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 calculate superconducting T c a numerical solution of self-consistent Eliashberg equations is proposed.


Introduction
Since the discovery of high temperature superconductivity in cuprates it has been believed by many researchers that an electronic mechanism could be responsible for high values of T c .Recent experimental evidences of a d-wave superconducting pairing in high-T c cuprates strongly support this idea (see, for example, 1,2]).At present various phenomenological models for the spin-uctuation pairing mechanism are known (for reference see, e.g., 2,3]).Numerical nite cluster calculations also suggest a d-wave superconducting instability for models with strong electron correlations 4].Anderson 5] was the rst who stressed the importance of strong electron correlations in copper oxides and proposed to take them into account within the framework of a one-band Hubbard model: H = ?tX hiji (a + i a j + H:c:) + U X i n i" n i# ; ( where t is an e ective transfer integral for the nearest neighbour sites, hiji, 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 site needs a large additional energy U: H t?J = ?tX hiji; (ã + i ãj + H:c:) + J X hiji (S i S j ? 1   4 n i n j ): (2) Here electron operators ã+ i = a + i (1 ?n i? ) act in the subspace without a double occupancy and n i = n i" + n i# is the number operator for electrons.The second term describes the spin-1/2 Heisenberg antiferromagnet (AFM) with the exchange energy J = 4t 2 =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): J ij X i X j ?X i X j ; (3)   where t ij = t; t 0 is the electron hopping energy for the nearest and the second neighbours on the 2D square lattice, respectively, and J ij is the exchange interaction.
We have also introduced chemical potential and number operator n i = P X i .
The HO are de ned as for three possible states at the lattice site i: ji; i = ji; 0i; ji; i for an empty site and for a site singly occupied by an electron with the spin =2 ( = 1; = ?).
They obey the completeness relation Here the energy levels E 1 = E 0 ?and E 2 = 2E 0 ? 2 + are introduced for singly and doubly occupied sites, respectively, where E 0 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 = p ? d between pand d-levels in the CuO 2 plane.The hopping integrals have di erent values for the LHB (t 11 ij ), the UHB (t 22 ij ) and the inter-band transitions (t 12 ij ).They can be written in the form t ij = ?K 2t ij where t = t pd is the p ? d hybridization integral and ij are the overlapping parameters for the Wannier oxygen states which are equal to: 1 = j j a x=y ' ?0:14 for the nearest neighbours and 2 = j j ax ay ' ?0:02 for the second neighbours, where a x=y are lattice constants.The coe cients K 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 de ned for 4 possible states at lattice site i: ji; i = ji; 0i ; ji; i ; ji; "#i for an empty site, a site singly occupied by an electron with the spin and for a doubly-occupied site, respectively.For these states the completeness relation for the Hubbard operators reads: X 00 i + X X i + X 22 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 ! 1) was rst 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 eld 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 eld approximation was considered for the t ?J model within the GF approach in 11,12] where the d-wave spin-uctuation 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 eld type approximation within the projection technique for the GF.Local superconducting pairings of the sand d-symmetry were obtained which, however, should disappear in the limit of strong correlations, U ! 1.Unfortunately, in this approximation the self-energy operator caused by kinematic and exchange interactions is ignored, though it results in nite life-time e ects 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 responsible for the non-local spin-uctuation 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 Eliashberg equations revealed a strong renormalization of the QP hole spectrum due to spin-uctuations and proved the dwave pairing.The maximum T c ' 0:01t was obtained at the optimal concentration of doped holes ' 0:2.However, a twosublattice 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  In the given paper we consider a paramagnetic state with only short-range dynamic spin uctuations at a moderate doping.We develop the theory of superconductivity 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 operators.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 Prelov sek 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].

Dyson equation for the t ? J model
To discuss the superconducting pairing within model (3) we consider the matrix Green function (GF) Ĝij; (t ?t 0 ) = hh i (t)j + j (t 0 )ii (8) in terms of the Nambu operators: 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: J il (B l 0 ? 0 )X 0 0 i ; (10) where B i 0 = (X 00 i + X i ) 0 + X i 0 = (1 ? 1 2 n i + S i ) 0 + S i 0 : (11) The boson-like operator B i 0 describes electron scattering on spin and charge uctuations caused by the nonfermionic commutation relations for the HO's (the rst term in (10) { the so-called kinematical interaction) and by the exchange spin-spin interaction (the second term in (10)).
By di erentiating the GF (8) with respect to time t and t 0 and employing the projection technique (see, e.g., 6]) we get the following Dyson equation: Ĝij for the Fourier component.Here the zero{order GF is calculated in the mean-eld approximation Ĝ0 ij (!) = Q f!^ 0 ij ?Êij g ?1 (13) with the frequency matrix Êij = hf i ; H]; + j gi Q ?1 and the correlation function Q = hX 00 i + X i i = 1 ?n=2 .In a paramagnetic state it depends only on the average number of electrons n = hn i i = P hX i i.The self-energy operator ^ kl (!) is de ned by the equation: where the irreducible part of the operator Ẑi = i ; H] is de ned by the projection equation Ẑ(irr Êil l ; hf Ẑ(irr) i ; + j gi = 0 : 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 uctuations.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: hX 0 0 j 0 B + j 0 X 0 0 i 0 (t)B i 0 (t)i (j6 =j 0 ; i6 =i 0 ) ' hX 0 0 j 0 X 0 0 i 0 (t)ihB + j 0 B i 0 (t)i : Using a spectral representation for the GF we obtain the following results for self-energy matrix elements in the k-representation: dzd N(!; z; ) 11 (12) (q; k ?q j )A 11 (12) (q; z); (17) with N(!; z; ) = 1 2 tanh(z=2T) + coth( =2T) !?z ?: Here we introduce a spectral density for the normal (G 11 ) and anomalous (G 12 ) GF: A 11 (q; z) = ? 1 Q Im hhX 0 q j X 0 q ii z+i ; A 12 (q; z) = ? 1 Q Im hhX 0 q j X 0 ?qii z+i (20) and the electron -electron interaction functions caused by spin and charge uctuations 11 (12) (q; k ?q j ) = g 2 (q; k ?q) D +(?) (k ?q; ); (21) 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; ) = ? 1 Im n hhS q j S ?q ii +i (1=4)hhn q j n + q ii +i o : ( A linearized system of Eliashberg equations close to T c can be written as selfconsistent equations for the normal GF and its self-energy operator and for the gap equation: f2J(k ?q)+ + 12 (q; k ?q j i! n ?i! m )g G 11 (q; i! m ) G 11 (q; ?i! m ) (q; i! m ): (24) In equation ( 24) we omit the k-independent part of the gap function k in the MFA (13) which is caused by the kinematic interaction 8], since it gives no contribution to the d-wave pairing ( 11]).Here we use the imaginary frequency representation, != i! n = i T(2n + 1).The energy of quasiparticles E k and the renormalized chemical potential ~ = ?in the MFA ( 13) is given by E k = ?(k)Q ?s (k)=Q ?4J N X q (k ?q)N q ; (25) where (k) = t(k) = 4t (k) + 4t 0 0 (k), s (k) = 4t (k) 1s + 4t 0 0 (k) 2s with (k) = (1=2)(cos a x q x + cos a y q y ); 0 (k) = cos a x q x cos a y q y .= 1 N X q (q)N q ?4J(n=2 ?1s =Q) : (26) The average number of electrons in the k-representation is written in the form: where which de nes function N q in equations ( 25), (26).When calculating the normal part of the frequency matrix (25) we neglect charge uctuations and introduce spin correlation functions for the nearest, a 1 = ( a x ; a y ), and the second, a 2 = (a x a y ), neighbour lattice sites : 1s = hS i S i+a 1 i ; 2s = hS i S i+a 2 i : (29) In the present calculations we take into account only the spin-uctuation contribution modelled by the spin-uctuation susceptibility (see, e. g., 18,19]): 00 s (q; !) = s (q) 00 s (!) with the characteristic AFM correlation length and spin-uctuation energy !s ' J.To x constant 0 in (30) we use the following normalization condition: In this approximation we get for the interaction functions (21) 11 (q; k ?q j i! ) = 12 (q; k ?q j i! ) = ?g 2 (q; k ?q) s (k ?q) For model (30) we can calculate static spin correlation functions (29) from the equations: 1s = hS i S i+a 1 i = 1 N X q (q)hS q S ?q i; 2s = hS i S i+a 2 i = 1 N X q 0 (q)hS q S ?q i; where hS q S ?q i = s (q) +1 Z ?1 dz exp (z=T) ? 1 00 s (z) = s (q) 2 !s : Therefore, we have obtained a closed system of equations which should be solved numerically.Preliminary calculations 20] con rm 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 occupation 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, t 2 , interactions in (24)) of the d-wave symmetry that occurs at high T c ' 0:06t.
We also introduce unity matrices ~ 0 (4 4) and ^ 0 ( 2 (44) where the mean eld spectrum is given by the dispersions 1 (k) and 2 (k) for a singly occupied d-hole-like band and a doubly occupied singlet band, respectively 6].

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 uctuations 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 tting 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 uctuations (equation (30)), was used.Nevertheless, we believe that numerical results should not depend considerably on the explicit form of the model for spin-charge uctuations.Being normalized (equation (31)), it cannot change substantially the sum over (q; !) 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 is in the singlet (d-hole) band, = 2 (1), and the main contribution to the integrand in equation (47) comes from the same band ( rst 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 rst term in equation (24).However, in the two-band model the spin-uctuation contribution to equation ( 47) is given by the frequency dependent susceptibility D ?(q; z) and the inter-band contributions / K 2  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.
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 subband (UHB) occupied by two-hole p?d singlet states 6].In terms of the Hubbard operators the asymmetric Hubbard model reads:

G
i i = 1 ? 1 : To solve the Dyson equation (38) which can be written in the general form as calculate the zero{order GF (39) and the self-energy matrix (41).The anomalous part of the zero{order GF in (39), F0 (k; !), vanishes if one disregards the mean-eld, k-independent gap function (due to the kinematic interaction) which violates the restriction hX 2 i X 2 i i = 0, see 22].For the normal part we can use a diagonal approximation Ĝ0 (k; !) = 2 =(! ? 2 (k))