Superconductivity in the t-J model

A comparison of microscopic theories for superconductivity in the limit of strong electron correlations is presented. We consider the results for the two-dimensional t-J model obtained within a projection technique for the Green functions in terms of the Hubbard operators and a slave-fermion rep...

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Дата:	2002
Автор:	Plakida, N.M.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2002
Назва видання:	Condensed Matter Physics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/120682
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Цитувати:	Superconductivity in the t-J model / N.M. Plakida // Condensed Matter Physics. — 2002. — Т. 5, № 4(32). — С. 707-727. — Бібліогр.: 49 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1206822017-06-13T03:06:26Z Superconductivity in the t-J model Plakida, N.M. A comparison of microscopic theories for superconductivity in the limit of strong electron correlations is presented. We consider the results for the two-dimensional t-J model obtained within a projection technique for the Green functions in terms of the Hubbard operators and a slave-fermion representation for the RVB state. It is argued that the latter approach resulting in an odd-symmetry p-wave pairing for fermions is inadequate. Представлено порівняння мікроскопічних теорій для надпровідності в границі сильних електронних кореляцій. Ми розглядаємо результати для двовимірної t-J моделі, отриманої в рамках методу проектування для функцій Гріна на мові операторів Габбарда, а також представленні допоміжних ферміонів для стану резонуючих валентних зв’язків. Стверджується, що останнє наближення, яке приводить до непарної симетрії p-хвильового спарювання для ферміонів є непридатним. 2002 Article Superconductivity in the t-J model / N.M. Plakida // Condensed Matter Physics. — 2002. — Т. 5, № 4(32). — С. 707-727. — Бібліогр.: 49 назв. — англ. 1607-324X PACS: 74.20.-z, 74.20.Mn, 74.72.-h DOI:10.5488/CMP.5.4.707 http://dspace.nbuv.gov.ua/handle/123456789/120682 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	A comparison of microscopic theories for superconductivity in the limit of strong electron correlations is presented. We consider the results for the two-dimensional t-J model obtained within a projection technique for the Green functions in terms of the Hubbard operators and a slave-fermion representation for the RVB state. It is argued that the latter approach resulting in an odd-symmetry p-wave pairing for fermions is inadequate.
format	Article
author	Plakida, N.M.
spellingShingle	Plakida, N.M. Superconductivity in the t-J model Condensed Matter Physics
author_facet	Plakida, N.M.
author_sort	Plakida, N.M.
title	Superconductivity in the t-J model
title_short	Superconductivity in the t-J model
title_full	Superconductivity in the t-J model
title_fullStr	Superconductivity in the t-J model
title_full_unstemmed	Superconductivity in the t-J model
title_sort	superconductivity in the t-j model
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2002
url	http://dspace.nbuv.gov.ua/handle/123456789/120682
citation_txt	Superconductivity in the t-J model / N.M. Plakida // Condensed Matter Physics. — 2002. — Т. 5, № 4(32). — С. 707-727. — Бібліогр.: 49 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT plakidanm superconductivityinthetjmodel
first_indexed	2025-07-08T18:23:31Z
last_indexed	2025-07-08T18:23:31Z
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fulltext	Condensed Matter Physics, 2002, Vol. 5, No. 4(32), pp. 707–727 Superconductivity in the t-J model N.M.Plakida Joint Institute for Nuclear Research, 141980 Dubna, Russia Received September 2, 2002 A comparison of microscopic theories for superconductivity in the limit of strong electron correlations is presented. We consider the results for the two-dimensional t-J model obtained within a projection technique for the Green functions in terms of the Hubbard operators and a slave-fermion representation for the RVB state. It is argued that the latter approach re- sulting in an odd-symmetry p-wave pairing for fermions is inadequate. Key words: strong electron correlations, high-temperature superconductors, Green functions, slave fermions PACS: 74.20.-z, 74.20.Mn, 74.72.-h 1. Introduction A mechanism of high-temperature superconductivity in cuprates is still unre- solved since strong electron correlations in copper-oxygen planes prevent us from applying the well established methods of band structure calculations developed for conventional metals. An important role of electron correlations in cuprates was ini- tially stressed by Anderson [1] who suggested to consider them within the framework of the Hubbard model or the so-called t-J model which follows from the Hubbard model in the limit of strong correlations. To study these models, a lot of numer- ical work have been done [2,3] though the obtained results are still controversial. For instance, a robust d-wave pairing was observed for the t-J model [4], while a long-range order was not found in the original Hubbard model [5]. In analytical approaches, mostly a mean-field theory (MFA) was applied in the studies of the Hubbard or the t-J models. The resonating valence bond (RVB) state in the t-J model was proposed by Baskaran et al. [6] where superconductivity was obtained as a result of spin correlations induced by the superexchange interaction. Similar results were found by Cyrot [7] for a superconducting pairing mediated by superexchange interaction. To overcome the problem of strong correlations in the t-J model and nonfermionic commutation relations for the physical electron operators, a number of auxiliary field representations were proposed (see, e.g., [8–32]). However, in these methods, a spin-charge separation is usually assumed for spinon and holon fields which violates rigorous restrictions imposed by nonfermionic commutation c© N.M.Plakida 707 N.M.Plakida relations as the “no double occupancy” constraint which may result in unphysical conclusions. A reliable analytical approach to deal with strong correlations in the Hubbard or the t-J model is based on the Green function methods in terms of the Hubbard operators which rigorously preserve the non-fermionic commutation relations [33]. Here, we may mention our results based on the Mori-type projection technique for the Green functions [34–36] and the diagram technique calculations by Izyumov et al. [37,38]. These approaches enable one to go beyond MFA by taking into account self-energy corrections. For instance, a numerical solution of the Dyson equation in [36] revealed a non-Fermi-liquid behavior in the normal state at low doping and the d-wave superconductivity mediated by the exchange and spin-fluctuation pairing. In the recent paper [39] a microscopical theory of superconductivity in CuO2 layer within the effective two-band p-d Hubbard model in the strong correlation limit was developed. It has been proved that the MFA for the Hubbard model results in the antiferromagnetic exchange d-wave pairing which is equivalent to the pairing observed in the t-J model in MFA. In the present paper we compare the results for the t-J model obtained within the Green function method in terms of the Hubbard operators [34–36] and by applying the slave-fermion hard-boson representation [27] for the RVB state. It will be shown that the latter approach results in an odd-symmetry p-wave pairing for the spinless fermions as in the path-integral representation in [32] which contradicts the known numerical and analytical calculations. It casts doubts on the validity of spin-charge separation approach in studying the superconducting pairing in the t-J model. In the next section we briefly present the results of the projection technique for the Green functions [36] for the t-J model. In section 3 superconducting pairing within the slave-fermion representation for the Hubbard operators is considered. Results and discussions are given in section 4. Concluding remarks are in section 5. 2. Green function method 2.1. Dyson equation for the t-J model In the present section we consider the superconducting pairing in the t-J model by applying the Green function technique [34–36]. The t-J model in the standard notation [1,42] reads: Ht−J = −t ∑ i6=j,σ ã+ iσãjσ + J ∑ 〈ij〉 (SiSj − 1 4 ninj), (1) where ã+ iσ = a+ iσ(1 − ni−σ) are the projected operators for physical electrons, ni = ∑ σ ã+ iσãiσ is the number operator and Sα i = (1/2) ∑ s,s′ ã + isσ α s,s′ ãis′ are spin-1/2 op- erators. Here, t is the effective transfer integral and J is antiferromagnetic exchange energy for a pair of the nearest neighbor spins, 〈ij〉, i > j. To take into account the projected character of electron operators on a rigorous basis we employ the Hubbard operator technique. The Hubbard operators (HO) are 708 Superconductivity in the t-J model defined as Xαβ i = \|i, α〉〈i, β\| for three possible states \|i, α〉 at a lattice site i: for an empty site \|i, 0〉 and for a singly occupied site \|i, σ〉 by an electron with the spin σ/2 (σ = ±1, σ̄ = −σ). They obey the completeness relation X00 i + ∑ σ Xσσ i = 1, (2) which rigorously preserves the constraint of no double occupancy. The spin and density operators in equation (1) are expressed by HO as Sσ i = Xσσ̄ i , Sz i = 1 2 ∑ σ σXσσ i , ni = ∑ σ Xσσ i . (3) The HO obey the following multiplication rule Xαβ i Xγδ i = δβγX αδ i and commutation relations [ Xαβ i , Xγδ j ] ± = δij ( δβγX αδ i ± δδαXγβ i ) , (4) where the upper sign stands for the Fermi-like HO (as, e. g., X0σ i ) and the lower sign stands for the Bose-like operators (as the spin and number operators in equation (3)). By using the Hubbard operator representation, we write the Hamiltonian of the t-J model (1) in a more general form: Ht−J = − ∑ i6=j,σ tijX σ0 i X0σ j − µ ∑ iσ Xσσ i + 1 4 ∑ i6=j,σ Jij ( Xσσ̄ i X σ̄σ j − Xσσ i X σ̄σ̄ j ) . (5) The electron hopping energy for the nearest neighbors, tij = t, and the second neighbors, tij = t′, on a 2D square lattice, and the exchange interaction Jij = J for the nearest neighbors1 can be considered as independent parameters if, starting from a more realistic for copper oxides three-band p-d model [43], we reduce it to the t-J model [42]. In that case the parameters t, t′ and J can be evaluated in terms of the original parameters of the p-d model (see, e.g., [44,45]). We also introduced the chemical potential µ which can be calculated from the equation for the average number of electrons n = 〈ni〉 = ∑ σ 〈Xσσ i 〉. (6) To discuss the superconducting pairing within the model (5), we introduce the Nambu notation for HO: Ψiσ = ( X0σ i X σ̄0 i ) , Ψ+ iσ = ( Xσ0 i X0σ̄ i ) , and consider the matrix Green function (GF) Ĝij,σ(t − t′) = 〈〈Ψiσ(t)\|Ψ+ jσ(t ′)〉〉 = ( G11 ijσ G12 ijσ G21 ijσ G22 ijσ ) , (7) 1Sometimes a coefficient (1/2) instead of (1/4) is written in the last sum in equation (5) that results in two times larger values of contributions given by the exchange energy J (see, e.g., [34]). 709 N.M.Plakida where Zubarev’s notation for the anticommutator GF is used [46]. By differentiating the GF (7) over the time t we get for the Fourier component the following equation ωĜijσ(ω) = δijQ̂σ + 〈〈Ẑiσ\|Ψjσ〉〉ω , (8) where Ẑiσ = [Ψiσ, H], Q̂σ = ( Qσ 0 0 Qσ̄ ) , Qσ = 〈X00 i + Xσσ i 〉. Since we consider a spin-singlet state, the correlation function Qσ = Q = 1 − n/2 depends only on the average number of electrons (6). Now, we project the many-particle GF in (8) on the original single-electron GF 〈〈Ẑiσ\|Ψ+ jσ〉〉 = ∑ l Êilσ〈〈Ψlσ\|Ψ+ jσ〉〉 + 〈〈Ẑ(irr) iσ \|Ψ+ jσ〉〉, (9) where the irreducible (irr) part of the many-particle operator Ẑiσ is defined by the equation 〈{Ẑ(irr) iσ , Ψ+ jσ}〉 = 〈{Ẑiσ − ∑ l ÊilσΨlσ, Ψ+ jσ}〉 = 0, which results in the definition of the frequency matrix Êijσ = 〈{[Ψiσ, H], Ψ+ jσ}〉 Q−1. (10) Now, we can introduce the zero-order GF in the generalized MFA which is given by the frequency matrix (10) Ĝ0 ijσ(ω) = Q{ωτ̂0δij − Êijσ}−1. (11) To derive the Dyson equation for the single-electron GF (7) we write down an equation of motion for the irreducible part of the GF in (9) with respect to the second time t′ for the right-hand side operator Ψ+ jσ(t ′). Then, performing the same projection procedure as in equation (9) we obtain the Dyson equation for the GF in the form Ĝijσ(ω) = Ĝ0 ijσ(ω) + ∑ kl Ĝ0 ikσ(ω) Σ̂klσ(ω) Ĝljσ(ω), (12) where the self-energy operator Σ̂klσ(ω) is defined by the equation T̂ijσ(ω) = Σ̂ijσ(ω) + ∑ kl Σ̂ikσ(ω) Ĝ0 klσ(ω) T̂ljσ(ω). (13) Here, the scattering matrix is given by T̂ijσ(ω) = Q−1〈〈Ẑ(irr) iσ \|Ẑ(irr)+ jσ 〉〉ω Q−1. (14) 710 Superconductivity in the t-J model From equation (13) it follows that the self-energy operator is given by the irreducible part of the scattering matrix (14) that has no parts connected by the single zero- order GF (11): Σ̂ijσ(ω) = Q−1〈〈Ẑ(irr) iσ \|Ẑ(irr)+ jσ 〉〉(irr)ω Q−1. (15) Equations (11), (12) and (15) give an exact representation for the single-electron GF (7). However, to solve the self-consistent system of equations, one has to introduce an approximation for the many-particle GF in the self-energy matrix (15) which describes inelastic scattering of electrons on spin and charge fluctuations. 2.2. Self-consistent equations Here, we derive a self-consistent system of equations in MFA. To calculate the frequency matrix (10) we use the equation of motion for the HO: ( i d dt + µ ) X0σ i = − ∑ l,σ′ tilBiσσ′X0σ′ l + 1 2 ∑ l,σ′ Jil(Blσσ′ − δσσ′)X0σ′ i , (16) where we introduced the operator Biσσ′ = (X00 i + Xσσ i )δσ′σ + X σ̄σ i δσ′σ̄ = (1 − 1 2 ni + σSz i )δσ′σ + S σ̄ i δσ′σ̄ . (17) The Bose-like operator (17) describes electron scattering on spin and charge fluc- tuations caused by the kinematic interaction (the first term in (16)) and by the exchange spin-spin interaction (the second term in (16)). By performing commutations in (10) we get the following for the normal and for the anomalous parts of the frequency matrix: E11 ijσ = −µδij + δij ∑ l {til〈Xσ0 i X0σ l 〉/Q + 1 2 Jil(Q − 1 + χcs il /Q)} − tij(Q + χcs ij /Q) − 1 2 Jij〈Xσ0 j X0σ i 〉/Q, (18) E12 ijσ = ∆ijσ = δij ∑ l til〈X0σ̄ i X0σ l − X0σ i X0σ̄ l 〉/Q − 1 2 Jij〈X0σ̄ i X0σ j − X0σ i X0σ̄ j 〉/Q. (19) Here, in the calculation of the correlation function for the normal component of the frequency matrix: ∑ σ′ 〈Biσσ′Bjσ′σ〉 = 〈(1 − 1 2 ni + σSz i )(1 − 1 2 nj + σSz j ) + Sσ i Sσ̄ j 〉 = 〈(1 − 1 2 ni)(1 − 1 2 nj)〉 + 〈SiSj〉 = Q2 + χcs ij , we introduce the charge- and spin-fluctuation correlation functions χcs ij = 1 4 〈δniδnj〉 + 〈SiSj〉, 711 N.M.Plakida with δni = ni − 〈ni〉. Further, we neglect the charge fluctuations, 〈δniδnj〉 ' 0 , but take into account spin correlations given by spin correlation functions for the nearest (χ1s) and for the next-nearest (χ2s) neighbor lattice sites: χ1s = 〈SiSi+a1 〉, χ2s = 〈SiSi+a2 〉, (20) where a1 = (±ax,±ay) and a2 = ±(ax ± ay). In a paramagnetic state assumed here they depend only on the distance between the lattice sites. In the k-representation for the GF Gαβ σ (k, ω) = ∑ j Gαβ ojσ(ω) e−ikj, we get for the zero-order GF (11): Ĝ(0) σ (k, ω) = Q {ωτ̂0 − (εk − µ̃)τ̂3 − ∆σ kτ̂1}−1 = Q ωτ̂0 + (εk − µ̃)τ̂3 + ∆σ kτ̂1 ω2 − (εk − µ̃)2 − \|∆σ k\|2 , (21) where τ̂0, τ̂1, τ̂3 are the Pauli matrices. The quasiparticle energy εk and the renor- malized chemical potential µ̃ = µ − δµ in the MFA are defined by the frequency matrix (18) εk = −t̃(k) − 2J N ∑ q γ(k − q)Nqσ, (22) δµ = 1 N ∑ q t(q)Nqσ − 2J(n/2 − χ1s/Q), (23) where J(q) = 4Jγ(q), t(k) = 4tγ(k) + 4t′γ′(k) with γ(k) = (1/2)(cos axqx + cos ayqy), γ′(k) = cos axqx cos ayqy , while the renormalized hopping integral is given by t̃(k) = 4t γ(k) Q (1 + χ1s/Q 2) + 4t′ γ′(k) Q (1 + χ2s/Q 2). (24) The average number of electrons in equations (22), (23) in the k-representation is written in the form: nk,σ = 〈Xσ0 k X0σ k 〉 = QNkσ. It should be pointed out that the renormalization of the hopping parameter (24) caused by the spin correlation functions (20) is essential at low doping when short- range antiferromagnetic correlations are strong. For instance, for the hole doping δ = 1− n ' 0.05 and Q = (1 + δ)/2 ' 0.53 the correlation functions are estimated 712 Superconductivity in the t-J model as [36]: χ1s ' −0.3, χ2s ' 0.2, which results in complete suppression of the nearest neighbors hopping, while the next-neighbor hopping is quite large: teff = t Q[1 + χ1s/Q 2] ' 0, t′eff = t′ Q[1 + χ2s/Q 2] ' 0.9t′. For a large doping, the antiferromagnetic correlations are suppressed and the nearest neighbor hopping prevails: teff/t′eff ' t/t′ � 1 . The superconducting gap ∆σ k in equation (21) is defined by the anomalous com- ponent of the frequency matrix (19): ∆σ k = 2 NQ ∑ q [ t(q) − 1 2 J(k − q ] 〈X0σ̄ −qX 0σ q 〉, (25) There are two contributions in equation (25) given by the k-independent kinematic interaction t(q) and the exchange interaction J(k − q). The kinematic interaction gives no contribution to the d-wave pairing in MFA, equation (25) (see [34]), and we disregard it in the subsequent equations.2 The anomalous correlation function in equation (25) can be easily calculated from the anomalous part of the GF (21): 〈X0σ̄ −qX 0σ q 〉 = −Q ∆σ q 2Eq tanh Eq 2T , (26) which results in the BCS-type equation for the gap function: ∆σ k = 1 N ∑ q J(k − q) ∆σ q 2Eq tanh Eq 2T , (27) where Ek = [ (εk−µ̃)2+\|∆σ k\|2 ]1/2 is the quasiparticle energy in the superconducting state. As was proved in [39], the retardation effects for the exchange interaction are negligible and therefore there is no restriction in integrating over the energy in equation (27). It means that all electrons in the conduction band participate in the superconducting pairing contrary to the BCS equation for the electron-phonon model where the energy integration and pairing are restricted to a narrow energy shell of the order of the phonon energy close to the Fermi energy. The equation (27) is identical to the results in MFA of the diagram technique [37], while the gap equation obtained in [40] has an additional factor Q = (1 − n/2) which is spurious. This factor appears if we apply a simple decoupling procedure in the equation of motion (16) for GF instead of the projection technique given by equation (10). Writing the bosonic operators in the exchange interaction as a product of two fermionic operators: X σ̄σ i = X σ̄0 i X0σ i and performing a decoupling of the fermionic operators: 〈〈X σ̄0 l X0σ l X0σ̄ i − X σ̄0 l X0σ̄ l X0σ i \|Xσ0 j 〉〉ω ' 〈X0σ l X0σ̄ i − X0σ̄ l X0σ i 〉〈〈X σ̄0 l \|Xσ0 j 〉〉ω , (28) 2Superconducting pairing mediated by the kinematic interaction was first proposed by Zaitsev and Ivanov [47]. 713 N.M.Plakida we obtain the same equation (27) for the gap function but with the additional Q- factor at the right hand side. In the decoupling we miss the normalization factor for the correlation functions in the denominator of the frequency matrix, equation (10), which cancels out with Q in the numerator of the corresponding GF (21) and there- fore there is no Q-factor in equation (27). So, a rigorous way to apply MFA with a proper account of the projected character of HO is to use the projection technique as discussed above. The self-energy contribution (15) in the second order of the kinematic interaction is considered in [36] while it is omitted in [37]. As discussed in [36,39], it mediates the spin-fluctuation pairing and results in finite life-time effects for the quasiparticle spectrum giving rise to an incoherent contribution to the single-particle density of states. Here, we shall not further discuss these self-energy effects since to compare the GF approach with the slave-fermion technique it will be sufficient to consider only MFA for the gap equation (27). 3. Slave-fermion approach 3.1. Slave-fermion representation A number of auxiliary field representations have been proposed so far (see, e.g., [8–32]). In the slave-boson method [8–10] the projected electron operator is expressed as a product of auxiliary Bose field for charge degree of freedom (holon) and Fermi field for spin degree of freedom (spinon). The main problem in this ap- proach is the so-called constraint imposed by the projected character of electronic operators in the t-J model which prohibits double occupancy of any lattice site. To treat the constraint, a site-dependent Lagrange multiplier is introduced. However, to solve the problem the MFA is usually applied and the Lagrange multiplier is taken to be independent of the lattice site, so the local constraint is replaced by a glob- al one with uncontrollable consequences. In the slave-fermion method, the charge degree of freedom is represented by spinless fermion operators, while to describe the spin degree of freedom the Bose field (Schwinger bosons [13–22] or spin opera- tors [23–27]) is used. The Schwinger boson representation though being physically meaningful for the Heisenberg model [11,12] gives poor results for the doped case: the antiferromagnetic ground state persists up to a very high doping and it does not reproduce the large Fermi surface as in the slave-boson method. In the slave fermion and in the spin operator representation, the magnetic properties of the model are described in a more reliable way (see, e.g., [25,29,31]). Below, we consider the slave-fermion hard-core boson representation proposed in [27] and later employed in [28–31] to investigate different physical properties of cuprates within the t-J model. It has some advantages since the constraint of no double occupancy can be fulfilled without introducing the Lagrange multiplier. To decouple the charge and spin degrees of freedom for physical electrons, the HO in the theory [27] is represented as a product of a spinless fermion h+ i for the charge degree of freedom (holon) and a hard-core boson biσ for the spin degree of freedom 714 Superconductivity in the t-J model (spinon): X0σ i = h+ i biσ , Xσ0 i = hi b + iσ , (29) which have the following commutation relations: h+ i hj + hjh + i = δi,j , biσb+ jσ − b+ jσbiσ = δi,j(1 − 2b+ iσbiσ). The hard-core bosons are Pauli operators which commute on different lattice sites and anticommute on the same lattice site prohibiting double occupancy. The Pauli operators can be also represented by the spin-lowering S− i and spin-raising S+ i op- erators for spin-1/2: b+ i↑ = S+ i = bi↓ , b+ i↓ = S− i = bi↑ , or b+ iσ = Sσ i , biσ = S σ̄ i . The on-site electron local constraint ∑ σ Xσ0 i X0σ i = ∑ σ Xσσ i = hih + i ∑ σ b+ iσ biσ = hih + i = 1 − h+ i hi 6 1 (30) is satisfied here since, for the Pauli operators at any lattice site, we have the equation ∑ σ b+ iσ biσ = S+ i S− i + S− i S+ i = 1, (31) and the spinless holon number n (h) i = h+ i hi can be equal to 1 or 0. However, the spin-charge separation imposed by the representation (29) results in extra degrees of freedom: a spin 1/2 is assigned to any lattice site including an empty site, while in the HO representation, equation (2), we have only 3 states: an empty state and a filled state with spin ±1/2. To cure this defect one should introduce a projection operator to exclude the unphysical states [27]3. Otherwise the commutation relations for the original HO, equation (4), and their representation, equation (29), will give different results. For instance, for the physical electrons which are described by HO we have {X0σ i , Xσ0 i } = X00 i + Xσσ i = 1 − X σ̄σ̄ i , ∑ σ 〈{X0σ i , Xσ0 i }〉 = 2 − 〈ni〉 = 1 + δ, (32) where the hole doping concentration δ = 1 − n . If we use the representation (29) then we can write the commutation relations as {X0σ i , Xσ0 i } = {h+ i biσ , hi b + iσ} = h+ i hi + (1 − 2 h+ i hi)b + iσbiσ , ∑ σ {h+ i biσ , hi b + iσ} = 1, (33) where we have used equation (31). For the average value in equation (33) we get, respectively, ∑ σ 〈{h+ i biσ , hi b + iσ}〉 = 1, (34) 3The same applies to other slave-fermion spin operator representations as in [23–26]. 715 N.M.Plakida which contradicts the rigorous result for HO, equation (32). For the average number of electrons in the representation (29) by using the definition (6) we can write n = 〈ni〉 = ∑ σ 〈Xσ0 i X0σ i 〉 = ∑ σ 〈hib + iσh+ i biσ〉 = 〈hih + i 〉 = 1 − δ, (35) which coincides with the definition (6) if we take the definition X00 i = h+ i hi for the hole number operator. However, this definition is not unique. For instance, we can write: X00 i = X0σ i Xσ0 i = X0σ̄ i X σ̄0 i , which results in the equation: X00 i = (1/2)h+ i hi if we use the representation (29) and the condition(31). Thus, the double counting of empty sites results in a controversial equation for an average number of electrons which is therefore valid only with accuracy of ±δ. 3.2. Mean-field approximation Let us consider the resonating valence bond (RVB) state in the original Hamilto- nian (5) as proposed by Baskaran et al. [6]. To this end we should write the Bose-like spin operators in the exchange energy in Ht−J (5) as a product of two single-particle Fermi-like operators: Xσσ̄ i = Xσ0 i X0σ̄ i and introduce the singlet operators b↑ij ≡ bij = 1√ 2 (X0+ i X0− j − X0− i X0+ j ). (36) Then, using MFA for the singlet operators in the exchange interaction of the t-J model, we get the RVB effective Hamiltonian: HJ = 1 2 ∑ i6=j Jij ( X+− i X−+ j − X++ i X−− j ) = −1 2 ∑ i6=j Jij b+ ijbij ' −1 2 ∑ i6=j Jij ( B+ ijbij + b+ ijBij − \|Bij\|2 ) , (37) where we introduced the RVB order parameter: B (↑) ij ≡ Bij = 〈bij〉 = 1√ 2 〈X0+ i X0− j − X0− i X0+ j 〉. (38) Here, we should point out that MFA in equation (37) employs a decoupling of the Hubbard operators on the same lattice site: X+− i X−+ j = X+0 i X0− i X−0 j X0+ j ⇒ 〈X+0 i X−0 j 〉 X0+ j X0− i , (39) which is not unique and results in an uncontrollable approximation since the Hub- bard operators obey the multiplication rule: Xαβ i Xβγ i = Xαγ i , and any intermediate state β can be used in the decoupling. To obtain a self-consistent equation for the order parameter, we assume a spin- charge separation as is usually done in the slave-particle methods by applying a 716 Superconductivity in the t-J model decoupling of spinon and holon degrees of freedom introduced in equation (29): Bij = 1√ 2 〈h+ i bi↑ h+ j bj↓ − h+ i bi↓ h+ j bj↑〉 ' 〈h+ i h+ j 〉 1√ 2 〈bi↑ bj↓ − bi↓ bj↑〉 ≡ 〈h+ i h+ j 〉ϕij , (40) bij = 1√ 2 (h+ i bi↑ h+ j bj↓ − h+ i bi↓ h+ j bj↑) ' h+ i h+ j ϕij . (41) Within these approximations we obtain the effective Hamiltonian for holons Hh ' − ∑ i6=j t̃ij hi h + j − µ ∑ i hi h + i − 1 2 ∑ i6=j J̃ij (〈hj hi〉 h+ i h+ j + hj hi〈h+ i h+ j 〉 − \|〈h+ i h+ j 〉\|2) , (42) where the effective hopping parameter and the exchange energy are given by t̃ij = tij 〈b+ i↑ bj↑ + b+ i↓ bj↓〉 = tij 〈S+ i S− j + S− i S+ j 〉, J̃ij = Jij \|ϕij\|2 = Jij 1 2 \|〈bi↑ bj↓ − bi↓ bj↑〉\|2 . (43) To obtain a phase diagram for the RVB order parameter Bij = F+ ij ϕij as a function of temperature T and hole doping δ one should solve a self-consistent system of equations for the both order parameters, holon Fij and spinon ϕij ones. Here we consider only the equations for the holon order parameter by suggesting that there exists a region of (T, δ) where the spinon order parameter is nonzero. Introducing k-vector representation for the correlation functions: Fij = 〈hi hj〉 = 1 N ∑ k ei(k(i−j)) F (k) = i N ∑ k sin(k(i − j)) 〈hk h−k〉, (44) ϕij = 1√ 2 〈bi↑ bj↓ − bi↓ bj↑〉 = 1 N ∑ k ei(k(i−j)) ϕ(k) = 2i N ∑ k sin(k(i − j)) 1√ 2 〈bk↑ b−k↓〉, (45) the Hamiltonian (42) can be written in k-space as Hh = ∑ k (ε(k) − µh) h+ k hk − 1 2 ∑ k {∆(k) h+ −k h+ k + ∆+(k) hk h−k}, (46) where the chemical potential for holons µh = −µ and the holon spectrum ε(k) according to equation (43) is written in the form (cf. equation (24)): ε(k) = t̃(k) = 4t γ(k) 2〈S+ i S− i+a1 〉 + 4t′ γ′(k) 2〈S+ i S− i+a2 〉, (47) 717 N.M.Plakida Here, we assumed that the spin correlation functions have s-wave symmetry, e.i., χi,i±ax = χi,i±ay . We also introduced the holon gap function defined by the equation ∆(k) = −∆(−k) = 1 N ∑ q J̃(k − q) 〈hq h−q〉. (48) In equations (47), (48) we used a notation ε(k), ∆(k) for the holon spectrum and the holon gap to distinguish them from those of physical electrons in section 2.2. The normal and the anomalous correlation functions can be easily calculated for the BCS-type Hamiltonian (46): 〈h+ q hq〉 = 1 2 ( 1 − ε(q) − µh E(q) tanh E(q) 2T ) , (49) F (q) = 〈hq h−q〉 = ∆(q) 2E(q) tanh E(q) 2T , (50) where the quasiparticle spectrum E(k) = [(ε(k) − µh) 2 + \|∆(k)\|2]1/2 . From these equations there follow the self-consistent equations for the gap function and the average number of holes δ which defines the holon chemical potential µh: ∆(k) = 1 N ∑ q J̃(k − q) ∆(q) 2E(q) tanh E(q) 2T , (51) δ = 1 N ∑ q 〈h+ q hq〉 = 1 N ∑ q 1 2 ( 1 − ε(q) − µh E(q) ) tanh E(q) 2T . (52) These equations are identical to the results obtained in [32] where the path-integral representation for the t-J model was applied and an effective BCS-type Hamiltonian analogous to equation (46) was derived. Starting from the MFA for RVB order parameter as in equation (37), the authors introduced MFA for the spinon auxiliary field that resulted in the spin-charge separation as in our equation (41). Therefore, we have proved that the results of the path-integral representation for the t-J employed in [32] are equivalent to the MFA for the slave fermion – hard-core boson approach considered in this section. 3.3. Holon Green functions To avoid uncontrollable approximation caused by the decoupling of the Hubbard operators on the same lattice site in equation (39) used in MFA for RVB state in the Hamiltonian (37), in the present section we consider a projection technique for the holon GF for the spinon-holon model. By using the spinless fermion hard-core boson representation (29) we write the Hamiltonian of the t-J model (5) as follows: HtJ = − ∑ i6=j tij hi h + j (S+ i S− j + S− i S+ j ) − µ ∑ i hi h + i − 1 4 ∑ i6=j Jij hi hj h+ j h+ i (S+ i S− j − S− i S+ j )(S+ j S− i − S− j S+ i ). (53) 718 Superconductivity in the t-J model To obtain an equation for the holon GF, we apply the projection technique described in section 2.1. By introducing the matrix GF for holon operators: Ĝh ij(t − t′) = ( 〈〈hi(t)\|h+ j (t′)〉〉〈〈hi(t)\|hj(t ′)〉〉〈〈h+ i (t)\|h+ j (t′)〉〉〈〈h+ i (t)\|hj(t ′)〉〉 ) , we can obtain an equation of motion for the GF as discussed in section 2.1 (see equations (8)–(11)). For the zero-order GF in MFA we get the following result: Ĝ(h,0)(k, ω) = ωτ̂0 + (ε̃(k) − µh)τ̂3 + ∆̃(k)τ̂1 ω2 − (ε̃(k) − µh)2 − \|∆̃(k)\|2 , (54) where the holon dispersion and the gap function are given by the equations: ε̃(k) = t̃(k) − 1 N ∑ q {J̃ ′(k − q) − J̃ ′(0)} 〈h+ q hq〉, (55) ∆̃(k) = 1 N ∑ q J̃ ′(k − q) 〈hq h−q〉. (56) In comparison with MFA for RVB state considered previously, in the GF approach we obtain additional renormalization for the holon dispersion, equation (47), while the gap equation has the same form, equation (48), but with a different renormalized exchange interaction: J̃ ′ ij = Jij 1 2 〈\|S+ i S− j − S− i S+ j \|2〉 = Jij ( 1 4 − 〈Sz i Sz j 〉). (57) By using an equation of motion method instead of MFA for RVB state, we managed to take into account spinon correlations in the effective holon interaction (57) and obtained a simple formula for it by using the identities: S+ i S+ i = 0, S− i S+ i = (1/2)− Sz i in the last equation. From equation (57) we get the following estimation for the effective exchange energy for the nearest neighbors: J̃ ′ ' (0.5−0.25) J if we assume the AFM Neél phase: 〈Sz i Sz j 〉 = −1/4 or completely neglect AFM correlations: 〈Sz i Sz j 〉 = 0. 4. Results and discussion 4.1. Spectral density Spectral density for physical electron excitations in the lower Hubbard subband within the t-J model which is defined by the equation Aσ(k, ω) = − 1 π Im〈〈X0σ k \|Xσ0 k 〉〉ω+iε, (58) satisfies the sum rule: ∑ σ +∞ ∫ −∞ dω Aσ(k, ω) = ∑ σ 〈{X0σ i , Xσ0 i }〉 = 2 Q = 1 + δ. (59) 719 N.M.Plakida In the MFA for the GF in terms of the HO, equation (21), we get: Aσ(k, ω) = Q uk δ(ω − Ek) + Q vk δ(ω + Ek), (60) where uk = 1 2 ( 1 + (εk − µ̃) Ek ) , vk = 1 2 ( 1 − (εk − µ̃) Ek ) . (61) In MFA, the spectral density (60) satisfies the sum rule (59) but it has no incoherent background which appears if one takes into account the self-energy corrections as shown in [36]. In the spinon-holon representation (29), the spectral density, as follows from equation (34), does not obey the sum rule (59): ∑ σ +∞ ∫ −∞ dω A(sh) σ (k, ω) = ∑ (i−j) e−i(k(i−j)) +∞ ∫ −∞ dω { − 1 π Im〈〈h+ i biσ \|hj b+ jσ〉〉ω+iε } = ∑ σ 〈{h+ i biσ , hi b + iσ}〉 = 1. (62) MFA for the spinon-holon GF in equation (62) results in the spin-charge separation which defines the spectral density (62) as a convolution of the anticommutator holon and the commutator spinon GF [28]: A(sh) σ (k, ω) = 1 N ∑ q 1 2π2 +∞ ∫ −∞ +∞ ∫ −∞ dω1dω2 tanh(ω1/2T ) + coth(ω2/2T ) ω − ω1 − ω2 × Im〈〈h+ q \|hq〉〉ω1 Im〈〈S σ̄ k−q\|Sσ k−q〉〉ω2 . (63) In this representation, the spectral density of physical electron excitations is given by a superposition of the spectral density for spinless fermion (holon) excitations and a background produced by spin excitations [28]. In MFA, the holon spectral density in the paired state below Tc is defined by the GF (54): A(h)(k, ω) = − 1 π Im〈〈h+ k \| hk〉〉ω+iε = ũk δ(ω + Ẽ(k)) + ṽk δ(ω − Ẽ(k)), (64) where the quasiparticle energy Ẽ(k) = [(ε̃(k) − µh) 2 + \|∆̃(k)\|2]1/2 and ũk, ṽk are given by equation (61) but with quasiparticle energies for holons. The spectral densi- ty of electron excitations is measured in the angle-resolved photoemission (ARPES) experiments which provide information concerning the symmetry of the gap func- tion. As discussed below, the symmetry of superconducting gap function ∆k defined by equation (27) appears to be d-wave, while the holon gap function ∆(k) in equa- tion (51) has an odd symmetry of the p-type as in the triplet pairing. This p-wave gap symmetry has never been observed in ARPES experiments in cuprates. 720 Superconductivity in the t-J model 4.2. Gap symmetry For the models with strong electron correlations, the s-wave component of su- perconducting gap should be strongly suppressed due to the on-site Coulomb corre- lations. For the t-J model it follows from the constraint of no double occupancy on a single site given by the identity: 〈ĉi,σĉi,−σ〉 = 1 N ∑ k 〈ĉk,σĉ−k,−σ〉 = 0, (65) for the physical electron operators ĉi,σ = ci,σ(1 − ni,−σ). Since the anomalous corre- lation function 〈ĉk,σĉ−k,−σ〉 is proportional to the gap function ∆(k), equation (65) imposes a certain constraint on the symmetry of the gap function. In particular, for the solution (26) in MFA equation (65) reads [34,35]: 〈X0σ i X0σ̄ i 〉 = 1 N ∑ q 〈X0σ q X0σ̄ −q〉 = Q N ∑ q ∆σ q 2Eq tanh Eq 2T = 0. (66) For a tetragonal lattice, the Fermi surface (FS) is invariant under the C4-axis rotation in the k-space and, therefore, to satisfy the condition (66) for Eq > 0 , the gap function ∆σ q should change its sign along the FS. It means that the symmetric s- wave solution, ∆s(kx, ky) ∝ (cos qx + cos qy) , does not fit equation (66), while the d-wave solution with B1g or B2g symmetry, ∆d(kx, ky) = −∆d(ky, kx), satisfies the condition (66). In general, equation (66) should be considered as a constraint on the symmetry of the gap function in the superconducting phase and the solutions which violate this constraint should be disregarded. Now we consider the symmetry of the spinon and the holon order parameters. The condition (65) for the RVB order parameter reads Bii = 1 N ∑ k B(k) = 1 N ∑ k,q ϕ(k − q) F +(q) = 0, (67) which also imposes a constraint on the symmetry of the spinon and holon order parameters. Since the symmetry of the holon pairing order parameter, equation (44), for spinless fermions is odd, being imposed by their anticommutation relations, it results in the odd symmetry of the gap ∆(k) in equation (48). For a tetragonal lattice, the symmetry is given by the odd two-dimensional irreducible representation Eu which can be modelled by the function: ∆(k) ∝ η± p (k) = (sin kx ± sin ky) as in the p-wave triplet pairing. The same holds for the spinon order parameter in equation (45), ϕ(k) ∝ η± p (k). The RVB order parameter Bij = 〈bij〉 = ϕijF + ij as a product of two antisym- metric order parameters with Eu symmetry has either A (s-wave) or B1g (d-wave) symmetry. Namely, if we adopt only the nearest neighbor pairing for both the order parameters: ϕij = ϕi,i+ax {(δj,i+ax − δj,i−ax ) ± (δj,i+ay − δj,i−ay )}, F+ ij = F+ i,i+ax {(δj,i+ax − δj,i−ax ) ± (δj,i+ay − δj,i−ay )}, (68) 721 N.M.Plakida then, for the Fourier transform of the RVB order parameter, we get B(k) = ∑ q ϕ(k − q) F +(q) = 2b ∑ q {sin(kx − qx) ± sin(ky − qy)} {sin qx ± sin qy} = −b (cos kx ± cos ky), (69) where b = ϕi,i+ax F+ i,i+ax and the sign + (−) in the last line corresponds to the same (different) signs in the first line. The solution (69) satisfies the condition (67) for both the s- and d-wave symmetry: ∑ k B(k) ∝ ∑ k(cos kx ± cos ky) = 0. However, the explicit solution for the holon order parameter (50) shows its more complicated k-dependence which proves that the frequently used model for the near- est neighbor pairing, equations (68), is inadequate. For the solution (50), the condi- tion (67) for the RVB order parameter reads: ∑ k,q ϕ(k − q) F +(q) = ∑ k,q ϕ(k − q) ∆+(q) 2E(q) tanh E(q) 2T = 0, (70) which also imposes a requirement on the symmetry of the singlet (RVB) order pa- rameter which should have only the d-wave symmetry, B(kx, ky) = −B(ky, kx) , to satisfy this condition. It is interesting to compare the results for the gap equation derived in MFA for the t-J model and that one for the Hubbard model. By applying the projection technique for the GF for the Hubbard model, one can get the following equation for the gap function (see, e.g., [39]): ∆ijσ ∝ 〈X02 i Nj〉 = 〈ci↓ci↑Nj〉, (71) where we have used the identity for the Hubbard operators, X02 i = X0↓ i X↓2 i = ci↓ci↑ . From equation (71) it follows that the pairing occurs on one lattice site but in the different Hubbard subbands. By using the equation of motion for the GF 〈〈X02 i (t)\|Nj(t ′)〉〉, the anomalous correlation function 〈X02 i Nj〉 can be calculat- ed without any decoupling that results in the same gap equation as in the MFA for the t-J model [39]): ∆ijσ = Jij 〈X0σ i X0σ̄ j 〉/Q where we have used the notation of the present paper. Therefore, the symmetry constraint considered above is also applicable to the gap solutions obtained in the Hubbard model. 4.3. Superconducting Tc In the present section we compare the pairing temperature Tc defined by the gap equation for physical electrons (27) and that one for holons, equation (51). We study Tc as a function of a corresponding chemical potential ν = µ/W for a given value of the pairing interaction λ = J/W where all energies are measured in units of the renormalized half bandwidth for pairing fermions W = 4 t̃ : e(k) = ε(k) W = γ(k) + τγ ′ (k), τ = t′eff teff , d(k) = ∆(k) W . (72) 722 Superconductivity in the t-J model The gap equations (27), (51) (or (56)) can be written in the form d(k) = 4λ N ∑ q γ(k − q) d(q) 2[(e(q) − ν)2 + d(q)]1/2 tanh [(e(q) − ν)2 + d(q)]1/2 2T , (73) Let us consider Tc(ν) for different pairing symmetry, d(k) = dα ηα(k): s-wave: ηs(k) = (cos kx + cos ky), d-wave: ηd(k) = (cos kx − cos ky), p-wave: η± p (k) = (sin kx ± sin ky). (74) By integrating equation (73) with the corresponding symmetry parameter ηα(k) we obtain equation (73) for T = Tc in the same form for any symmetry: 1 λ = 1 N ∑ k (ηα(k))2 1 2(e(k) − ν) tanh e(k) − ν 2T (α) c = 1 2 ∫ +1 −1 dε ε − ν Nα(ε) tanh ε − ν 2T (α) c , (75) if we introduce an effective density of state (DOS) for the corresponding symmetry, α = s, d, p: Nα(ε) = 1 N ∑ k (ηα(k))2 δ(ε − e(k)) , (76) which is normalized ∫ +1 −1 dε Nα(ε) = 1, since 1 N ∑ k (ηα(k))2 = 1. The results of calculation of the effective DOS, equation (76), for different symmetry α = s, d, p is presented in figure 1 for t′ = 0. From this dependence it is easy to draw a conclusion that the Tc(ν) function will follow the dependence Nα(ν) for a corresponding symmetry since the effective coupling constant V ' λNα(ν) reaches its maximum value at the maximum value of Nα(ν). In the logarithmic approximation a solution for Tc(µ), equation (75), can be written in the conventional BCS form: Tc ' √ µ(W − µ)exp(−1/V ), V ' J Nµ , (77) but with the prefactor proportional to the Fermi energy, µ = EF which can result in high-Tc. The highest Tc(ν) appears for the d-wave pairing and the lowest Tc(ν) – for the p- wave pairing since in the former case the van Hove singularity gives a strong contribution for DOS ( [ηd(k = (±π, 0))]2 = 4 ), while in the latter case it is completely suppressed ( [ηp(k = (±π, 0))] = 0 ). This estimation can be checked by a direct numerical solution of equation (75) (see, e.g. [35,36] and [48] for p-wave symmetry). 723 N.M.Plakida −1 −0.5 0 0.5 1 ε 0 2 4 6 8 N α( ε) Figure 1. Effective density of states (76) for s (solid line,) d (dot-dashed line) and p (dashed line) symmetry. 5. Conclusion In the present paper we consider a superconducting pairing mediated by the exchange interaction which is generic for the system with strong electron correlations such as cuprates. The mechanism of the exchange pairing is the lowering of kinetic energy of electron pairs due to their coherent hopping between different Hubbard subbands. Since the excitation energy of this hopping is much larger than the Fermi energy the retardation effects in the exchange interaction are negligible [39] which results in the pairing of all electrons (holes) in the conduction band and a high-Tc proportional to the Fermi energy, equation (77). To obtain the estimation for the superconducting Tc it is tempting to use a mean-field approximation for the exchange interaction within the t-J model. How- ever, meaningful physical results can be obtained only if one takes into account strong electron correlations on a rigorous basis which is provided by the Hubbard operator (HO) technique. Any auxiliary field representations applied within MFA inevitably violate rigorous commutation relations for HO which may result in un- physical conclusions. In the present paper we have proved this by comparing the results for superconducting gap equation derived within the slave fermion – hard- core boson representation, equation (29), and HO technique for the Green functions (section 2.1). In the former method, the projected character of physical electron operators is neglected that results in a double counting of empty states and in vi- olation of the sum rule, equation (62). The spin-charge separation which occurs in 724 Superconductivity in the t-J model MFA, equations (40), (41), results in separate equations for two order parameters for spinons and holons instead of one equation for physical electrons as in the GF method. The gap equation (51) for spinless fermions has only antisymmetric solu- tions which results in an unphysical p-wave gap for the quasiparticle excitations in superconducting state never observed in ARPES experiments and much lower Tc than for the d-wave pairing given by equation (27) for physical electrons. The obtained results within MFA for the spinless fermion – hard-core bosons in section 2 appear to be identical to the path-integral representation for the t-J model employed in [32]. Therefore, the latter results have the same flaws as discussed above which casts doubts on the approaches based on the idea of spin-charge separation treated in MFA which results in violation of the local constraint. 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