Spin-fluctuation superconductivity in the Hubbard model

The theory of the superconductivity mediated by kinematic and exchange interactions in t−J and two-band Hubbard models in a paramagnetic state is formulated. The Dyson equations for the matrix Green functions in terms of the Hubbard operators are obtained in the non-crossing approximation. To ca...

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Datum:	1998
1. Verfasser:	Plakida, N.M.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 1998
Schriftenreihe:	Condensed Matter Physics
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/118580
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Zitieren:	Spin-fluctuation superconductivity in the Hubbard model / N.M.Plakida // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 57-68. — Бібліогр.: 22 назв. — англ.

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spelling	irk-123456789-1185802017-05-31T03:04:22Z Spin-fluctuation superconductivity in the Hubbard model Plakida, N.M. The theory of the superconductivity mediated by kinematic and exchange interactions in t−J and two-band Hubbard models in a paramagnetic state is formulated. The Dyson equations for the matrix Green functions in terms of the Hubbard operators are obtained in the non-crossing approximation. To calculate superconducting Tc a numerical solution of self-consistent Eliashberg equations is proposed. Формулюється теорія надпровідності за посередництвом кінематичної та обмінної взаємодій в t−J моделі та двозонній моделі Хаббарда в парамагнітному стані. Отримано рівняння Дайсона для матрич них функцій Гріна в термінах операторів Хаббарда у неперехресному наближенні. Для розрахунку температури переходу в надпровідний стан Tc запрпоновано числовий розв’язок самоузгоджених рівнянь Еліасберга. 1998 Article Spin-fluctuation superconductivity in the Hubbard model / N.M.Plakida // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 57-68. — Бібліогр.: 22 назв. — англ. 1607-324X PACS: 74.20.Mn, 74.20.-z, 74.72.-h DOI:10.5488/CMP.1.1.57 http://dspace.nbuv.gov.ua/handle/123456789/118580 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The theory of the superconductivity mediated by kinematic and exchange interactions in t−J and two-band Hubbard models in a paramagnetic state is formulated. The Dyson equations for the matrix Green functions in terms of the Hubbard operators are obtained in the non-crossing approximation. To calculate superconducting Tc a numerical solution of self-consistent Eliashberg equations is proposed.
format	Article
author	Plakida, N.M.
spellingShingle	Plakida, N.M. Spin-fluctuation superconductivity in the Hubbard model Condensed Matter Physics
author_facet	Plakida, N.M.
author_sort	Plakida, N.M.
title	Spin-fluctuation superconductivity in the Hubbard model
title_short	Spin-fluctuation superconductivity in the Hubbard model
title_full	Spin-fluctuation superconductivity in the Hubbard model
title_fullStr	Spin-fluctuation superconductivity in the Hubbard model
title_full_unstemmed	Spin-fluctuation superconductivity in the Hubbard model
title_sort	spin-fluctuation superconductivity in the hubbard model
publisher	Інститут фізики конденсованих систем НАН України
publishDate	1998
url	http://dspace.nbuv.gov.ua/handle/123456789/118580
citation_txt	Spin-fluctuation superconductivity in the Hubbard model / N.M.Plakida // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 57-68. — Бібліогр.: 22 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT plakidanm spinfluctuationsuperconductivityinthehubbardmodel
first_indexed	2025-07-08T14:16:16Z
last_indexed	2025-07-08T14:16:16Z
_version_	1837088563036946432
fulltext	Condensed Matter Physics, 1998, Vol. 1, No 1(13), p. 57–68 Spin-fluctuation superconductivity in the Hubbard model N.M.Plakida Joint Institute for Nuclear Research, 141980 Dubna, Russia Received May 14, 1998 The theory of the superconductivity mediated by kinematic and exchange interactions in t−J and two-band Hubbard models in a paramagnetic state is formulated. The Dyson equations for the matrix Green functions in terms of the Hubbard operators are obtained in the non-crossing approximation. To calculate superconducting Tc a numerical solution of self-consistent Eliashberg equations is proposed. Key words: superconductivity, strong electron correlations, Hubbard model, t− J model, spin fluctuations PACS: 74.20.Mn, 74.20.-z, 74.72.-h 1. Introduction Since the discovery of high temperature superconductivity in cuprates it has been believed by many researchers that an electronic mechanism could be respon- sible for high values of Tc. Recent experimental evidences of a d-wave supercon- ducting pairing in high-Tc cuprates strongly support this idea (see, for example, [1,2]). At present various phenomenological models for the spin-fluctuation pairing mechanism are known (for reference see, e.g., [2,3]). Numerical finite cluster cal- culations also suggest a d-wave superconducting instability for models with strong electron correlations [4]. Anderson [5] was the first who stressed the importance of strong electron cor- relations in copper oxides and proposed to take them into account within the framework of a one-band Hubbard model: H = −t ∑ 〈ij〉σ (a+iσajσ +H.c.) + U ∑ i ni↑ni↓, (1) where t is an effective transfer integral for the nearest neighbour sites, 〈ij〉, and U is the Coulomb single-site energy. He also considered the so-called t− J model which results from the Hubbard model (1) in the strong coupling limit, U ≫ t, when only singly occupied sites are taken into account, since a doubly occupied c© N.M.Plakida 57 N.M.Plakida site needs a large additional energy U : Ht−J = −t ∑ 〈ij〉,σ (ã+iσãjσ +H.c.) + J ∑ 〈ij〉 (SiSj − 1 4 ninj). (2) Here electron operators ã+iσ = a+iσ(1− ni−σ) act in the subspace without a double occupancy and ni = ni↑ + ni↓ is the number operator for electrons. The second term describes the spin-1/2 Heisenberg antiferromagnet (AFM) with the exchange energy J = 4t2/U for the nearest neighbours. To allow for the constraint of no double occupancy on a rigorous basis it is convenient to rewrite the t−J model (2) in terms of the Hubbard operators (HO): Ht−J = − ∑ i 6=j,σ tijX σ0 i X0σ j − µ ∑ iσ Xσσ i + 1 2 ∑ i 6=j,σ Jij ( Xσσ̄ i X σ̄σ j −Xσσ i X σ̄σ̄ j ) , (3) where tij = t, t ′ is the electron hopping energy for the nearest and the second neigh- bours on the 2D square lattice, respectively, and Jij is the exchange interaction. We have also introduced chemical potential µ and number operator ni = ∑ σ X σσ i . The HO are defined as Xαβ i = \|i, α〉〈i, β\| (4) for three possible states at the lattice site i: \|i, α〉 = \|i, 0〉, \|i, σ〉 for an empty site and for a site singly occupied by an electron with the spin σ/2 (σ = ±1, σ̄ = −σ). They obey the completeness relation X00 i + ∑ σ Xσσ i = 1, (5) which rigorously preserves the constraint of no double occupancy. Below we consider a more realistic for copper-oxide compounds two-band p−d model. It can be reduced to the asymmetric Hubbard model with the lower Hubard sub-band (LHB) occupied by one-hole Cu-d like states and the upper Hubbard sub- band (UHB) occupied by two-hole p−d singlet states [6]. In terms of the Hubbard operators the asymmetric Hubbard model reads: H = H0 +Ht = E1 ∑ iσ Xσσ i + E2 ∑ i X22 i − ∑ i 6=jσ {t11ij X σ0 i X0σ j + t22ij X 2σ i Xσ2 j + σt12ij (X 2σ̄ i X0σ j +Xσ0 i X σ̄2 j )}. (6) Here the energy levels E1 = E0 − µ and E2 = 2E0 − 2µ + ∆ are introduced for singly and doubly occupied sites, respectively, where E0 is a reference energy. In the singlet-hole model (6) the Coulomb repulsion energy U in the standard Hubbard model (1) is substituted by the charge transfer energy ∆ = ǫp − ǫd between p- and d-levels in the CuO2 plane. The hopping integrals have different values for the LHB (t11ij ), the UHB (t22ij ) and the inter-band transitions (t12ij ). They can be written in the form tαβij = −Kαβ2tνij where t = tpd is the p− d hybridization integral and 58 Superconductivity in Hubbard model νij are the overlapping parameters for the Wannier oxygen states which are equal to: ν1 = νj j±ax/y ≃ −0.14 for the nearest neighbours and ν2 = νj j±ax±ay ≃ −0.02 for the second neighbours, where ax/y are lattice constants. The coefficients Kαβ depend on the dimensionless parameter t/∆ and for a realistic value of ∆ = 2t they are of the order 0.5− 0.9 [6] (also see [7]). The Hubbard operators (4) for a two-band model (6) are defined for 4 possible states at lattice site i: \|i, α〉 = \|i, 0〉 , \|i, σ〉 , \|i, ↑↓〉 for an empty site, a site singly occupied by an electron with the spin σ and for a doubly-occupied site, respectively. For these states the completeness relation for the Hubbard operators reads: X00 i + ∑ σ Xσσ i +X22 i = 1 . (7) A number of attempts have been made to obtain a superconducting pairing within the microscopical theory for the Hubbard models discussed above. It should be pointed out that the superconducting pairing due to the kinematic interaction in the Hubbard model (1) in the limit of strong electron correlations (U → ∞) was first proposed by Zaitsev and Ivanov [8]. Close results were obtained by Plakida and Stasyuk [9] by applying an equation of the motion method to two-time Green functions (GF) [10]. However, in these papers only the mean field approximation was considered which results in the s-wave pairing irrelevant to strongly correlated systems (for the discussion see [11]). Later on the theory in the mean field approx- imation was considered for the t − J model within the GF approach in [11,12] where the d-wave spin-fluctuation superconducting pairing due to the exchange interaction J was studied. Superconductivity in the original Hubbard model (1) was discussed in [13,14] in the mean field type approximation within the projection technique for the GF. Local superconducting pairings of the s- and d-symmetry were obtained which, however, should disappear in the limit of strong correlations, U → ∞. Unfor- tunately, in this approximation the self-energy operator caused by kinematic and exchange interactions is ignored, though it results in finite life-time effects and gives a substantial contribution to the renormalization of the quasiparticle (QP) spectrum in the normal state. The self-energy of the anomalous GF is also respon- sible for the non-local spin-fluctuation d-wave superconducting pairing. Recently it was demonstrated for the spin-polaron representation of the t− J model in [15]. A self-consistent numerical treatment of the strong coupling Eliash- berg equations revealed a strong renormalization of the QP hole spectrum due to spin-fluctuations and proved the d- wave pairing. The maximum Tc ≃ 0.01t was obtained at the optimal concentration of doped holes δ ≃ 0.2. However, a two- sublattice representation used in [15] can be rigorously proved only for a small doping with a long-range AFM order. At a moderate doping one has to consider a paramagnetic (spin-rotationally invariant) state in the t− J model. The opposite limit of low electron densities in the t − J model was studied by M. Kagan and Rice [16]. They observed various forms of electron pairing at low temperatures including the d-wave instability at the values J/t > 1. A special diagram technique for the Hubbard operators was also applied by Izyumov et 59 N.M.Plakida al. [17] to consider spin fluctuations and a superconducting pairing in the t − J model. However, no numerical results were presented. In the given paper we consider a paramagnetic state with only short-range dynamic spin fluctuations at a moderate doping. We develop the theory of super- conductivity for the t − J model (3) and the asymmetric Hubbard model (6) by applying the projection technique to the GF [10] in terms of the Hubbard oper- ators. Contrary to the above mentioned papers, the self-energy operators due to kinematic and exchange interactions are explicitly calculated in the non-crossing approximation. The QP spectrum in the normal state of the t − J model within the GF approach at T = 0 was also studied recently by Prelovśek [18]. Below, the Dyson equations for the matrix Green function are presented for the t− J model in section 2 and for the Hubbard model in section 3, which are a direct generalization of the theory developed earlier in collaboration with Professor I.V.Stasyuk [9,11]. 2. Dyson equation for the t − J model To discuss the superconducting pairing within model (3) we consider the matrix Green function (GF) Ĝij,σ(t− t′) = 〈〈Ψiσ(t)\|Ψ + jσ(t ′)〉〉 (8) in terms of the Nambu operators: Ψiσ = ( X0σ i X σ̄0 i ) , Ψ+ iσ = ( Xσ0 i X0σ̄ i ) , (9) where Zubarev notation [10] for the anticommutator Green function (8) is used. To calculate the GF (8) we use the equation of motion for the HO: ( i d dt + µ ) X0σ i = − ∑ l tilBiσσ′X0σ′ l + ∑ l Jil(Blσσ′ − δσσ′)X0σ′ i , (10) where Biσσ′ = (X00 i +Xσσ i )δσ′σ +X σ̄σ i δσ′σ̄ = (1− 1 2 ni + σSσ i )δσ′σ + Sσ̄σ i δσ′σ̄ . (11) The boson-like operator Biσσ′ describes electron scattering on spin and charge fluctuations caused by the nonfermionic commutation relations for the HO’s (the first term in (10) – the so-called kinematical interaction) and by the exchange spin-spin interaction (the second term in (10)). By differentiating the GF (8) with respect to time t and t ′ and employing the projection technique (see, e.g., [6]) we get the following Dyson equation: Ĝijσ(ω) = Ĝ0 ijσ(ω) + ∑ kl Ĝ0 ikσ(ω) Σ̂klσ(ω) Ĝljσ(ω) (12) 60 Superconductivity in Hubbard model for the Fourier component. Here the zero–order GF is calculated in the mean-field approximation Ĝ0 ijσ(ω) = Q {ωτ̂0δij − Êijσ} −1 (13) with the frequency matrix Êijσ = 〈{[Ψiσ, H],Ψ+ jσ}〉 Q −1 and the correlation func- tion Q = 〈X00 i + Xσσ i 〉 = 1 − n/2 . In a paramagnetic state it depends only on the average number of electrons n = 〈ni〉 = ∑ σ〈X σσ i 〉. The self-energy operator Σ̂klσ(ω) is defined by the equation: Σ̂ijσ(ω) = Q−1 Σ̃ijσ(ω) = Q−1 〈〈Ẑ (irr) iσ \| Ẑ (irr)+ jσ 〉〉(irr)ω Q−1 (14) where the irreducible part of the operator Ẑiσ = [Ψiσ, H] is defined by the projection equation Ẑ (irr) iσ = Ẑiσ − ∑ l ÊilσΨlσ , 〈{Ẑ (irr) iσ ,Ψ+ jσ}〉 = 0 . (15) Equations (12) - (14) give an exact representation for the one-electron GF (8). To calculate it, however, one has to apply approximations to many-particle GF in the self-energy matrix (14) which describes inelastic scattering of electrons on a spin and charge fluctuations. Here we employ a non-crossing approximation (or a self-consistent Born approximation) for the irreducible part of many-particle Green functions in (14). It neglects vertex corrections and is given by the following two-time decoupling for the correlation functions: 〈Xσ′0 j′ B+ jσσ′X0σ′ i′ (t)Biσσ′(t)〉(j 6=j′, i 6=i′) ≃ 〈Xσ′0 j′ X0σ′ i′ (t)〉〈B+ jσσ′Biσσ′(t)〉 . (16) Using a spectral representation for the GF we obtain the following results for self-energy matrix elements in the k-representation: Σ̃σ 11(12)(k, ω) = 1 N ∑ q ∫ +∞ ∫ −∞ dzdΩN(ω, z,Ω)λ11(12)(q, k − q \| Ω)Aσ 11(12)(q, z), (17) with N(ω, z,Ω) = 1 2 tanh(z/2T ) + coth(Ω/2T ) ω − z − Ω . (18) Here we introduce a spectral density for the normal (G11) and anomalous (G12) GF: Aσ 11(q, z) = − 1 Qπ Im 〈〈X0σ q \| Xσ0 q 〉〉z+iδ , (19) Aσ 12(q, z) = − 1 Qπ Im 〈〈X0σ q \| X0σ̄ −q〉〉z+iδ (20) and the electron - electron interaction functions caused by spin and charge fluctu- ations λ11(12)(q, k − q \| Ω) = g2(q, k − q) D+(−)(k − q,Ω), (21) 61 N.M.Plakida where g(q, k− q) = t(q)− J(k − q) and the spectral density of bosonic excitations are given by the imaginary part of the spin and charge susceptibilities: D±(q,Ω) = − 1 π Im { 〈〈Sq \| S−q〉〉Ω+iδ ± (1/4)〈〈nq \| n + q 〉〉Ω+iδ } . (22) A linearized system of Eliashberg equations close to Tc can be written as self- consistent equations for the normal GF and its self-energy operator G̃σ 11(k, iωn) = {iωn − Ek + µ̃− Σ̃σ 11(k, iωn)} −1 , Σ̃σ 11(k, iωn) = − T N ∑ q ∑ m G̃σ 11(q, iωm)λ11(q, k − q \| iωn − iωm) (23) and for the gap equation: Φσ(k, iωn) = ∆σ k + Σ̃σ 12(k, iωn) = T N ∑ q ∑ m {2J(k − q)+ +λ12(q, k − q \| iωn − iωm)}G̃ σ 11(q, iωm)G̃ σ̄ 11(q,−iωm)Φ σ(q, iωm). (24) In equation (24) we omit the k-independent part of the gap function ∆σ k in the MFA (13) which is caused by the kinematic interaction [8], since it gives no con- tribution to the d-wave pairing ([11]). Here we use the imaginary frequency repre- sentation, ω = iωn = iπT (2n+ 1). The energy of quasiparticles Eσ k and the renormalized chemical potential µ̃ = µ− δµ in the MFA (13) is given by Eσ k = −ǫ(k)Q− ǫs(k)/Q− 4J N ∑ q γ(k − q)Nqσ, (25) where ǫ(k) = t(k) = 4tγ(k) + 4t′γ′(k), ǫs(k) = 4tγ(k)χ1s + 4t′γ′(k)χ2s with γ(k) = (1/2)(cosaxqx + cos ayqy), γ ′(k) = cos axqx cos ayqy. δµ = 1 N ∑ q ǫ(q)Nqσ − 4J(n/2− χ1s/Q) . (26) The average number of electrons in the k-representation is written in the form: n = 1 N ∑ k,σ 〈Xσ0 k X0σ k 〉 = Q N ∑ k,σ Nkσ, (27) where Nkσ = {1 + 2T N ∑ k ∞ ∑ n=−∞ G̃σ 11(k, iωn)}, (28) which defines function Nqσ in equations (25), (26). When calculating the normal part of the frequency matrix (25) we neglect charge fluctuations and introduce 62 Superconductivity in Hubbard model spin correlation functions for the nearest, a1 = (±ax,±ay), and the second, a2 = ±(ax ± ay), neighbour lattice sites : χ1s = 〈SiSi+a1〉 , χ2s = 〈SiSi+a2〉 . (29) In the present calculations we take into account only the spin-fluctuation con- tribution modelled by the spin-fluctuation susceptibility (see, e. g., [18,19]): χ ′′ s (q, ω) = χs(q) χ ′′ s (ω) = χ0 1 + ξ2(q − QAF ) 2 tanh ω 2T 1 1 + (ω/ωs)2 (30) with the characteristic AFM correlation length ξ and spin-fluctuation energy ωs ≃ J . To fix constant χ0 in (30) we use the following normalization condition: 1 N ∑ i 〈SiSi〉 = 1 N ∑ q χs(q) +∞ ∫ −∞ dz exp z T − 1 χ ′′ s (z) = πωs 2N ∑ q χs(q) = 3 4 n . (31) In this approximation we get for the interaction functions (21) λ11(q, k − q \| iων) = λ12(q, k − q \| iων) = −g2(q, k − q)χs(k − q) +∞ ∫ 0 2zdz z2 + ω2 ν tanh(z/2T ) 1 + (z/ωs)2 . (32) For model (30) we can calculate static spin correlation functions (29) from the equations: χ1s = 〈SiSi+a1〉 = 1 N ∑ q γ(q)〈SqS−q〉, χ2s = 〈SiSi+a2〉 = 1 N ∑ q γ′(q)〈SqS−q〉, where 〈SqS−q〉 = χs(q) +∞ ∫ −∞ dz exp (z/T )− 1 χ ′′ s (z) = χs(q) π 2 ωs. (33) Therefore, we have obtained a closed system of equations which should be solved numerically. Preliminary calculations [20] confirm the existence of narrow QP peaks for the one-electron spectral density (19) near the Fermi surface (FS). The latter has a characteristic behaviour for strongly correlated systems with the occu- pation numbers N(k) ≥ 0.5 throughout the whole Brillouin zone which results in the large FS even at a small doping. A direct numerical solution by the fast Fourier transformation of gap equation (24) proves a superconducting pairing (caused by the exchange, J , and kinematic, t2, interactions in (24)) of the d-wave symmetry that occurs at high Tc ≃ 0.06t. 63 N.M.Plakida 3. Dyson equation for the p − d model In this section we discuss the results for the two band p−d model (6). To study the two–band problem we have to introduce a matrix Green function concerning the normal state properties Ĝijσ(t− t′) = 〈〈X̃iσ(t); X̃ + jσ(t ′)〉〉, (34) where we use two–component operators X̃iσ = ( Xσ2 i X0σ̄ i ) , X̃+ jσ = ( X2σ j X σ̄0 j ) . (35) By differentiating the GF (34) with respect to time t and t ′ and using the projection technique described above, we get the Dyson equation in the form analogous to (12). In [6] only a zero order GF was calculated in the form analogous to the one band GF (13). The two-band spectrum for d-like holes and p − d singlets as well as the density of states were calculated. It was found that hybridization between d-like holes and singlets results in a substantial renormalization of the spectrum. In addition, the dispersion relation depends strongly on antiferromagnetic short- range spin correlations (given by the static spin correlation functions, equations (29) ) in the spin-singlet state. For large spin correlations at small doping values one finds a next-nearest neighbour dispersion. With the doping decreasing spin correlations, the dispersion changes to an ordinary nearest neighbour one. However, to consider the superconducting properties of the two–band model we introduce the 4×4 matrix Green function [21]: G̃ijσ(t− t′) = 〈〈X̂iσ(t); X̂ + jσ(t ′)〉〉 (36) for the four–component operators. For example, the H.c. one is given by the row- vector: X̂+ jσ = ( X2σ j X σ̄0 j X σ̄2 j X0σ j ) . (37) By differentiating the GF (36) with respect to time t and t ′ and using the projection technique described above, we get the following Dyson equation in (k, ω) space: G̃σ(k, ω) −1 = G̃0 σ(k, ω) −1 − Σ̃σ(k, ω) . (38) The zero–order matrix GF is given by the generalized mean–field approximation: G̃0 ij,σ(ω) = {ωδi,j τ̃0 − Ãij,σ} −1χ̃, (39) where Ãij,σ = 〈{[X̂iσ, H] , X̂+ jσ}〉 χ̃ −1 (40) is a frequency matrix. The self–energy operator is given by the irreducible part of the scattering matrix: Σ̃ij,σ(ω) = χ̃−1〈〈[X̂iσ, H]\|[H, X̂+ jσ]〉〉 (irr) ω χ̃−1. (41) 64 Superconductivity in Hubbard model We also introduce unity matrices τ̃0 (4× 4) and τ̂0 (2× 2) and the matrix χ̃ = τ̂0χ̂ with χ̂ = ( χ2 0 0 χ1 ) , (42) where χ2 = 〈X22 i +Xσσ i 〉 = 1−χ1. To solve the Dyson equation (38) which can be written in the general form as G̃σ(k, ω) = ( Ĝσ(k, ω) F̂σ(k, ω) (F̂σ(k, ω)) + −Ĝσ̄(k,−ω) ) (43) we have to calculate the zero–order GF (39) and the self-energy matrix (41). The anomalous part of the zero–order GF in (39), F̂ 0 σ (k, ω), vanishes if one disregards the mean-field, k-independent gap function (due to the kinematic interaction) which violates the restriction 〈Xσ2 i X σ̄2 i 〉 = 0, see [22]. For the normal part we can use a diagonal approximation Ĝ0 σ(k, ω) = ( χ2/(ω − Ω2(k)) 0 0 χ1/(ω − Ω1(k)) ) , (44) where the mean field spectrum is given by the dispersions Ω1(k) and Ω2(k) for a singly occupied d-hole-like band and a doubly occupied singlet band, respec- tively [6]. To calculate the self-energy matrix (41) we use the non-crossing approximation described above (see equation (16)). Writing down the self-energy matrix as Σ̃σ(k, ω) = χ̃−1 ( M̂σ(k, ω) Φ̂σ(k, ω) Φ̂+ σ (k, ω) −M̂σ̄(k,−ω) ) χ̃−1 (45) we obtain for the normal part of the matrix M̂σ(k, ω) = t2 N ∑ q γ2(q) π ∫ ∞ −∞ ∫ ∞ −∞ dνdzN(ω, ν, z)D+(k − q, z), ×{P̂2[−ImG22(q, ν + iδ)] + P̂1[−ImG11(q, ν + iδ)]} (46) where γ(q) = ∑ j exp(iqj)ν0j , N(ω, ν, z) is given by (18) and the spin-charge sus- ceptibility D+(k − q, z) is given by (22). The contributions from the singlet band and from the d-hole band are defined by the matrices P̂2 and P̂1, respectively. Their diagonal terms are: P 11 2 = (K21) 2 , P 22 2 = (K22) 2 , P 11 1 = (K11) 2 , P 22 1 = (K12) 2 . Here we use a notation for hopping integrals in the two-band model (6): tαβij = −Kαβ2tνij where α, β = 2 or 1 for the singlet and the d-hole band, respectively. Matrix equation (46) defines the renormalization of quasiparticle spectra in the two bands due to spin and charge fluctuations, while the analogous equation for 65 N.M.Plakida matrix Φ̂σ(k, ω) gives a gap equation. In the diagonal approximations for the zero- order GF (44) and self-energy (45), (46), the Dyson equation (43) can be solved, which enables one to write an equation for superconducting gaps φαα σ (k, ω) = Φαα σ (k, ω)/χα in the bands α = 2, 1 in the closed form: φαα σ (k, ω) = t2 N ∑ q γ2(q) π ∫ ∞ −∞ ∫ ∞ −∞ dνdzN(ω, ν, z)D−(k − q, z) ×{K2 αα[−Im φαα σ (q, ν) Qα(q, ν) ]− χβK 2 12 χα [−Im φββ σ (q, ν) Qβ(q, ν) ]}, (47) where α 6= β and the denominator Qα(q, ω) = (ω−Ωα(q)−Mαα σ (q, ω)/χα)(ω+Ωα(q)+Mαα σ̄ (q,−ω)/χα)−\|φαα σ \|2 (48) gives a spectrum of excitation in the superconducting phase. 4. Conclusions To summarize, we would like to stress that starting from the microscopical t−J (equation (2)) or the two-band p−d (equation (6)) model we obtain a self-consistent system of equations for the Green functions and the corresponding self-energies. The frequency matrices in the zero- order Green functions (equations (13), (39)) and the renormalization of the quasiparticle spectra given by self-energies, (23) for the t − J model and (46) for the two-band model, and the superconducting pairing in gap equations, (24) for the t − J model and (47) for the two-band model, are caused by spin and charge fluctuations which arise from nonfermionic commutation relations for the Hubbard operators in the models (see the equation of motion (11)). Therefore, in our microscopical theory we have no fitting parameters for the electron-spin interaction as in phenomenological approaches. However, the theory is not fully self-consistent in the respect that the phenomenological model for dynamical spin fluctuations (equation (30)), was used. Nevertheless, we believe that numerical results should not depend considerably on the explicit form of the model for spin-charge fluctuations. Being normalized (equation (31)), it cannot change substantially the sum over (q, ω) in the equations for self-energies. The non-crossing approximation for self-energies (equation (16)) also seems to be quite reliable as has explicitly been proved for the spin-polaron t−J model where vertex corrections are small. It is also interesting to compare the results for the one-band t − J model and the two-band Hubbard model. In the two-band model for the hole (electronically) doped case the chemical potential µ is in the singlet (d-hole) band, α = 2 (1), and the main contribution to the integrand in equation (47) comes from the same band (first term), while the contribution from the other band is proportional to t/∆2. The latter is analogous to the static spin-exchange contribution of order J ≃ (t/∆2) in the one-band t − J model, i.e. to the first term in equation (24). However, in the two-band model the spin-fluctuation contribution to equation (47) is given by 66 Superconductivity in Hubbard model the frequency dependent susceptibility D−(q, z) and the inter-band contributions ∝ K2 12 cannot be fully allowed for within the framework of the one-band t − J model. It would be interesting to compare the solutions of gap equations in the t − J model, equation (24), and in the two-band model, equation (47). However, it demands rather complicated numerical work and will be considered in future publications. Acknowledgements Partial financial support by the INTAS–RFBR Program, Grant No. 95–591 and by NREL, Subcontract AAX-6-16763-01 is acknowledged. References 1. Tsuei C.C., Kirtley J.R., Rupp M., et al. Pairing symmetry in single-layer tetragonal Tl2Ba2CuO6+δ superconductors. // Science, 1996, vol. 271, p. 329-332. 2. Scalapino D.J. The case for dx2−y2 pairing in the cuprate superconductors. // Phys. Reports, 1995, vol. 250, p. 329-365. 3. Plakida N.M. 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R11997- R12000. 16. Kagan M.Yu., Rice T.M. Superconductivity in the two-dimensional t − J model at low electron density. // J. Phys.: Cond. Matt., 1994, vol. 6, p. 3771-3780. 17. Izyumov Yu.A., Letfulov B.M. Superconductivity in the Hubbard model with strong Coulomb repulsion. // Intern. J. Modern Phys. B, 1992, vol. 6, p. 321-357. 18. Prelovśek P. Electron Green’s function in the planar t− J model. // Zeit. Physik B, 1997, vol. 103, p. 363-368. 19. Jaklič J., Prelovśek P. Anomalous spin dynamics in doped quantum antiferromagnets. // Phys. Rev. Lett., 1995, vol. 74, No 17, p. 3411-3414; Universal charge and spin dynamics in optimally doped antiferromagnets. // Phys. Rev. Lett., 1995, vol. 75, No 17, p. 1340-1343. 20. Plakida N.M., Oudovenko V.S. Spin-fluctuation pairing in the t-J model. Proceedings of the NATO ARW “Symmetry and pairing in superconductors”, Yalta (Ukraine), April 28 - May 2, 1998. 21. Plakida N.M. Superconductivity in the two-band singlet-hole model for copper-oxide plane. // Physica C, 1997, vol. 282-287, p. 1737-1738. 22. Lovtsov S.V., Plakida N.M. Yushankhai V.Yu. On superconducting pairing in the two-band Hubbard model. // Zeit. Phys. B, 1991, vol. 82, p. 1-4. Спін-флуктуаційна надпровідність у моделі Хаббарда М.М.Плакіда Об’єднаний інститут ядерних досліджень, 141980 м. Дубна, Росія Отримано 14 травня 1998 р. Формулюється теорія надпровідності за посередництвом кінематич- ної та обмінної взаємодій в t−J моделі та двозонній моделі Хаббар- да в парамагнітному стані. Отримано рівняння Дайсона для матрич- них функцій Гріна в термінах операторів Хаббарда у неперехресному наближенні. Для розрахунку температури переходу в надпровідний стан Tc запрпоновано числовий розв’язок самоузгоджених рівнянь Еліасберга. Ключові слова: надпровідність, сильні електронні кореляції, модель Хаббарда, t− J модель, спінові флуктуації PACS: 74.20.Mn, 74.20.-z, 74.72.-h 68

Spin-fluctuation superconductivity in the Hubbard model

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