Green function approach to the theory of superconductivity in the t - J model

The theory of superconducting pairing due to the exchange and kinematical interactions in the t - J model in a paramagnetic state is developed. The Dyson equation for the matrix Green functions in terms of the Hubbard operators is obtained in the noncrossing approximation. The linearized self-сo...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
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/123456789/119898
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Green function approach to the theory of superconductivity in the t - J model / N.M. Plakida // Condensed Matter Physics. — 1998. — Т. 1, № 4(16). — С. 905-917. — Бібліогр.: 19 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-119898
record_format	dspace
spelling	irk-123456789-1198982017-06-11T03:03:21Z Green function approach to the theory of superconductivity in the t - J model Plakida, N.M. The theory of superconducting pairing due to the exchange and kinematical interactions in the t - J model in a paramagnetic state is developed. The Dyson equation for the matrix Green functions in terms of the Hubbard operators is obtained in the noncrossing approximation. The linearized self-сonsistent system of Eliashberg equations is proposed to study the temperature and doping dependence of the quasi-particle hole spectrum in the normal state and to calculate the temperature of the superconducting phase transition and the symmetry of the gap function. Pозвинуто теоpiю надпpовiдного спаpювання, що вiдбувається завдяки обмiнним та кiнематичним взаємодiям у t - J моделi в паpамагнiтному станi. Отpимано piвняння Дайсона для матpицi функцiй Ґpiна чеpез опеpатоpи Хабаpда у непеpехpесному наближеннi. Запpопоновано лiнеаpизовану самоузгоджену систему piвнянь Елiашбеpга для вивчення темпеpатуpної та концентpацiйної залежностi квазiчастинкового спектpу дipок у ноpмальному станi, а також для знаходження темпеpатуpи надпpовiдного фазового пеpеходу i симетpiї щiлини. 1998 Article Green function approach to the theory of superconductivity in the t - J model / N.M. Plakida // Condensed Matter Physics. — 1998. — Т. 1, № 4(16). — С. 905-917. — Бібліогр.: 19 назв. — англ. 1607-324X DOI:10.5488/CMP.1.4.905 PACS: 74.20.-z, 74.20.Mn, 74.72.-h http://dspace.nbuv.gov.ua/handle/123456789/119898 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 superconducting pairing due to the exchange and kinematical interactions in the t - J model in a paramagnetic state is developed. The Dyson equation for the matrix Green functions in terms of the Hubbard operators is obtained in the noncrossing approximation. The linearized self-сonsistent system of Eliashberg equations is proposed to study the temperature and doping dependence of the quasi-particle hole spectrum in the normal state and to calculate the temperature of the superconducting phase transition and the symmetry of the gap function.
format	Article
author	Plakida, N.M.
spellingShingle	Plakida, N.M. Green function approach to the theory of superconductivity in the t - J model Condensed Matter Physics
author_facet	Plakida, N.M.
author_sort	Plakida, N.M.
title	Green function approach to the theory of superconductivity in the t - J model
title_short	Green function approach to the theory of superconductivity in the t - J model
title_full	Green function approach to the theory of superconductivity in the t - J model
title_fullStr	Green function approach to the theory of superconductivity in the t - J model
title_full_unstemmed	Green function approach to the theory of superconductivity in the t - J model
title_sort	green function approach to the theory of superconductivity in the t - j model
publisher	Інститут фізики конденсованих систем НАН України
publishDate	1998
url	http://dspace.nbuv.gov.ua/handle/123456789/119898
citation_txt	Green function approach to the theory of superconductivity in the t - J model / N.M. Plakida // Condensed Matter Physics. — 1998. — Т. 1, № 4(16). — С. 905-917. — Бібліогр.: 19 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT plakidanm greenfunctionapproachtothetheoryofsuperconductivityinthetjmodel
first_indexed	2025-07-08T16:52:31Z
last_indexed	2025-07-08T16:52:31Z
_version_	1837098391172022272
fulltext	Condensed Matter Physics, 1998, Vol. 1, No. 4(16), p. 905–917 Green function approach to the theory of superconductivity in the t − J model N.M.Plakida Joint Institute for Nuclear Research, 141980 Dubna, Russia Received January 28, 1998 The theory of superconducting pairing due to the exchange and kinemati- cal interactions in the t − J model in a paramagnetic state is developed. The Dyson equation for the matrix Green functions in terms of the Hubbard operators is obtained in the noncrossing approximation. The linearized self- consistent system of Eliashberg equations is proposed to study the tem- perature and doping dependence of the quasi-particle hole spectrum in the normal state and to calculate the temperature of the superconducting phase transition and the symmetry of the gap function. Key words: superconductivity, quasi-particle, Dyson equation, Green functions, Hubbard operators PACS: 74.20.-z, 74.20.Mn, 74.72.-h 1. Introduction In 1960 D.N. Zubarev [1] proposed a theory of superconductivity for an elect- ron-phonon system based on the equation of motion method for two-time Green functions [2]. The paper was published simultaneously with the famous papers by G.M. Eliashberg [3] where the temperature diagram technique was used to obtain the Gorkov type equations for an electron-phonon system. However, Zubarev’s formulation did not draw proper attention of the following investigators, while the Eliashberg theory was cited frequently in many papers and his formulation became known as the Eliashberg (or Migdal-Eliashberg) theory of superconductivity for an electron-phonon system. The real advantage of the Eliashberg formulation is that it permits one to consider a strong coupling limit by using the skeleton diagram technique. In the Zubarev formulation based on a successive differentiation of the Green functions over the same time one cannot employ the skeleton diagram technique. However, in the two-time differentiation method for the Green functions this problem can be easily overcome and the Eliashberg type equations, as it has been shown in [4], can be formulated in a very simple and transparent way for a general model of an electron boson-field interaction. c© N.M.Plakida 905 N.M.Plakida In the present paper the theory of superconductivity is formulated for the t− J model by applying an equation of motion method for the Green functions in terms of the Hubbard operators. It should be pointed out that superconducting pairing due to the kinematical interaction in the Hubbard model in the limit of strong electron correlations (U → ∞) was first obtained by Zaitsev and Ivanov [5]. However, they considered only the mean field approximation which results in the s-wave pairing irrelevant for strongly correlated systems (for a discussion see [6]). Later on the theory in the mean field approximation was considered for the t− J model within the Green function approach in [6,7] where d-wave spin fluctuation superconducting pairing was obtained due to exchange interaction J . In the present formulation we have calculated a matrix self-energy operator in the noncrossing approximation for kinematical and exchange interactions which neglects vertex corrections, as in the Migdal-Eliashberg theory. The self-energy operator allowing for the finite life-time effects for electrons plays an essential role both in the renormalization of the quasiparticle spectrum and in supercon- ducting pairing. It was clearly demonstrated in [8] where the t − J model in the polaron representation was considered. A self-consistent numerical solution of the Eliashberg equations has proved a strong renormalization of the quasiparticle hole spectrum due to spin-fluctuations and d-wave pairing at a finite concentration of doped holes. However, the two-sublattice representation used in [8] can be rigor- ously proved only for a small doping. At a moderate doping one has to consider the paramagnetic (spin-rotationally invariant) state in the t − J model. In that case the Hubbard operator technique is useful. However, due to unconventional commutation relations the Hubbard operators cannot be treated within the stan- dard diagram technique (see, e.g., [9]). To overcome this problem one can employ different types of the slave-boson (-fermion) technique to use the standard fermion- boson diagram technique. However, due to crude treatment of constraints which should be observed in the slave-field techniques the results of that calculation appear to be quite unreliable. The two-time Green function approach in terms of the Hubbard operators was also used to consider superconducting pairing in the t−J model with an electron- phonon interaction in [10] and to discuss the electron and hole spectra in the normal state in [11]. A general discussion concerning the t−J model and its appli- cations in the description of physical properties of the systems with strong electron correlations, as copper-oxide superconductors, can be found, e.g., in [12–14]. The paper is organized as follows. In the next section the t−J model in terms of the Hubbard operators is introduced. In section 3 the Dyson equation for the matrix Green function is obtained by the projection technique in the equation of motion method. In section 4 a self-consistent system of Eliashberg equations in the noncrossing (self-consistent) approximation is formulated and the linearized gap equation for the calculation of Tc is presented. Conclusions are given in section 5. 906 Superconductivity in the t− J model 2. The t − J model The simplest model allowing for the electron correlations in copper oxides is the one-band Hubbard model [15]: H = −t ∑ ijσ a+iσajσ + U ∑ i ni↑ni↓, (2.1) where t is an effective transfer integral and U is the Coulomb one-site energy. In the strong coupling limit, U ≫ t, we can reduce the Hubbard model (2.1) (or a more realistic for copper oxides p− d model [16]) to the t− J model [17]: Ht−J = − ∑ i 6=j,σ tij c̃ + iσ c̃jσ + J ∑ 〈ij〉 (SiSj − 1 4 ninj), (2.2) where the first term describes electron hopping with the energy tij for the nearest neighbours, tij = t, for the second neighbours, tij = t ′ , etc., on a two dimensional square lattice. The electron operators c̃+iσ = c+iσ(1− ni−σ) act in the space without 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 for the nearest neighbours which is equal to J = 4t2/U for the Hubbard model (2.1) or can be considered as an independent parameter in the case of the p − d model. In the model (2.2) two main features of a doped hole motion in copper-oxides are properly taken into account: a constraint on no double occupancy for the holes on lattice sites due to strong electron correlations and an interaction of holes with AFM spin fluctuations which result in strong renormalization of the quasiparticle spectrum (for a review, see [14]). To take into account on a rigorous basis the exclusion of doubly occupied states in electronic hopping, we employ the Hubbard operator (HO) technique. The HOs are defined as Xαβ i = \|i, α〉〈i, β\| (2.3) for the three possible states at the lattice site i \|i, α〉 = \|i, 0〉, \|i, σ〉, (2.4) for an empty site and for a singly occupied site by an electron with spin σ/2 (σ = ±1). In the t − J model only singly occupied sites are retained and the completeness relation for the HOs reads: X00 i + ∑ σ Xσσ i = 1. (2.5) The spin and density operators in equation (2.2) are expressed by HOs as Sσ i = Xσσ̄ i , Sz i = 1 2 ∑ σ σXσσ i , ni = ∑ σ Xσσ i , (2.6) 907 N.M.Plakida where σ̄ = −σ. The HOs obey the following multiplication rules Xαβ i Xγδ i = δβγX αδ i (2.7) and commutation relations [ Xαβ i , Xγδ j ] ± = δij ( δβγX αδ i ± δδαX γβ i ) . (2.8) In equation (2.8) the upper sign stands for the case when both HOs are Fermi-like ones (as, e.g., X0σ i ). The spin and density operators (2.6) are Bose-like and for them the lower sign in equation (2.8) should be taken. The Hamiltonian of the t− J model (2.2) in terms of HOs reads 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 ) , (2.9) where the exchange interaction is written in a more general form with the exchange energy Jij for the lattice sites (i, j). The unconventional commutation relations (2.8) hamper the treatment of the model within the standard diagrammatic technique. To overcome this problem we will use the equation of motion method for the two-time Green functions [2] in terms of the HOs (2.3) which rigorously preserve the constraint of no-double occupancy. 3. Dyson equation for the matrix Green function To discuss superconducting pairing within the model (2.9) we consider the matrix Green function (GF) Ĝij,σ(t− t′) = 〈〈Ψiσ(t)\|Ψ + jσ(t ′)〉〉 (3.1) in terms of the Nambu operators: Ψiσ = ( X0σ i X σ̄0 i ) , Ψ+ iσ = ( Xσ0 i X0σ̄ i ) . (3.2) Here Zubarev’s notations for the anticommutator Green function (3.1) are used [2]. By differentiating the GF (3.1) over the time t we get the following equation for the Fourier component: ωĜijσ(ω) = δijQ̂σ + 〈〈Ẑiσ \| Ψjσ〉〉ω, (3.3) where Ẑiσ = [Ψiσ, H] and Q̂σ = ( Qσ 0 0 Qσ̄ ) , (3.4) with the matrix elements Qσ = 〈X00 i +Xσσ i 〉. Here and in what follows we consider a spin-singlet state for which the correlation functions do not depend on the spin σ. 908 Superconductivity in the t− J model In that case, by using equation (2.5) we get Qσ = Q = 1− n/2 where the average number of electrons is given by the equation n = 〈ni〉 = ∑ σ 〈Xσσ i 〉 . (3.5) Now, we project the many–particle GF in (3.3) on the one–hole one by intro- ducing an irreducible (irr) part of the Ẑiσ operator 〈〈Ẑiσ \| Ψ+ jσ〉〉 = ∑ l Êilσ〈〈Ψlσ \| Ψ+ jσ〉〉+ 〈〈Ẑ (irr) iσ \| Ψ+ jσ〉〉 . (3.6) The projection is defined by the condition 〈{Ẑ (irr) iσ ,Ψ+ jσ}〉 = 0, (3.7) that results in the equation for the frequency matrix Êijσ = 〈{[Ψiσ, H],Ψ+ jσ}〉 Q̂ −1 σ . (3.8) Here {A,B} and [A,B] are the anticommutator and the commutator for the A,B operators, respectively. To calculate the matrix (3.8) we use the equation of motion for the HOs as, e.g., ( i d dt + µ ) X0σ i = ∑ l tilBiσσ′X0σ′ l + ∑ l Jil(Blσσ′ − δσσ′)X0σ′ i , (3.9) where we introduced the operator Biσσ′ = (X00 i +Xσσ i )δσ′σ +X σ̄σ i δσ′σ̄ . (3.10) The Bose-like operator (3.10) describes electron scattering on spin and charge fluctuations caused by the nonfermionic commutation relations for the HOs (the first term in (3.9) – the so-called kinematical interaction) and by the exchange spin- spin interaction (the second term in (3.9)). It can be demonstrated explicitly by using the completeness relation (2.5) that results in the following representation: X00 i +Xσσ i +X σ̄σ i = 1− 1 2 ∑ σ Xσσ i + 1 2 (Xσσ i −X σ̄σ̄ i ) +X σ̄σ i = 1− 1 2 ni + σSz i + Sσ̄ i . By performing commutations in (3.8), we get for the normal and the anomalous parts of the frequency matrix: E11 ijσ = δij ∑ l {til〈X σ0 i X0σ l 〉/Qσ + Jil(Qσ − 1 + χcs il /Qσ)} −tij(Qσ + χcs ij/Qσ)− Jij〈X σ0 j X0σ i 〉/Qσ, (3.11) E12 ijσ = δij ∑ l til〈X 0σ̄ i X0σ l +X0σ̄ l X0σ i 〉/Qσ −Jij〈X 0σ̄ i X0σ j +X0σ̄ j X0σ i 〉/Qσ . (3.12) 909 N.M.Plakida Here we introduce the charge- and spin-fluctuation correlation functions χcs ij = 1 4 〈δniδnj〉+ 〈SiSj〉, (3.13) with δni = ni − 〈ni〉. Now we introduce the zero–order GF in the generalized mean–field approxi- mation by neglecting the finite lifetime effects described by the operator Ẑ (irr) iσ in equation (3.6) Ĝ0 ijσ(ω) = {ωτ̂0δij − Êijσ} −1Q̂σ, (3.14) where τ̂0 is the unity matrix. By writing the equation of motion for the irreducible part of the GF in (3.6) with respect to the second time t′ for the right–hand side operator Ψ+ jσ(t ′) and performing the same projection procedure as in (3.6), we get 〈〈Ẑ (irr) iσ \| Ψ+ jσ〉〉ω = ∑ l 〈〈Ẑ (irr) iσ \| (Z (irr) lσ )+〉〉ω Q̂−1 σ Ĝ0 ljσ(ω) . (3.15) By using (3.3), (3.6) and (3.15), we can obtain the Dyson equation for the GF (3.1) in the form: Ĝijσ(ω) = Ĝ0 ijσ(ω) + ∑ kl Ĝ0 ikσ(ω) Σ̂klσ(ω) Ĝljσ(ω), (3.16) where the self–energy operator Σ̂klσ(ω) is defined by the equation T̂ijσ(ω) = Σ̂ijσ(ω) + ∑ kl Σ̂ikσ(ω) Ĝ 0 klσ(ω) T̂ljσ(ω) . (3.17) Here the scattering matrix is given by the equation T̂ijσ(ω) = Q̂σ −1 〈〈Ẑ (irr) iσ \| Ẑ (irr)+ jσ 〉〉ω Q̂σ −1 . (3.18) From equation (3.17) it follows that the self-energy operator is given by the ir- reducible part of the scattering matrix (3.18) which has no single zero-order GF (3.14) lines: Σ̂ijσ(ω) = Q̂σ −1 〈〈Ẑ (irr) iσ \| Ẑ (irr)+ jσ 〉〉(irr)ω Q̂σ −1 . (3.19) Equations (3.14), (3.16) and (3.19) give an exact representation for the one– hole GF (3.1). To calculate it, however, one has to apply some approximations for the many–particle GF in the self-energy matrix (3.19) which describes inelastic scattering of electrons on spin and charge fluctuations. 4. Self-consistent Eliashberg equations To solve the Dyson equation (3.16) we introduce the k-representation for the GF Gαβ σ (k, ω) = ∑ j Gαβ ojσ(ω) e −ikj . (4.1) 910 Superconductivity in the t− J model For the zero-order GF (3.14) we get: Ĝ(0) σ (k, ω)−1 = {ωτ̂0 − (Eσ k − µ̃)τ̂3 −∆σ k τ̂1}Q̂ −1 σ , (4.2) where τ̂0, τ̂1, τ̂3 are the Pauli matrix. The energy of the quasiparticles Eσ k , the renormalized chemical potential µ̃ = µ− δµ and the gap function ∆σ k in the MFA, equations (3.11), (3.12), are given by Eσ k = −ǫ(k)Qσ − ǫs(k)/Qσ − 4J N ∑ q γ(k − q)Nqσ, (4.3) where ǫ(k) = t(k) = 4tγ(k) + 4t′γ′(k), ǫs(k) = 4tγ(k)χ1s + 4t′γ′(k)χ2s, γ(k) = (1/2)(cos axqx + cos ayqy), γ′(k) = cos axqx cos ayqy, with δµ = 1 N ∑ q ǫ(q)Nqσ − 4J(n/2− χ1s/Qσ) , (4.4) ∆σ k = 2 NQσ ∑ q J(k − q)〈X0σ̄ −qX 0σ q 〉. (4.5) The average number of electrons (3.5) in the k-representation is written in the form: n = 1 N ∑ k,σ 〈Xσ0 k X0σ k 〉 = 1 N ∑ k,σ QσNkσ , (4.6) which defines the function Nqσ in equations (4.3), (4.4). In the calculation of the normal part of the frequency matrix (4.3) we neglected the charge fluctuation (the first term in equation (3.13)) and introduced the spin correlation functions for the nearest (χ1s) and the next-nearest (χ2s) neighbour lattice sites χ1s = 〈SiSi+a1〉 , χ2s = 〈SiSi+a2〉, (4.7) where a1 = (±ax,±ay) is the nearest and a2 = ±(ax ± ay) — the next-nearest neighbour lattice sites. In the gap equation (4.5) we omitted the k-independent part caused by the kinematical interaction (the first term in equation (3.12)) since it gives no contribution to d-wave pairing [6]. To calculate the self–energy operator Σ̂(k, ω) we employ a noncrossing approx- imation (or the self-consistent Born approximation) for the irreducible part of the many–particle Green functions in (3.19). In this approximation vertex corrections are neglected as in the Migdal-Eliashberg approximation and it is given by the two-time decoupling for the correlation functions in (3.19) as, e.g., given below: 〈Xσ′0 j′ B+ jσσ′X 0σ′ i′ (t)Biσσ′(t)〉 ≃ 〈Xσ′0 j′ X0σ′ i′ (t)〉〈B+ jσσ′Biσσ′(t)〉 . (4.8) 911 N.M.Plakida The proposed decoupling does not violate equal time correlations since in equation (4.8) j 6= j′ and i 6= i′ . Using a spectral representation for the GF we obtain the following result for the self-energy in the noncrossing approximation: Σσ 11(k, ω) = −Σσ̄ 22(−k,−ω) = 1 N ∑ q ∫ +∞ ∫ −∞ dzdΩN(ω, z,Ω)λ11(q, k − q \| Ω)Aσ 11(q, z), (4.9) Σσ 12(k, ω) = (Σσ 21(k, ω)) ∗ = − 1 N ∑ q ∫ +∞ ∫ −∞ dzdΩN(ω, z,Ω)λ12(q, k − q \| Ω)Aσ 12(q, z), (4.10) where N(ω, z,Ω) = 1 2 tanh(z/2T ) + coth(Ω/2T ) ω − z − Ω . (4.11) Here we introduce the spectral density: Aσ 11(q, z) = − 1 Qσπ Im 〈〈X0σ q \| Xσ0 q 〉〉z+iδ = Aσ̄ 22(q,−z), (4.12) Aσ 12(q, z) = − 1 Qσπ Im 〈〈X0σ q \| X0σ̄ −q〉〉z+iδ = Aσ 21(q, z), (4.13) and the electron – electron interaction functions caused by spin-charge fluctuations λ11(q, k − q \| Ω) = g2(q, k − q)D+(k − q,Ω), (4.14) λ12(q, k − q \| Ω) = g2(q, k − q)D−(k − q,Ω), (4.15) where g(q, k − q) = t(q) − J(k − q) and the spectral density for the spin-charge fluctuations is defined by the commutator Green functions D±(q,Ω) = − 1 π Im { 1 4 〈〈nq \| n + q 〉〉Ω+iδ ± 〈〈Sq \| S−q〉〉Ω+iδ } . (4.16) The solution of the Dyson equation ( 3.16) can be written in the Eliashberg nota- tions as Ĝσ(k, ω) = QσG̃ σ(k, ω) = Qσ ωZσ k (ω)τ̂0 + (Eσ k + ξσk (ω)− µ̃)τ̂3 + Φσ k(ω)τ̂1 (ωZσ k (ω)) 2 − (Eσ k + ξσk (ω)− µ̃)2− \| Φσ k(ω) \| 2 , (4.17) where ω(1− Zσ k (ω)) = 1 2 [Σσ 11(k, ω) + Σσ 22(k, ω)] , ξσk (ω)) = 1 2 [Σσ 11(k, ω)− Σσ 22(k, ω)] , (4.18) Φσ k(ω) = ∆σ k +Σσ 12(k, ω) . 912 Superconductivity in the t− J model For the numerical solution of the system of equations (4.9)–(4.18) it is useful to introduce an imaginary frequency representation for the Green function (4.17) with ω = iωn = iπT (2n + 1) and the spin-charge Green functions (4.16) with Ω = iωn = iπT2n where n = 0,±1,±2, ... . By using the representation for the function (4.11) N(iωn, z,Ω) = −T ∑ m 1 iωm − z 1 i(ωn − ωm)− Ω (4.19) after integration in equations (4.9), (4.10) we get: Σσ 11(k, iωn) = − T N ∑ q ∑ m G̃σ 11(q, iωm)λ11(q, k − q \| iωn − iωm), (4.20) Σσ 12(k, iωn) = T N ∑ q ∑ m G̃σ 12(q, iωm)λ12(q, k − q \| iωn − iωm). (4.21) The interaction functions are given by λ11(q, k − q \| iων) = g2(q, k − q)D+(q, iων) , (4.22) λ12(q, k − q \| iων) = g2(q, k − q)D−(q, iων). (4.23) Here we have Gσ 11(k, iωm) = −Gσ̄ 22(k,−iωm), G σ 12(k, iωm) = Gσ 21(k,−iωm). A linearized system of the Eliashberg equations (4.18) for T 6 Tc has the following form: G̃σ 11(k, iωn) = 1 iωn − Ek + µ̃− Σσ 11(k, iωn) , (4.24) Φσ(k, iωn) = ∆σ k + φσ(k, iωn) = T N ∑ q ∑ m {2J(k − q) (4.25) −λ12(q, k − q \| iωn − iωm)}G̃ σ 11(q, iωm)G̃ σ̄ 11(q,−iωm)Φ σ(q, iωm). At first the system of equations for the normal GF (4.24), (4.20) should be solved for a given concentration of electrons n 1− n/2 = 1 + 2T N ∑ k ∞ ∑ n=−∞ G̃11(k, iωn) . (4.26) Then, the eigenvalues and eigenfunctions of the linear equation for the gap func- tion (4.26) should be calculated to obtain the superconducting transition temper- ature Tc and the (q, ω) dependence gap function. For numerical calculations one has to introduce a model for the charge–spin– fluctuation functions in (4.16). By taking into account only the spin-fluctuation contribution, we can write them in the form: D− s (q, iων) = −D+ s (q, iων) = χs(q) +∞ ∫ 0 2zdz z2 + ω2 ν χ ′′ s (z), (4.27) 913 N.M.Plakida where we have introduced for the spin-fluctuation susceptibility a model represen- tation (see, e.g., [11,18]) χ ′′ s (q, ω) = − 1 π Im 〈〈Sq \| S−q〉〉ω+iδ = χs(q) χ ′′ s (ω) = χ0 1 + ξ2(q−QAF) 2 tanh ω 2T 1 1 + (ω/ωs)2 (4.28) with the characteristic AFM correlation length ξ and the spin-fluctuation energy ωs ≃ J . To fix the constant χ0 in (4.28) we can use for the spin-fluctuation sus- ceptibility the following normalizing condition: 1 N ∑ i 〈SiSi〉 = 1 N ∑ q χs(q) +∞ ∫ −∞ dz exp (z/T )− 1 χ ′′ s (z) = 3 4 n. (4.29) For the low temperature, T ≪ ωs, we can integrate over frequency in equation (4.29): +∞ ∫ −∞ dz exp (z/T )− 1 χ ′′ s (z) ≃ +∞ ∫ 0 dzχ ′′ s (z) ≃ +∞ ∫ 0 dz 1 + (z/ωs)2 = π 2 ωs. Therefore, for the constant χ0 we get from (4.29) χ0 = 3n 2πωsC1 , C1 = 1 N ∑ q 1 1 + ξ2q2 ≃ π ξ2 ln(1 + ξ2/π). (4.30) Here ξ and q are dimensionless quantities (ξ/a and qa where a is the lattice con- stant). In the approximation (4.27), we get for the interaction functions (4.22), (4.23): −λ11(q, k − q \| iων) = λ12(q, k − q \| iων) = g2(q, k − q)χs(k − q)Fs(iων), (4.31) where Fs(iων) = +∞ ∫ 0 2zdz z2 + ω2 ν 1 1 + (z/ωs)2 tanh z 2T . (4.32) By using the model (4.28), we get for the static spin correlation functions (4.7): χ1s = 〈SiSi+a1〉 = 1 N ∑ q γ(q)〈SqS−q〉, χ2s = 〈SiSi+a2〉 = 1 N ∑ q γ′(q)〈SqS−q〉, (4.33) where 〈SqS−q〉 = χs(q) +∞ ∫ −∞ dzχ ′′ s (z) exp (z/T )− 1 ≃ χs(q) π 2 ωs ≃ 3n 4C1 1 1 + ξ2(q−Q)2AF . Therefore, we have obtained a closed system of equations which should be solved numerically as it was recently done for the spin-polaron model in [8]. 914 Superconductivity in the t− J model 5. Conclusions In the present paper the theory of the quasiparticle spectrum and supercon- ducting pairing in the t − t′ − J model (2.2) in a paramagnetic state is formu- lated. By employing the equation of motion method for the two-time GF [2] and differentiating it over two times, t and t′, we easily obtained a self-consistent sys- tem of equations for the matrix GF (4.17) and self-energy (4.9), (4.10) in the noncrossing approximation, equation (4.8). The latter is equivalent to the Migdal- Eliashberg approximation which neglects vertex corrections. Though in the t−t′−J model (2.2) we have no small parameter as in the electron-phonon model consid- ered by Eliashberg [3], the vertex renormalization in the former may be not so important as it was proved for the spin-polaron representation of the t − t′ − J model (2.2) (see, e.g., [19]). All the calculations are performed in the real time representation, though the imaginary frequency representation which is more con- venient for a numerical study is given for the linearized system of Eliashberg equations (4.24), (4.26). However, the theory is not fully self-consistent in that respect that a phe- nomenological model for effective electron-electron coupling due to spin fluctua- tions is to be proposed, equation (4.27), to enable a numerical study of temperature and doping dependence of the quasiparticle spectrum and superconducting pairing in the model (2.2). The results of numerical calculations and the comparison with other approaches will be presented in a separate paper. Acknowledgements Partial financial support by the INTAS–RFBR Program, Grant No 95–591, is acknowledged. References 1. Zubarev D.N. On the theory of superconductivity. // Doklady AN SSSR, 1960, vol. 132, No. 5, p. 1055–1058 (in Russian). 2. Zubarev D.N. Two-time Green functions in the statistical physics. // Uspekhi Fiz. Nauk, 1960, vol. 71, No. 1, p. 71–116 (in Russian). 3. Eliashberg G.M. Interaction of electrons with lattice vibrations in a superconductor. // J. Exp. Teor. Fiz., 1960, vol. 38, No. 3, p. 966–976 (in Russian); Temperature Green’s function for electrons in a superconductor. // ibid., 1960, vol. 39, No. 5, p. 1437–1441 (in Russian). 4. Vujičić G.M., Petru Z.K., Plakida N.M. On the equations of superconductivity for electron-ion model of metals. // Teor. Mat. Fiz., 1981, vol. 46, No. 1, p. 91–98 (in Russian). 5. Zaitsev R.O., Ivanov V.A. On the possibility of pair condensation in the Hubbard model. // Fiz. Tverd. Tela, 1987, vol. 29, No. 8, p. 2554–2556 (in Russian); Supercon- ductivity in the Hubbard model. // ibid., No. 10, p. 3111–3119 (in Russian). 915 N.M.Plakida 6. Plakida N.M., Yushankhai V.Yu., Stasyuk I.V. On the role of kinematical and ex- change interactions in superconducting pairing of electrons in the Hubbard model. // Physica C, 1989, vol. 160, p. 80–88. 7. Yushankhai V.Yu., Plakida N.M., Kalinay P. Superconducting pairing in the mean- field approximation for the t − J model: Numerical analysis. // Physica C, 1991, vol. 174, p. 401–406. 8. Plakida N.M., Oudovenko V.S., Horsch P., Liechtenstein A.I. Superconducting pairing of spin polarons in the t−J model. // Phys. Rev. B, 1997, vol. 55, No. 18, p. R11997– R12000. 9. Izyumov Yu.A., Letfulov B.M. Spin fluctuations and superconductivity in the Hubbard model with a strong Coulomb repulsion. // J. Phys.: Cond. Matt., 1991, vol. 3, p. 5373– 5391. 10. Plakida N.M., Hayn R. Superconducting pairing in the singlet band of the Emery model. // Zeit. Physik B, 1994, vol. 93, p. 313–319. 11. Prelovśek P. Electron Green’s function in the planar t− J model. // Zeit. Physik B, 1997, vol. 103, p. 363–368. 12. Plakida N.M. High-temperature Superconductivity. Heidelberg, Springer, 1995. 13. Plakida N.M. Spin fluctuation superconducting pairing in copper oxides. // Philo- sophical Magazine B, 1997, vol. 76, No. 5, p. 771–795. 14. Izyumov Yu.A. The t− J model for strongly correlated electrons and high-Tc super- conductors. // Uspekhi Fiz. Nauk, 1997, vol. 167, No. 5, p. 465–497 (in Russian). 15. Anderson P.W. The resonating valence bond state in La2CuO4 and superconductiv- ity. // Science, 1987, vol. 235, p. 1196–1198. 16. Emery V. Theory of high-Tc superconductivity in oxides. // Phys. Rev. Lett., 1987, vol. 58, No. 26, p. 2794–2797. 17. Zhang F.C., Rice T.M. Effective Hamiltonian for the superconductiviting Cu oxides. // Phys. Rev. B, 1988, vol. 37, No. 7, p. 3759–3761. 18. Jaklič J., Prelovśek P. Anomalous spin dynamics in doped quantum antiferromag- nets. // 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. 19. Liu Z., Manousakis E. Dynamical properties of a hole in a Heisenberg antiferromag- net. // Phys. Rev. B, 1992, vol. 45, No. 5, p. 2425–2437. 916 Superconductivity in the t− J model Підхід функцій Ґріна в теорії надпровідності для t − J моделі М.М.Плакіда Об’єднаний інститут ядерних досліджень, 141980 Дубна, Росія Отримано 28 січня 1998 р. Pозвинуто теоpiю надпpовiдного спаpювання, що вiдбувається зав- дяки обмiнним та кiнематичним взаємодiям у t− J моделi в паpа- магнiтному станi. Отpимано piвняння Дайсона для матpицi функцiй Ґpiна чеpез опеpатоpи Хабаpда у непеpехpесному наближеннi. За- пpопоновано лiнеаpизовану самоузгоджену систему piвнянь Елiаш- беpга для вивчення темпеpатуpної та концентpацiйної залежностi квазiчастинкового спектpу дipок у ноpмальному станi, а також для знаходження темпеpатуpи надпpовiдного фазового пеpеходу i си- метpiї щiлини. Ключові слова: надпpовiднiсть, квазiчастинка, piвняння Дайсона, функцiї Ґpiна, опеpатоpи Хабаpда PACS: 74.20.-z, 74.20.Mn, 74.72.-h 917 918

Green function approach to the theory of superconductivity in the t - J model

Institution

Ähnliche Einträge