + � � � � �f U U Z c sc U2 4 2 2 8 0 2 1 2 � � � C � � �( )( ) ( )( ) tanh(( )e e e � � �c/ c c d/ d z / z 2 2 21 1 ) tanh( ) tanh( ) � � � �� Interaction of strongly correlated electrons and acoustical phonons Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 627 � � � tanh( ) tanh( ) tanh( ) tanh(� � � �z / z z / z z / z c2 2 1 1 2 2 2 2 2 2 / c d/ d c d 2 2 1 2 2 ) tanh( )�� ! ! � � � �� tanh( ) tanh( ) tanh( ) tanh� � �c/ c d/ d z / z 2 2 21 1 ( ) ( ) ( ) � C C z / z c d c d 2 2 2 2 2 4 2 2 2 4 2 2 4 � � �� 3 4 1 51 , (115) where z c d c d / 2 1 1 2 42 2 2 2 2 4 1 2 � � A � �� ( ) C . (116) For half-filling,U � 0 and C4 2 2� c d t we have <� C � sc c U U f� � � � � � � � 4 4 2 2 1 3 1 3 1 e e \| \|/ \| \|/ (117) for sufficiently large \| \|U , i.e., ln \| \| lnU� � 9. In case of very small values C corresponding to the condition ln \| \| ln \| \| ln� � �� D 1 2 3U (118) Eq. (115) simplifies to (C F> ) 1 1 1 41 2 2� � � � � � � � � � � 3 4 5 �f A Ac sc�� e � � �4 02 6 2 1 3 2 2f A A Ac sc� �( ) ( ) , (119) where A n d d / n n n � � � �� 1 2 2! ( ) tanh( ) � �� . (120) Because� �\| \| is assumed to be larger than 3, the coef- ficients in Eq. (120) can be approximated by A A A1 3 2 5 3 7 1 2 3 8 5 16 � � �� \| \| , \| \| , \| \|� � � (121) allowing to replace Eq. (119) by 1 3 144 6 0 2 � � � � f f fc c c � �\| \| . (122) For negative chemical potential, � � 0, and fc � � �1 1/ c( )�� the equation for the critical temperature has the simple form: t t y t y3 246 24 191 24 0� � � � �( ) , with t k T yB c c c � � � � � , \| \| , (124) yielding for small t the solution t y y y y � 1 46 191 1 2 3� � �( ) , (125) where y y� 24 191 , (126) with the requirement that y �� 1 is fulfilled. In the simplest approximation Tc is of the order k TB c � 24 191 \| \|,� (127) showing that Tc is proportional to the renormalized chemical potential in Eq. (13) and increases linearly with increasing strength of the electron-phonon cou- pling parameter. 6. Conclusions The interaction of correlated electrons and acousti- cal phonons has been discussed by using the canonical transformation of Lang–Firsov which results in mobile polarons consisting of electrons surrounded by the acoustical phonon fields (clouds). A kind of general- ized Wick’s theorem is used to handle the strong Cou- lomb repulsion between the electrons emerged into the see of phonon fields. In the strong-coupling limit of the electron-phonon interaction chronological thermodynamic averages of products of acoustical phonon-field operators are ex- pressed by averages of one-cloud operators. For the normal one-cloud propagator the Lorentzian form in Eq. (25) while for anomalous one the Gaussian form in Eq. (28) has been found. The superconducting phase transition is determined as usual by the appearance of electronic Cooper pairs. For the system of renormalized electronic Green‘s functions in Eq. (48) the diagrammatic structure is an- alyzed and the Dyson equations have been derived, see Eqs. (52)–(54). Besides the full Green’s functions the equations contain also three correlation functions and three mass operators. These quantities have been cal- culated by summing infinite series of diagrams after performing appropriate approximations of Eq. (53). Resulting Eqs. (60) and (62) for the superconducting state are then linearized in terms of the order parame- 628 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 V.A. Moskalenko, P. Entel, and D.F. Digor terY�� leading to the final Eq. (88) which determines the superconducting transition temperature Tc, which is invariant with respect to particle-hole transforma- tion. Further analysis shows that the problem of su- perconductivity in the frame of the Hubbard—Hol- stein model is analogous to the discussion of superconductivity in the frame of the single band Hubbard model with appropriately renormalized paramaters. For further discussion the normal state properties given by Eq. (77) are investigated with respect to the metal-insulator transition. For half-filling, i.e., one electron per lattice site and � �U/2, the model yields a metallic state provided the renormalized value ofU is smaller than 3 2/ W~ , where ~W is the electron en- ergy band width, which is narrowed by the effect of the phonon fields, see Eqs. (95) and (98). The param- eter b in Eq. (95) determines the deviation from half-filling. It has been shown that away from half-filling a metallic state is favored forU / W� 3 2 ~ and b larger than a critical value bc. The search for superconductivity has then been per- formed on the basis of Eq. (88) for s-wave supercon- ductivity for two cases. In the first case, at half-filling ( )� �U/2 with positiveU, Eq. (88) has been reduced to Eq. (114), which has no solution. In the second case of negativeU the nontrivial solution in Eq. (127) has been found. As mentioned before, in a real solid with correlated electrons the quantityU should be re- placed by an effective screened parameter. For an overall attractive interaction mediated by the phonons the Hubbard–Holstein model can have a supercon- ducting solution, although the bare valueU0 can still be substantially large. Acknowledgments This work was supported by the Heisenberg—Lan- dau Program. It is a pleasure to acknowledge discus- sions with Prof. N. Plakida and Dr. S. Cojocaru and to thank Vadim Shulezhko for asistance in diagram drawing. V.A.M. would like to thank the University of Duisburg-Essen for financial support. P.E. thanks the Bogoliubov Laboratory of Theoretical Physics, JINR, for the hospitality he received during his stay in Dubna. Appendix A: Laplace approximation In this section we provide calculational details of the Fourier representations ~( )� i� and ~( )� i� defined in Eqs. (24a) and (24b) by making use of the Laplace method of approximation [29]. In the strong-coupling limit the integrand is maximal at the end points � � 0 and � �� and is considerably smaller at other points of the interval of integration ( , )0 � . Therefore, we can re- place the initial integral in Eq. (24a) by one in which the region of integration is limited to the two small in- tervals ( , )0 0� and ( , )� � �� 0 . In these intervals inser- tion of the expansions (19) lead to ~( ) .( )� � � � � � � � � i d di ic c� � � � 0 0 0 � �� e e (A1) Then, because in the strong-coupling limit the collec- tive frequency �c is large, we can replace �0 by infin- ity, which yields a Lorentzian for (A1), ~( ) ( ) � � � i i c c � � � � 2 2 2 . (A2) If we take into account the space dependence and ex- pand in terms of small distances, \| \|x , we obtain � � ( \| ) ( \| ) ,x x� �0 1 2 1 2 (A3) where �� 1 2 21 6 2 2 � �N g / / k k k k k \| ( )\| coth( ) sinh( ) . (A4) For small values of \| \|x the function in Eq. (21) can then be written in factorized form, � � � � �( \| ) ( ( \| )x x)� 0 , (A5) where � � � �� ( \| ) ~( )( \| ) ( \| )0 100 0� �� e ei i� � � , (A6) � ��( ) ~( ) ,x x qx� �� e e1 2 2 1/ N q i � (A7) ~( ) ( ) .� � �q / / q /� �2 1 3 2 2 1e (A8) For the Fourier representation of the anomalous phonon-cloud propagator � �( \| )x we consider first \| \|x � 0. In this case we need the � expansion of 0 �20\| near the midpoint of the interval � �� /2 where this function is minimal. Near this minimum we use the following expression: � � � � �( \| ) ( \| ) ( \| ) ( )0 0 1 2 0 2 2 2 � � �� / / . This approximation can be used in the integral (24b), which allows to rewrite it as ~( ) exp[ ( \| ) ( \| )]� � � � � i d /n / / � � 2 2 0 0 00 0 2 � � � � � ! Interaction of strongly correlated electrons and acoustical phonons Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 629 ! � � ��exp[i / /�� 1 2 2 0 22( ) ( \| )] . (A9) The width of the small interval, 2 0� , is now extended to infinity because the second derivative, �� ( \| )0 2/ , is large in the strong-coupling limit, which yields a Gaussian distribution, ~( ) exp [ ( \| ) ( \| )� � �i / /n� � � � �2 00 0 22 � �i / /� �� 2 22 ( )], � �� ( \| )0 2/ . (A10) This allows to get an approximate expression for Eq. (38). With Eqs. (30) and (32) we have, for example, to evaluate �2 1 1 2 2 3 3 4 4( , ; , \| , ; , )x x x xi i i i� � � � � � � � � !�d d i i i i� � � � � � 1 0 4 1 1 2 2 3 3 4 4� exp [ ]� � � � ! � � � � � �exp { ( \| \| \| ) ( \| \| \| ) � � � �x x x x1 3 1 3 1 4 1 4 � � � � � � � � � � �( \| \| \| ) ( \| \| \| )x x x x2 3 2 3 4 2 42 � � � � � � � � � � �( \| \| \| ) ( \| \| \| )x x x x1 2 1 2 3 4 3 4 � 2 00 ( \| )}. (A11) Equation (A11) leads to a sum of 24 fourfold integrals with different chronological order of �n , n � 1 4� (for example, � � � � � 2 1 2 3 4� � � is defined by from � � � � �� 1 2 3 4 0), of which only 16 make an essential contribution in the strong-coupling limit. It is convenient to combine the 16 terms pairwise like [ & ],� � � � � � � �1 3 2 4 2 4 1 3� � � � � � [ & ],� � � � � � � �1 3 4 2 4 2 1 3� � � � � � [ & ],� � � � � � � �1 4 2 3 2 3 1 4� � � � � � [ & ].� � � � � � � �1 4 3 2 3 2 1 4� � � � � � Further pairwise terms are obtained from the chro- nological orders by changing in the first two groups the order of �1 and �3, and in the last two groups the order of �1 and � 4. All integrals are then calculated by using the maximum possible value of in Eq. (32) in the Laplace approximation, i.e., maximal contributions arise from positive terms in Eq. (32) and coinciding ar- guments in each function. In addition the � space, for which the integrand is maximal, should be large enough. For example, in the integration over the first pairwise terms we take \| \|� �1 3� and \| \|� �2 4� as well as \| \|x x1 3� and \| \|x x2 4� as small quantities. This leads with � �1 3 1� � t and � �2 4 2� � t and, assuming for simplicity, x x1 3� and x x2 4� , to the following ar- gument of the exponential function in Eq. (A11): � �( \| ) ( \| ) ( \| )0 01 2 1 2 1 2 2t t t� � � � � �x x � � � � �( \| ) ( \| )x x x x2 1 1 2 1 1 2 1 2� � � � � � �t � � � � �( \| ) ( \| )x x1 2 1 2 1 2 2 00� � � � �t t , (A12) which simply reduces to � ��c t t( )1 2 when expanding in t1 and t2. The corresponding integrations over t1 and t2 in the interval ( , )0 0� > B( , )0 lead to � � � � � 2 1 3 2 4� � � � d d i i� � � � � 1 0 2 0 1 1 1 3 2 2 4� � � � � !e ( ) ( )� � � � !� � � � � � �dt dt t i t ic c 1 0 2 0 0 0 1 3 2 4 1 3 2 4 � � � �x x x x, , ( ) (e � � ) � � � � � � ! � � � � x x x x1 3 2 4 3 4 , , ( )( )c ci i� � !� � � � �d d i i� � � � � 1 0 2 0 1 1 1 3 2 2 4e ( ) ( ).� � � � (A13) � � � � � 2 2 4 1 3� � � differs from � � � � � 2 1 3 2 4� � � by the last twofold integrals, which can be written as d d i i� � � � � 2 0 1 0 2 1 1 3 2 2 4� � � � � �e ( ) ( )� � � � � � � � � �d d i i� � � � � 1 0 2 1 1 1 3 2 2 4e ( ) ( ).� � � � (A14) By combining the last two integrals, (A13) and (A14), we obtain the law of conservation for the fre- quencies, � �� 2 2 1 3 2 4 2 4 1 3� � � � � �� x x x x1 3 2 4 1 3 2 4 2 3 4 , , , , ( )( ) . � � � � � �c ci i (A15) The same procedure can be used for the other 7 groups of integrals, which finally leads to �2 1 1 2 2 3 3 4 4( , ; , \| , ; , )x x x xi i i i� � � � � � �{ , , , , � � � � �x x x x1 3 2 4 1 3 2 4 2 � � � � � !� � � � �x x x x1 4 2 3 1 4 2 3 2 , , , , }� � � � ! � � ( ) [( ) ][( ) ] 2 2 1 2 2 2 2 2 � � � c c ci i� � . (A16) 630 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 V.A. Moskalenko, P. Entel, and D.F. Digor This equation can be rewritten in the form �2 1 1 2 2 3 3 4 4( , ; , \| , ; , )x x x xi i i i� � � � � � �� ( , \| , ) ( , \| , )x x x x1 1 3 3 2 2 4 4i i i i� � � � � � �( , \| , ) ( , \| , ),x x x x1 1 4 4 2 2 3 3i i i i� � � � (A17) � � � � �( , \| , ) ~ ( ) ., ,x x x x1 1 3 3 1 1 3 1 3 i i i� � � � �� (A18) From these equations we finally obtain Eq. (36). In similar manner we calculated the Fourier representation of the two-cloud function �2, which leads to Eq. (37). Appendix B: the critical temperature With help of the Poisson summation formula, 1 1 4 1 2� � � � f i i dz z f zn Cn ( ) tanh( ) ( ),� �� (B1) where C is the usual counterclockwise contour of the imaginary axis, we obtain for case that the parameters in Eq. (106) obey : 4 2 2 2 4� �( )c d / : I Q i c d 1 1 4 2 2 2 1 1 1 4 � � � � � !�� : ( ) ( ) ! � � � � �� Im tanh( ( ) ) , � E E E E 1 2 1 2 2i / i (B2a) I c d I c/ c 2 2 2 4 1 42 2 2 � � � � � : � : tanh( ) � � � 1 2 2 4 1 2 1 2c i / i: � E E E E Re tanh( ( ) ) , (B2b) I c d I d/ d 3 2 2 4 1 42 2 2 � � � � � : � : tanh( ) � � � 1 2 2 4 1 2 1 2d i / i: � E E E E Re tanh( ( ) ) , (B2c) with E : 2 1 1 2 2 4 2 2 2 2 1 2 � � A �� c d c d / , (B3) and Re tanh( ( ) )� E E E E 1 2 1 2 2� � � i / i � � � � E �E E �E E E �E �E 1 1 2 2 1 2 2 2 1 2 sinh sin ( )[cosh cos ] , (B4) Im tanh( ( ) )� E E E E 1 2 1 2� � � i / i � � � � � E �E E �E E E �E �E 2 1 1 2 1 2 2 2 1 2 sinh sin ( )[cosh cos ] . (B5) Inserting the results in Eq. (B2) into Eq. (106) al- lows us to obtain expressions for the parameters =11, =22 and =12. Furthermore, this allows to decompose the Tc equation with the help of the following abbre- viations, A U U Z U � � � � �� 1 2 12 2 2 0 2( ) ( )( )( ) � � � �e e e ( )= = =11 22 12 2� � � � � � � � � � � � � � 1 2 12 2 2 0 2 U U Z U ( ) ( )( )( ) � � � �e e e ( ) ( ) tanh( ) tanh( ) U c d c/ c d/ d sc� : � �2 2 2 2 84 2 2� � � � �� ! ! � � � � �� Re tanh( ( ) ) tanh( ) tanh� E E E E �1 2 1 2 2 2i / i c/ c ( )�c/ c 2 � tanh( ( ) ) ( ) ( � E E E E : : 1 2 1 2 2 4 2 2 2 4 2 2 2 4 � � " # $ % & ' � � � � i / i c d c � ! d2 2) ! � � � � �� Im tanh( ( ) ) tanh( ) tanh� E E E E �1 2 1 2 2 2i / i d/ d ( )�c/ c 2� � �� 1 2 2c d� 3 4 5 , (B6) and B U U Z � � � � � �� 1 2 1 0 11� = �e 1 2 2 0 22� � � � �� U U Z U � = � �e e ( ) Interaction of strongly correlated electrons and acoustical phonons Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 631 � � � � � � � � � � �U c d U Z U sc U� : : � � 2 2 1 1 24 4 2 2 2 2 0 [ ( ) ] ( ) ( ) ( )e � � � � 1 �1 �I1 � � � � � �� Im tanh( ( ) )� E E E E 1 2 1 2 2i / i U U Z U � � �� ( )( )( )2 1 2 0 � �e � � �� " # $ $ � � 2 2 2 0 U U d/ d Z U ( ) tanh( ) ) � � � � �e e � � �tanh( ) ( ) , c/ c Z 2 1 0 � � � % & ' ' 3 4 1 51 e (B7) where the value of I1 is determined by Eq. (B2a). Equation (88), which determines Tc, can then be written as 1 02� � �Bf Afc c , (B8) where the quantity fc is given by Eq. (79). Equation (B7) would allow to investigate in detail the inter- play of the different parameters and renormalized quantities obtained after eliminating the elec- tron-phonon interaction by the Lang–Firsov transfor- mation. The influence of these parameters on Tc can however only be obtained by numerical work, which has yet to be undertaken. The simplified discussion in the main text shows that Tc is proportional to the typical energy scale involved, which is the renor- malized chemical potential and not the bare quanti- ties of the original model. This means that due to the Lang–Firsov transformation the proportionality of Tc to a typical phonon frequency as in BCS theory or in the Eliashberg formulation is lost. This is an interest- ing observation but must be tested numerically by an- alyzing the more complex expressions of this paper. 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