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Eliashberg-type equations for correlated superconductors

The derivation of the Eliashberg -- type equations for a superconductor with strong correlations and electron--phonon interaction has been presented. The proper account of short range Coulomb interactions results in a strongly anisotropic equations. Possible symmetries of the order parameter include s, p and d wave. We found the carrier concentration dependence of the coupling constants corresponding to these symmetries. At low hole doping the d-wave component is the largest one.


I. INTRODUCTION
Recently there appeared in the literature a number of papers dealing with the problem of electron-phonon interaction in strongly correlated superconductors 1−9 . It turns out that interplay between Coulomb and electron-phonon interactions is very subtle and may result in an unexpected behaviour of various quantities 10−13 .
The main motivation for all these studies come from experiments on high T c superconductors which in one or other way have pointed out the importance of electron-phonon interaction.
There have been found Raman and infrared-active modes 14 with strongly temperature dependent linewidth. Small, but nonzero, isotope shift exponent α (T c ∝ M α , where M the ion isotope mass) 15,16 is the strong indication of the electron-phonon coupling. Similar conclusions can be inferred from tunneling and photoemission 17 , neutron 18,19 , specific heat 20 , thermal conductivity 21,22 and other experiments.
It is the purpose of this paper to derive Eliashberg type 23 equations valid for the strongly correlated superconductor with electron-phonon interaction. To this end we assume the validity of Migdal 24 theorem which means that we assume relatively weak electron-phonon interaction. In Section II we present the model Hamiltonian for strongly correlated systems and discuss the application of slave boson approximation to it in the U = ∞ limit, both in mean field and beyond it. The derivation of the Eliashberg equations on the real frequency axis is described in considerable detail in Section III. Section IV contains presentation of results and discussion.

II. THE MODEL
We start with the one band Hubbard model with general form of electron-phonon interaction Herec + iσ (c iσ ) is the creation (annihilation) operator for a spin σ electron at site i of the lattice, u α i is the α-th component of the displacement vector of the ion,ñ iσ =c + iσciσ , µ denotes chemical potential, t ij is the hopping integral assumed to take on nonzero value −t for i, j being nearest neighbour sites, U is the Hubbard on-site repulsion of carriers. H ph denotes the Hamiltonian of the lattice. The electron-lattice interaction described by the second term in (1) has two components T ′α the first of which, T α ijs has been derived from modulation of the hopping integral 25 t ij in the deformed lattice. It is related to the derivative of t ij taken at equilibrium position of an ion The second part of the interaction i.e. the term V α js is connected with fluctuations of the crystal field. Due to ionic character of high temperature superconductors this term is expected to be more important and has to be taken into account even if we have assumed the equilibrium value of crystal field ε i = t ii to be zero. The systematic derivation and the discussion of the electron-ion interaction in context of superconducting oxides can be found in Ref. 26 .
We are interested in the strong correlation limit, characterised by the large U values (U ≫ W , where W = 8t is the width of electron band in a two dimensional square lattice). In this limit and for less than half filled band (i.e. concentration of carriers n < 1) it is very inconvenient for two carriers to occupy the same site i. It means that double occupation of sites is prohibited and the dynamics of carriers is limited to empty and singly occupied states.
Thus we have to project out all the doubly occupied states of the system. This procedure is most conveniently carried out with the help of auxiliary particles 27 and in the U = ∞ limit. One rewrites the electron operators in (1) in terms of new fermion operators c + iσ (c iσ ) and auxiliary The term U may then be dropped out at the expense of introducing at each site a constraint via Lagrange multiplier Λ i . The constraint allows for at most single occupation of each size.
The Hamiltonian (1) then becomes Note that on-site terms are not modified. It is connected with the meaning of the boson operators b + i as operators creating empty states at site i. The hopping of a real electron from occupied state j to an empty state at i, described by termc + iσcjσ in (1) consists of motion of a fermion between these sites (c + iσ c jσ ) and at the same time the empty site moves from site i to j (b i b + j ). It has been shown that mean field description can be obtained by assuming average values b i = b + j = r and Λ i = Λ at each site. Two parameters r and Λ entering the mean field Hamiltonian can be choosen so as to make minimal the ground state energy E GS = H . One gets 27 Here n = 1 N iσ c + iσ c σ denotes the concentration of electrons in the band (n < 1). There are two modifications of the spectrum of electrons encountered on the mean field level. First is the band narrowing described by r 2 an its shift described by Λ. The spectrum of noninteracting fermions in the mean field is given by (r 2 ǫ k − µ + Λ) instead of (ǫ k − µ) of original electrons (at U = 0). For the half filled band n = 1 the system is localised (r = 0). Here ǫ k is the Fourier To go beyond mean field approximation for slave bosons it is convenient to define boson fluctuating fields via: b + i = r + δb + i , b i = r + δb i . In view of our main goal in next section which is derivation of the Eliashberg equations, we shall write down the Hamiltonian in terms of Gorkov-Nambu 28 field operators: Using this in (1) we obtain where in site representation First term of H ′ describes fermion-boson interaction, while the last one fermion-phonon-boson interaction. More complicated interactions are contained in H ′′ . They will be treated in a mean field type of approximation. The important point is that H ′′ significantly contributes to dynamics of fluctuating boson field. In the mean field approximation for electrons the boson fluctuations are described by effective Hamiltonian For the sake of completeness we write down the Hamiltonian for phonons which we assume to be described in the harmonic approximation. In the second quantised form it reads

III. EQUATIONS OF SUPERCONDUCTIVITY
To properly describe the superconducting state in the system at hand one has to work in site representation. The important point is that in considered U = ∞ limit the double occupation of a given site is strictly forbidden. This means inter alia that correlation functions c i↑ c j↓ describing superconducting pairs vanish exactly for i = j, i.e. the on-site pairing is forbidden.
On the other hand the correlations of the type c + iσ c iσ measure the average number of carriers at site i, and are allowed to enter into formula. This important fact has first been noted by Zieli/nski and coworkers 3,7 and leads, as we shall see, to severe changes in the form of Eliashberg equations.
To derive them we use the equation of motion method for the double time thermodynamic Green's functions. For arbitrary operators A an B it reads (we employ here the Zubariev where ≪ A|B ≫ ω is the Fourier transformed, frequency dependent, retarded Green's function. [A, B] ± denotes anticommutator (+) or commutator (−).
Standard procedure 30 leads to the following equation for the matrix Green's function (GF) The contribution from fermion-phonon scattering is given by The contribution from linear boson-fermion scattering reads with φ B ij = δb i + δb + j . Related, but probably less important contribution from quadratic boson-fermion scattering This contribution is absent if one approximates H ′′ in mean field like manner.
There are also two contributions from three particle scattering events: fermion-phononslave boson. From H ′ part of the Hamiltonian we get while from second term in H ′′ To get real frequency axis Eliashberg equations in a standard form 31 one has to express the higher order Green's function appearing on the rhs' of various selfenergy pieces by the GF's of fermions ≪ ψ k |ψ + k ≫ ω , bosons ≪ δb p |δb + p ≫ ω and ,,phonons" ≪ φ qν |φ − qν ≫ ω . This is easily achieved with help of spectral representation and decoupling of various time correlation functions as e.g.
This approximation neglets vertex corrections. In the present situations the vertex corrections stem from fermion-phonon, fermion-boson and also more complicated interactions. As already mentioned the assumption of small electron-phonon coupling allows us to neglect fermionphonon vertex corrections. The same is not true for the fermion-boson vertices. They do not contain small parameters and should be taken into account. Their importance has also been stressed in previous studies 2, 4 . We shall postpone discussion of vertex corrections and concentrate on the anisotropy of Eliashberg equations.
The next step is Fourier transform of the various selfenergy parts. Defining the Fourier Here φ qν = a qν + a + −qν denotes the phonon field, a + qν is creation operator of the qν phonon. Electron-phonon matrix elements M ν kq andM ν kqp are defined as The factors of the type are to be treated very carefully. As already mentioned the sum goes over all sites i ′ and j ′ if we evaluate the normal contributions to the self-energy matrix i.e. components (1,1) and (2,2) of the self-energy matrix which are proportional to the GF of the type ≪Ôc i ′ σ |Ô ′ c + j ′ σ ≫ (Ô,Ô ′ are arbitrary operators). In such a case the above sum reduces to Nδ k,0 . When calculating off-diagonal elements of the selfenergy which depends on the GF's like ≪Ôc i ′ ↑ |Ô ′ c j ′ ↓ ≫ it is reasonable to assume i ′ and j ′ to be nearest neighbours, as in the U = ∞ limit only nearest neigbhour pairs will probably survive. In this situation the sum reduces to γ( k) = −ǫ( k)//2t.
The approximation of the sum to (i ′ , j ′ ) being nearest neighbour sites relies on the experimental fact that the superconducting coherence length is small: of order of lattice spacing. In an effective theory like the one presented here it translates to nearest neighbour pairs.
To proceed we expand matrix selfenergyΣ k (ω) aŝ and write down the equations for various parts of it. We get where we denoted Completely analogous set of equations is obtained for contributions toΣ B k (ω). The only difference is that K ph k, k ′ andK ph k, k ′ will be replaced by K B k, k ′ andK B k,k ′ given by Due to complicated structure of the GF entering the expression forM ph−B k (ω) the corresponding Eliashberg equations on the real frequency axis have more complicated form with and Equations (17)(18)(19)(20)(21)(22)(23)(24) supplemented with expressions for phonon and slave -boson Greens functions form a complete set and have to be solved selfconsistently.
There are few differences between them and usual Eliashberg equations for superconductors.
The most important is the presence of two different types of kernels K kk ′ andK kk ′ determining normal (Z k ′ , χ k ′ ) and anomalous (φ k ′ ) parts of the self-energy, respectively.
Other point worth to mention is strong renormalisation of the bare electron spectrum ǫ k = −2tγ( k). At the mean field level of the theory it is both shifted and mass renormalised to r 2 ǫ k + Λ. Beyond mean field level for bosons the nontrivial energy dependence comes from renormalisation of the fermion (matrix) Greens function given by the χ B k (ω) factor. Note, that from the equation for χ B k (ω) and definition (20) where C(ω) and C 1 (ω) are weakly k dependent functions to be calculated from equation (17).
To take this effect into account one has to solve the equations at each point of the Brillouin zone -a very difficult task.

IV. RESULTS AND DISCUSSION
The complete solution of Eliashberg equations which are integral equations with complicated kernels would require finding all functions Z k (ω), φ k (ω), χ k (ω) at all frequencies and all points in the Brillouin zone. In the standard approach the valid argument was that most important changes are to be expected at the Fermi level. The various pieces of selfenergies were thus averaged over the wave vectors lying on the Fermi surface. In this way it was possible to reduce the equations to the integral equation of the single variable: the frequency. It is not obvious that the same arguments do apply in the present case especially to the boson contribution to self-energy.
Sticking to the mean field approximation for slave bosons leads to δb q = δb + q ≡ 0. In this approximation only phonon contribution to Σ k survives,Σ k (ω) =Σ ph k (ω). In this case most of arguments developed previously 31 apply and one can use known parametrisations of solutions to Eliashberg equations. The relevant parameters describing superconducting transition temperature T c are the typical phonon frequency ω D and electron-phonon parameter λ e−ph . The last parameter is determined by the kernel K ph . In our theory we do have two different kernels as also found previously by Zieli/nski and coworkers 3 7 .
We thus define two coupling constants λ e−ph andλ e−ph . The first one is related to wave function renormalisation and other describes genuine superconducting coupling in strongly correlated system. We define λ el−ph by 31 where Similar expression definesλ e−ph except thatK ph kk ′ (ω) enters. In the general case there exist four more parameters: λ e−B ,λ e−B , λ e−ph−B andλ e−ph−B .
For numerical purposes it is convenient to take optical phonons with ω qν = ω o . In this case the frequency integral in (26) can be calculated very easily.
Strong wave vector dependence of the kernelsK ph k k ′ makes the question of symmetry of the order parameter very important. To get some information on this point we calculate various symmetry components λ i e−ph of the pairing interaction. We define them as We have taken g 1 = cos k x a + cos k y a, g 2 = cos k x a − cos k y a, g 3 = sin k x a + sin k y a and g 4 = sin k x a − sin k y a; A i = k g 2 i ( k)δ(r 2 ǫ k − µ + λ). There are few parameters which control the behaviour of the system. These are: carrier concentration n, strength of the electron -phonon interaction parameters V c and V i , the frequency ω 0 of the optical phonon etc. All the energies are measured in units of D = 2t, and for numerical purposes we have taken ω 0 = 0.02, and the value of the ionic mass M appropriate for oxygen.
In this work we shall concentrate on the results obtained from the kernelK ph k k′ , describing anomalous part of the self-energy. In the previous work 8 we have analysed in some detail the properties of the system resulting from the kernel entering normal self-energy.
We start the discussion by showing the results obtained for V i = 0. The covalent part of the interaction is strongly k dependent. It is described by the matrix element |M k q | 2 in equation (15). In figure (1a)  In figure (1b) we show similar set of data obtained with |M k q | 2 replaced by its Fermi surface average M 2 F . Due to oscilatory dependence of |M k q | on wave vectors the average value of it is strongly concentration dependent (solid line in figure (1b)) and reduced in comparison to 1 (the value arbitrarily assumed for data shown in Fig.(1a)). Small values of M F at large n are due again to r 4 factor, while at small n to phase space restrictions.
Taking the full wave vector dependence of |M k q | leads to the results shown in figure (1c).
Solid line in this figure shows total couplingλ el−ph calculated from (26) Fig.2(b)) . The is due to interference effects between interaction channels (cf. expression (15) for |M k q | 2 ). In this work we have concentrated on the calculation of the electron -phonon coupling constants.
To obtain realistic estimations of the concentration dependence of the superconducting transition temperature one has to take simultaneously into account the condensation of the (slave) at low hole concentration. In this case the additional condition will modify the T c vs.n phase diagram making it qualitatively more similar to that obtained for covalent electron -phonon interaction.
In conclusion we have presented the derivation of the Eliashberg equations for the system with strong electron -electron and electron -phonon interactions. The results show the importance of the correlations which make the superconducting order parameter very anisotropic. Its symmetry depends on the carrier concentration. We have illustrated the theory by showing the results of calculations of coupling constants in the mean field approximation for slave bosons.
As already mentioned beyond mean field level there are important changes of the structure of the theory. The mixed fermion -phonon -slave boson contribution to the selfenergy and corresponding coupling constants will overtake the fermion -phonon contributions considered here.
The χ component of the selfenergy due to fermion -slave boson scattering acquires nontrivial energy ǫ k dependence. The study of both these effects will be reported in the forthcomming paper.