Bosonic Spectral Function and The Electron-Phonon Interaction in HTSC Cuprates

In Part I we discuss accumulating experimental evidence related to the structure and origin of the bosonic spectral function in high-temperature superconducting (HTSC) cuprates at and near optimal doping. Some global properties of the spectral function, such as number and positions of peaks, are extracted by combining optics, neutron scattering, ARPES and tunnelling measurements. These methods give convincing evidence for strong electron-phonon interaction (EPI) with the coupling constant between 1-3 in cuprates near optimal doping. Here we clarify how these results are in favor of the Eliashberg-like theory for HTSC cuprates near optimal doping. In Part II we discuss some theoretical ingredients - such as strong EPI, strong correlations - which are necessary to explain the experimental results related to the mechanism of d-wave pairing in optimally doped cuprates. These comprise the Migdal-Eliashberg theory for EPI in strongly correlated systems which give rise to the forward scattering peak. The latter is further supported by the weakly screened Madelung interaction in the ionic-metallic structure of layered cuprates. In this approach EPI is responsible for the strength of pairing while the residual Coulomb interaction (by including spin fluctuations) triggers the d-wave pairing.


Experimental Evidence for Strong EPI
1.1. Introduction. In spite of an unprecedented intensive experimental and theoretical study after the discovery of high-temperature superconductivity (HTSC) in cuprates, there is, even twenty-three years after, no consensus on the pairing mechanism in these materials. At present there are two important experimental facts which are not under dispute: (1) the critical temperature T c in cuprates is high, with the maximum T max c ∼ 160 K in the Hg-1223 compounds; (2) the pairing in cuprates is d-wave like, that is, Δ(k, ω) ≈ Δ d (ω)(cos k x − cos k y ). On the contrary there is a dispute concerning the scattering mechanism which governs normal state properties and pairing in cuprates. To this end, we stress that in the HTSC cuprates, a number of properties can be satisfactorily explained by assuming that the quasiparticle dynamics is governed by some electronboson scattering and in the superconducting state bosonic quasiparticles are responsible for Cooper pairing. Which bosonic quasiparticles are dominating in the cuprates is the subject which will be discussed in this work. It is known that the electron-boson (phonon) scattering is well described by the Migdal-Eliashberg theory if the adiabatic parameter A ≡ α · λ(ω B /W b ) fulfills the condition A 1, where λ is the electron-boson coupling constant, ω B is the characteristic bosonic energy, W b is the electronic band width, and α depends on numerical approximations [1,2]. The important characteristic of the electron-boson scattering is the Eliashberg spectral function α 2 F(k, k , ω) (or its average α 2 F(ω)) which characterizes scattering of quasiparticle from k to k by exchanging bosonic energy ω. Therefore, in systems with electron-boson scattering the knowledge of the spectral function is of crucial importance.
There are at least two approaches differing in assumed pairing bosons in the HTSC cuprates. The first one is based on the electron-phonon interaction (EPI), with the main proponents in [3][4][5][6][7][8][9][10][11], where mediating bosons are phonons and where the average spectral function α 2 F(ω) is similar 2 Advances in Condensed Matter Physics to the phonon density of states F ph (ω). Note that α 2 F(ω) is not the product of two functions although sometimes one defines the function α 2 (ω) = α 2 F(ω)/F(ω) which should approximate the energy dependence of the strength of the EPI coupling. There are numerous experimental evidences in cuprates for the importance of the EPI scattering mechanism with a rather large coupling constant in the normal scattering channel 1 < λ ep 3, which will be discussed in detail below. In the EPI approach α 2 F ph (ω) is extracted from tunnelling measurements in conjunction with IR optical measurements. The HTSC cuprates are on the borderline and it is a natural question-under which condition can high T c be realized in the nonadiabatic limit A ≈ 1?
The second approach [12][13][14][15][16][17] assumes that EPI is too weak to be responsible for high T c in cuprates and it is based on a phenomenological model for spin-fluctuation interaction (SFI) as the dominating scattering mechanism, that is, it is a nonphononic mechanism. In this (phenomenological) approach the spectral function is proportional to the imaginary part of the spin susceptibility Im χ(k − k , ω), that is, α 2 F(k, k , ω) ∼ g 2 sf Im χ(k − k , ω) where g sf is the SFI coupling constant. NMR spectroscopy and magnetic neutron scattering give evidence that in HTSC cuprates χ(q, ω) is peaked at the antiferromagnetic wave vector Q = (π/a, π/a) and this property is favorable for d-wave pairing. The SFI theory roots basically on the strong electronic repulsion on Cu atoms, which is usually studied by the Hubbard model or its (more popular) derivative the t-J model. Regarding the possibility to explain high T c solely by strong correlations, as it is reviewed in [18], we stress two facts. First, at present there is no viable theory as well as experimental facts which can justify these (nonphononic) mechanisms of pairing with some exotic pairing mechanism such as RVB pairing [18], fractional statistics, anyon superconductivity, and so forth. Therefore we will not discuss these, in theoretical sense interesting approaches. Second, the central question in these nonphononic approaches is the following-do models based solely on the Hubbard Hamiltonian show up superconductivity at sufficiently high critical temperatures (T c ∼ 100 K)? Although the answer on this important question is not definitely settled, there are a number of numerical studies of these models which offer negative answers. For instance, the sign-free variational Monte Carlo algorithm in the 2D repulsive (U > 0) Hubbard model gives no evidence for superconductivity with high T c , neither the BCS-like nor the Berezinskii-Kosterlitz-Thouless-(BKT-) like [19]. At the same time, similar calculations show that there is a strong tendency to superconductivity in the attractive (U < 0) Hubbard model for the same strength of U, that is, at finite temperature in the 2D model with U < 0 the BKT superconducting transition is favored. Concerning the possibility of HTSC in the t-J model, various numerical calculations such as Monte Carlo calculations of the Drude spectral weight [20] and hightemperature expansion for the pairing susceptibility [21] give evidence that there is no superconductivity at temperatures characteristic for cuprates and if it exists T c must be rather low-few Kelvins. These numerical results tell us that the lack of high T c (even in 2D BKT phase) in the repulsive (U > 0) single-band Hubbard model and in the t-J model is not only due to thermodynamical 2D-fluctuations (which at finite T suppress and destroy superconducting phase coherence in large systems) but it is also mostly due to an inherent ineffectiveness of strong correlations to produce solely high T c in cuprates. These numerical results signal that the simple single-band Hubbard and its derivative the t-J model are insufficient to explain solely the pairing mechanism in cuprates and some additional ingredients must be included.
Since EPI is rather strong in cuprates, then it must be accounted for. As it will be argued in the following, the experimental support for the importance of EPI in cuprates comes from optics, tunnelling, and recent ARPES measurements [22,23]. It is worth mentioning that recent ARPES activity was a strong impetus for renewed experimental and theoretical studies of EPI in cuprates. However, in spite of accumulating experimental evidence for importance of EPI with λ ep > 1, there are occasionally reports which doubt its importance in cuprates. This is the case with recent interpretation of some optical measurements in terms of SFI only [24][25][26][27] and with the LDA-DFT (local density approximation-density functional theory) band-structure calculations [28,29], where both claim that EPI is negligibly small, that is, λ ep < 0.3. The inappropriateness of these statements will be discussed in the following sections.
The paper is organized as follows. In Section 1 we will mainly discuss experimental results in cuprates at and near optimal doping by giving also minimal theoretical explanations which are related to the bosonic spectral function α 2 F(ω) as well as to the transport spectral function α 2 tr F(ω) and their relations to EPI. The reason that we study only cuprates at and near optimal doping is that in these systems there are rather well-defined quasiparticlesalthough strongly interacting-while in highly underdoped systems the superconductivity is perplexed and possibly masked by other phenomena, such as pseudogap effects, formation of small polarons, interaction with spin and (possibly charge) order parameters, pronounced inhomogeneities of the scattering centers, and so forth. As the ARPES experiments confirm, there are no polaronic effects in systems at and near the optimal doping, while there are pronounced polaronic effects due to EPI in undoped and very underdoped HTSC [8][9][10][11]. In this work we consider mainly those direct one-particle and two-particle probes of low-energy quasiparticle excitations and scattering rates which give information on the structure of the spectral functions α 2 F(k, k , ω) and α 2 tr F(ω) in systems near optimal doping. These are angle-resolved photoemission (ARPES), various arts of tunnelling spectroscopy such as superconductor/insulator/normal metal (SIN) junctions, break junctions, scanning-tunnelling microscope spectroscopy (STM), infrared (IR) and Raman optics, inelastic neutron and Xray scattering, and so forth. We will argue that these direct probes give evidence for a rather strong EPI in cuprates. Some other experiments on EPI are also discussed in order to complete the arguments for the importance of EPI in cuprates. The detailed contents of Section 1 are the following. In Section 1.2 we discuss some prejudices related to the strength of EPI as well as on the Fermi-liquid behavior of Advances in Condensed Matter Physics 3 HTSC cuprates. We argue that any nonphononic mechanism of pairing should have very large bare critical temperature T c0 T c in the presence of the large EPI coupling constant, λ ep ≥ 1, if the EPI spectral function is weakly momentum dependent, that is, if α 2 F(k, k , ω) ≈ α 2 F(ω) like in lowtemperature superconductors. The fact that EPI is large in the normal state of cuprates and the condition that it must be conform with d-wave pairing imply that EPI in HTSC cuprates should be strongly momentum dependent. In Section 1.3 we discuss direct and indirect experimental evidences for the importance of EPI in cuprates and for the weakness of SFI in cuprates. These are the following.
(a) Magnetic Neutron Scattering Measurements. These measurements provide dynamic spin susceptibility χ(q, ω) which is in the SFI phenomenological approach [12][13][14][15][16][17] related to the Eliashberg spectral function, that is, α 2 F sf (k, k , ω) ∼ g 2 sf Im χ(q = k − k , ω). We stress that such an approach can be theoretically justified only in the weak coupling limit, g sf W b , where W b is the band width and g sf is the phenomenological SFI coupling constant. Here we discuss experimental results on YBCO which give evidence for strong rearrangement (with respect to ω) of Imχ(q, ω) (with q at and near Q = (π, π)) by doping toward the optimal doped HTSC [30,31]. It turns out that in the optimally doped cuprates with T c = 92.5 K Im χ(Q, ω) is drastically suppressed compared to that in slightly underdoped ones with T c = 91 K. This fact implies that the SFI coupling constant g sf must be small.
(b) Optical Conductivity Measurements. From these measurements one can extract the transport relaxation rate γ tr (ω) and indirectly an approximative shape of the transport spectral function α 2 tr F(ω). In the case of systems near optimal doping we discuss the following questions. (i) First is the physical and quantitative difference between the optical relaxation rate γ tr (ω) and the quasiparticle relaxation rate γ(ω). It was shown in the past that equating these two (unequal) quantities is dangerous and brings incorrect results concerning the quasiparticle dynamics in most metals by including HTSC cuprates too [3][4][5][6][32][33][34][35][36][37][38]. (ii) Second are methods of extraction of the transport spectral function α 2 tr F(ω). Although these methods give at finite temperature T a blurred α 2 tr F(ω) which is (due to the ill-defined methods) temperature dependent, it turns out that the width and the shape of the extracted α 2 tr F(ω) are in favor of EPI. (iii) Third is the restricted sum rule for the optical weight as a function of T which can be explained by strong EPI [39,40]. (iv) Fourth is the good agreement with experiments of the Tdependence of the resistivity ρ(T) in optimally doped YBCO, where ρ(T) is calculated by using the spectral function from tunnelling experiments. Recent femtosecond time-resolved optical spectroscopy in La 2−x Sr x CuO 4 which gives additional evidence for importance of EPI [41] will be shortly discussed.
(c) ARPES Measurements and EPI. From these measurements the self-energy Σ(k, ω) is extracted as well as some properties of α 2 F(k, k , ω). Here we discuss the following items: (i) the existence of the nodal and antinodal kinks in optimally and slightly underdoped cuprates, as well as the structure of the ARPES self-energy (Σ(k, ω)) and its isotope dependence, which are all due to EPI; (ii) the appearance of different slopes of Σ(k, ω) at low (ω ω ph ) and high energies (ω ω ph ) which can be explained by the strong EPI; (iii) the formation of small polarons in the undoped HTSC which was interpreted to be due to strong EPI-this gives rise to phonon side bands which are clearly seen in ARPES of undoped HTSC [10,11].
(d) Tunnelling Spectroscopy. It is well known that this method is of an immense importance in obtaining the spectral function α 2 F(ω) from tunnelling conductance. In this part we discuss the following items: (i) the extracted Eliashberg spectral function α 2 F(ω) with the coupling constant λ (tun) = 2-3.5 from the tunnelling conductance of break-junctions in optimally doped YBCO and Bi-2212 [42][43][44][45][46][47][48][49][50][51][52][53][54][55] which gives that the maxima of α 2 F(ω) coincide with the maxima in the phonon density of states F ph (ω); (ii) the existence of eleven peaks in −d 2 I/dV 2 in superconducting La 1.84 Sr 0.16 CuO 4 films [56], where these peaks match precisely with the peaks in the intensity of the existing phonon Raman scattering data [57]; (iii) the presence of the dip in dI/dV in STM which shows the pronounced oxygen isotope effect and important role of these phonons.
(e) Inelastic Neutron and X-Ray Scattering Measurements. From these experiments one can extract the phonon density of state F ph (ω) and in some cases the strengths of the quasiparticle coupling with various phonon modes. These experiments give sufficient evidence for quantitative inadequacy of LDA-DFT calculations in HTSC cuprates. Here we argue that the large softening and broadening of the half-breathing Cu-O bond-stretching phonon, of the apical oxygen phonons and of the oxygen B 1g buckling phonons (in LSCO, BSCO, YBCO), cannot be explained by LDA-DFT. It is curious that the magnitude of the softening can be partially obtained by LDA-DFT but the calculated widths of some important modes are an order of magnitude smaller than the neutron scattering data show. This remarkable fact confirms that additionally the inadequacy of LDA-DFT in strongly correlated systems and a more sophisticated manybody theory for EPI is needed. The problem of EPI will be discussed in more details in Section 2.
In Section 1.4 brief summary of Section 1 is given. Since we are dealing with the electron-boson scattering in cuprates near the optimal doping, then in Appendix A (and in Section 2) we introduce the reader briefly to the Migdal-Eliashberg theory for superconductors (and normal metals) where the quasiparticle spectral function α 2 F(k, k , ω) and the transport spectral function α 2 tr F(ω) are defined. Finally, one can pose a question-do the experimental results of the above enumerated spectroscopic methods allow a building of a satisfactory and physically reasonable microscopic theory for basic scattering and pairing mechanism in cuprates? The posed question is very modest compared to the much stringent request for the theory of everythingwhich would be able to explain all properties of HTSC materials. Such an ambitious project is not realized even in those low-temperature conventional superconductors where it is definitely proved that in most materials the pairing is due to EPI and many properties are well accounted for by the Migdal-Eliashberg theory. For an illustration, let us mention only two examples. First, the experimental value 4 Advances in Condensed Matter Physics for the coherence peak in the microwave response σ s (T < T c , ω = const) at ω = 17 GHz in the superconducting Nb is much higher than the theoretical value obtained by the strong coupling Eliashberg theory [58]. So to say, the theory explains the coherence peak at 17 GHz in Nb qualitatively but not quantitatively. However, the measurements at higher frequency ω ∼ 60 GHz are in agreement with the Eliashberg theory [59]. Then one can say that instead of the theory of everything we deal with a satisfactory theory, which allows us qualitative and in many aspects quantitative explanation of phenomena in superconducting state. Second example is the experimental boron (B) isotope effect in MgB 2 (T c ≈ 40 K) which is smaller than the theoretical value, that is, α exp B ≈ 0.3 < α th B = 0.5, although the pairing is due to EPI for boron vibrations [60]. Since the theory of everything is impossible in the complex materials such as HTSC cuprates in Section 1, we will not discuss those phenomena which need much more microscopic details and/or more sophisticated many-body theory. These are selected by chance: (i) large ratio 2Δ/T c which is on optimally doped YBCO and BSCO ≈ 5 and 7, respectively, while in underdoped BSCO one has even (2Δ/T c ) ≈ 20; (ii) peculiarities of the coherence peak in the microwave response σ(T) in HTSC cuprates, which is peaked at T much smaller than T c , contrary to the case of LTSC where it occurs near T c ; (iii) the dependence of T c on the number of CuO 2 in the unit cell; (iv) the temperature dependence of the Hall coefficient; (v) distribution of states in the vortex core, and so forth.
The microscopic theory of the mechanism for superconducting pairing in HTSC cuprates will be discussed in Section 2. In Section 2.1 we introduce an ab initio manybody theory of superconductivity which is based on the fundamental (microscopic) Hamiltonian and the manybody technique. This theory can in principle calculate measurable properties of materials such as the critical temperature T c , the critical fields, the dynamic and transport properties, and so forth. However, although this method is in principle exact, which needs only some fundamental constants e, D, m e , M ion , k B and the chemical composition of superconducting materials, it was practically never realized in practice due to the complexity of many-body interactions-electron-electron and electron-lattice-as well as of structural properties. Fortunately, the problem can be simplified by using the fact that superconductivity is a lowenergy phenomenon characterized by the very small energy parameters (T c /E F , Δ/E F , ω ph /E F ) 1. It turns out that one can integrate high-energy electronic processes (which are not changed by the appearance of superconductivity) and then solve the low-energy problem by the (so-called) strongcoupling Migdal-Eliashberg theory. It turns out that in such an approach the physics is separated into the following: (1) the solution of the ideal band-structure Hamiltonian with the nonlocal exact crystal potential (sometimes called the excitation potential) V IBS (r, r ) (IBS-the ideal band structure) which includes the static self-energy (Σ (h) c0 (r, r , ω = 0)) due to high-energy electronic processes, that is, V IBS (r, r ) = [V e-i (r)+V H (r)]δ(r−r )+Σ (h) c0 (r, r , ω = 0), with V e-i and V H being the electron-ion and Hartree potential, respectively; (2) solving the low-energy Eliashberg equations. However, the calculation of the (excited) potential V IBS (r, r ) and the real EPI coupling g ep (r, r ) = δV IBS (r, r )/δR n , which include high-energy many-body electronic processes-for instance, the large Hubbard U effects-is extremely difficult at present, especially in strongly correlated systems such as HTSC cuprates. Due to this difficulty the calculations of the EPI coupling in the past were usually based on the LDA-DFT method which will be discussed in Section 2.2 in the contest of HTSC cuprates, where the nonlocal potential is replaced by the local potential V LDA (r)-the ground-state potential-and the real EPI coupling by the "local" LDA one g ep (r) = δV LDA (r)/δR n . Since the exchange-correlation effects enter V LDA (r) = V e-i (r) + V H (r) + V XC (r) via the local exchange-correlation potential V XC (r), it is clear that the LDA-DFT method describes strong correlations scarcely and it is inadequate in HTSC cuprates (and other strongly correlated systems such as heavy fermions) where one needs an approach beyond the LDA-DFT method. In Section 2.3 we discuss a minimal theoretical model for HTSC cuprates which takes into account minimal number of electronic orbitals and strong correlations in a controllable manner [6]. This theory treats the interplay of EPI and strong correlations in systems with finite doping in a systematic and controllable way. The minimal model can be further reduced (in some range of parameters) to the single-band t-J model, which allows the approximative calculation of the excited potential V IBS (r, r ) and the nonlocal EPI coupling g ep (r, r ). As a result one obtains the momentum-dependent EPI coupling g ep (k F , q) which is for small hole-doping (δ < 0.3) strongly peaked at small transfer momenta-the forward scattering peak. In the framework of this minimal model it is possible to explain some important properties and resolve some puzzling experimental results, like the following, for instance. (a) Why is d-wave pairing realized in the presence of strong EPI? (b) Why is the transport coupling constant (λ tr ) rather smaller than the pairing one λ, that is, λ tr λ/3? (c) Why is the mean-field (one-body) LDA-DFT approach unable to give reliable values for the EPI coupling constant in cuprates and how many-body effects can help? (d) Why is d-wave pairing robust in the presence of nonmagnetic impurities and defects? (e) Why are the ARPES nodal and antinodal kinks differently renormalized in the superconducting states, and so forth? In spite of the encouraging successes of this minimal model, at least in a qualitative explanation of numerous important properties of HTSC cuprates, we are at present stage rather far from a fully microscopic theory of HTSC cuprates which is able to explain high T c . In that respect at the end of Section 2.3 we discuss possible improvements of the present minimal model in order to obtain at least a semiquantitative theory for HTSC cuprates.
Finally, we would like to point out that in real HTSC materials there are numerous experimental evidences for nanoscale inhomogeneities. For instance, recent STM experiments show rather large gap dispersion, at least on the surface of BSCO crystals [61][62][63], giving rise to a pronounced inhomogeneity of the superconducting order parameter Δ(k, R), where k is the relative momentum of the Cooper Advances in Condensed Matter Physics 5 pair and R is the center of mass of Cooper pairs. One possible reason for the inhomogeneity of Δ(k, R) and disorder on the atomic scale can be due to extremely high doping level of ∼ (10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)% in HTSC cuprates which is many orders of magnitude larger than in standard semiconductors (10 21 versus 10 15 carrier concentration). There are some claims that high T c is exclusively due to these inhomogeneities (of an extrinsic or intrinsic origin) which may effectively increase pairing potential [64], while some others try to explain high T c solely within the inhomogeneous Hubbard or t-J model. Here we will not discuss this interesting problem but mention only that the concept of T c increase by inhomogeneity is not well-defined, since the increase of T c is defined with respect to the average value T c . However, T c is experimentally not well defined quantity and the hypothesis of an increase of T c by material inhomogeneities cannot be tested at all. In studying and analyzing HTSC cuprates near optimal doping we assume that basic effects are realized in nearly homogeneous systems and inhomogeneities are of secondary role, which deserve to be studied and discussed separately.

EPI versus Nonphononic Mechanisms.
Concerning the high T c values in cuprates, two dilemmas have been dominating after its discovery: (i) which interaction is responsible for strong quasiparticle scattering in the normal state? This question is related also to the dilemma of Fermi versus non-Fermi liquid; (ii) What is the mediating (gluing) boson responsible for the superconducting pairing, that is, phonons or nonphonons? In the last twenty-three years, the scientific community was overwhelmed by numerous proposed pairing mechanisms, most of which are hardly verifiable in HTSC cuprates.
(1) Fermi versus Non-Fermi Liquid in Cuprates. After discovery of HTSC in cuprates there was a large amount of evidence on strong scattering of quasiparticles which contradicts the canonical (popular but narrow) definition of the Fermi liquid, thus giving rise to numerous proposals of the so called non-Fermi liquids, such as Luttinger liquid, RVB theory, marginal Fermi liquid, and so forth. In our opinion there is no need for these radical approaches in explaining basic physics in cuprates at least in optimally, slightly underdoped and overdoped metallic and superconducting HTSC cuprates. Here we give some clarifications related to the dilemma of Fermi versus non-Fermi liquid. The definition of the canonical Fermi liquid (based on the Landau work) in interacting Fermi systems comprises the following properties: (1) there are quasiparticles with charge q = ±e, spin s = 1/2, and low-energy excitations ξ k (= k − μ) which are much larger than their inverse life-times, that is, ξ k 1/τ k ∼ ξ 2 k /W b . Since the level width Γ = 2/τ k of the quasiparticle is negligibly small, this means that the excited states of the Fermi liquid are placed in one-to-one correspondence with the excited states of the free Fermi gas; (2) at T = 0 K there is an energy level ξ kF = 0 which defines the Fermi surface on which the Fermi quasiparticle distribution function n F (ξ k ) has finite jump at k F ; (3) the number of quasiparticles under the Fermi surface is equal to the total number of conduction particles (we omit here other valence and core electrons)-the Luttinger theorem; (4) the interactions between quasiparticles are characterized by the set of Landau parameters which describe the low-temperature thermodynamics and transport properties. Having this definition in mind one can say that if fermionic quasiparticles interact with some bosonic excitation, for instance, with phonons, and if the coupling is sufficiently strong, then the former are not described by the canonical Fermi liquid since at energies and temperatures of the order of the characteristic (Debye) temperature k B Θ D (≡ Dω D ) (for the Debye spectrum ∼ Θ D /5), that is, for ξ k ∼ Θ D , one has τ −1 k ξ k and the quasiparticle picture (in the sense of the Landau definition) is broken down. In that respect an electron-boson system can be classified as a noncanonical Fermi liquid for sufficiently strong electron-boson coupling. It is nowadays well known that, for instance, Al, Zn are weak coupling systems since for ξ k ∼ Θ D one has τ −1 k ξ k and they are well described by the Landau theory. However, in (the noncanonical) cases where for higher energies ξ k ∼ Θ D one has τ −1 k ξ k , the electron-phonon system is satisfactory described by the Migdal-Eliashberg theory and the Boltzmann theory, where thermodynamic and transport properties depend on the spectral function α 2 F sf (k, k , ω) and its higher momenta. Since in HTSC cuprates the electron-boson (phonon) coupling is strong and T c is large, then it is natural that in the normal state (at T > T c ) we deal with a strong interacting noncanonical Fermi liquid which is for modest nonadiabaticity parameter A < 1 described by the Migdal-Eliashberg theory, at least qualitatively and semiquantitatively. In order to justify this statement we will in the following elucidate some properties in more details by studying optical, ARPES, tunnelling and other experiments in HTSC oxides.
(2) Is There Limit of the EPI Strength? In spite of the reached experimental evidence in favor of strong EPI in HTSC oxides, there was a disproportion in the research activity (especially theoretical) in the past, since the investigation of the SFI mechanism of pairing prevailed in the literature. This trend was partly due to an incorrect statement in [65,66] on the possible upper limit of T c in the phonon mechanism of pairing. Since in the past we have discussed this problem thoroughly in numerous papers-for the recent one see [67]-we will outline here the main issue and results only.
It is well known that in an electron-ion crystal, besides the attractive EPI, there is also repulsive Coulomb interaction. In case of an isotropic and homogeneous system with weak quasiparticle interaction, the effective potential V eff (k, ω) in the leading approximation looks like as for two external charges (e) embedded in the medium with the total longitudinal dielectric function ε tot (k, ω) (k is the momentum and ω is the frequency) [68,69], that is, In case of strong interaction between quasiparticles, the state of embedded quasiparticles changes significantly due 6 Advances in Condensed Matter Physics to interaction with other quasiparticles, giving rise to V eff (k, ω) / = 4πe 2 /k 2 ε tot (k, ω). In that case V eff depends on other (than ε tot (k, ω)) response functions. However, in the case when (1) holds, that is, when the weak-coupling limit is realized, T c is given by T c ≈ ω exp(−1/(λ ep − μ * )) [68][69][70]. Here, λ ep is the EPI coupling constant, ω is an average phonon frequency, and μ * is the Coulomb pseudopotential, μ * = μ/(1 + μ ln E F /ω) (E F is the Fermi energy). The couplings λ ep and μ are expressed by ε tot (k, ω = 0): where N(0) is the density of states at the Fermi surface and k F is the Fermi momentum-see more in [3][4][5]. In [65,66] it was claimed that the lattice stability of the system with respect to the charge density wave formation implies the condition ε tot (k, ω = 0) > 1 for all k. If this were correct, then from (2) it would follow that μ > λ ep , which limits the maximal value of T c to the value T max In typical metals E F < (1-10) eV, and if one accepts the statement in [65,66] that λ ep ≤ μ(≤0.5), one obtains T c ∼ (1-10) K. The latter result, if it would be correct, means that EPI is ineffective in producing not only high-T c superconductivity but also low-temperature superconductivity (LTS with T c 20 K). However, this result is in conflict first of all with experimental results in LTSC, where in numerous systems one has μ ≤ λ ep and λ ep > 1. For instance, λ ep ≈ 2.6 is realized in PbBi alloy which is definitely much higher than μ(<1), and so forth. Moreover, the basic theory tells us that ε tot (k / = 0, ω) is not the response function [68,69] (contrary to the assumption in [65,66]). Namely, if a small external potential δV ext (k, ω) is applied to the system (of electrons and ions in solids), it induces screening by charges of the medium and the total potential is given by δV tot (k, ω) = δV ext (k, ω)/ε tot (k, ω), which means that 1/ε tot (k, ω) is the response function. The latter obeys the Kramers-Kronig dispersion relation which implies the following stability condition [68,69]: that is, either or This important theorem invalidates the restriction on the maximal value of T c in the EPI mechanism given in [65,66]. We stress that the condition ε tot (k / = 0, ω = 0) < 0 is not in conflict with the lattice stability at all. For instance, in inhomogeneous systems such as crystal, the total longitudinal dielectric function is matrix in the space of reciprocal lattice vectors (Q), that is, ε tot (k + Q, k + Q , ω), and ε tot (k, ω) is defined by ε −1 tot (k, ω) = ε −1 tot (k + 0, k + 0, ω). In dense metallic systems with one ion per cell (such as metallic hydrogen) and with the electronic dielectric function ε el (k, 0), the macroscopic total dielectric function ε tot (k, 0) is given by [71][72][73] At the same time the energy of the longitudinal phonon ω l (k) is given by where Ω 2 p is the ionic plasma frequency, and G ep is the local (electric) field correction-see [71][72][73]. The right condition for lattice stability requires that ω 2 l (k) > 0, which implies that for ε el (k, 0) > 0 one has ε el (k, 0)G ep (k) < 1. The latter condition gives automatically ε tot (k, 0) < 0. Furthermore, the calculations [71][72][73] show that in the metallic hydrogen (H) crystal ε tot (k, 0) < 0 for all k / = 0. Note that in metallic H the EPI coupling constant is very large, that is, λ ep ≈ 7 and T c may reach very large value T c ≈ 600 K [74]. Moreover, the analyses of crystals with more ions per unit cell [71][72][73] give that ε tot (k / = 0, 0) < 0 is more a rule than an exception-see Figure 1. The physical reason for ε tot (k / = 0, 0) < 0 is local field effects described by G ep (k). Whenever the local electric field E loc acting on electrons (and ions) is different from the average electric field E, that is, E loc / = E, there are corrections to ε tot (k, 0) which may lead to ε tot (k, 0) < 0.
The above analysis tells us that in real crystals ε tot (k, 0) can be negative in the large portion of the Brillouin zone thus giving rise to λ ep − μ > 0 in (2). This means that analytic properties of the dielectric function ε tot (k, ω) do not limit T c in the phonon mechanism of pairing. This result does not mean that there is no limit on T c at all. We mention in advance that the local field effects play important role in HTSC cuprates, due to their layered structure with very unusual ionic-metallic binding, thus opening a possibility for large EPI.
In conclusion, we point out that there are no serious theoretical and experimental arguments for ignoring EPI in HTSC cuprates. To this end it is necessary to answer several important questions which are related to experimental findings in HTSC cuprates. (1) If EPI is important for pairing in HTSC cuprates and if superconductivity is of d-wave type, how are these two facts compatible? (2) Why is the transport EPI coupling constant λ tr (entering resistivity) rather smaller than the pairing EPI coupling constant λ ep (>1) (entering T c ), that is, why one has λ tr (≈0. 6-1.4) λ ep (∼2-3.5)? (3) If EPI is ineffective for pairing in HTSC oxides, in spite of λ ep > 1, why is it so?
(3) Is a Nonphononic Pairing Realized in HTSC? Regarding EPI one can pose a question about whether it contributes significantly to d-wave pairing in cuprates. Surprisingly, despite numerous experiments in favor of EPI, there is a belief that EPI is irrelevant for pairing [12][13][14][15][16][17]. This belief is mainly based, first, on the above discussed incorrect lattice stability criterion related to the sign of ε tot (k, 0), which implies small EPI and, second, on the well-established experimental fact that d-wave pairing is realized in cuprates [75], which is believed to be incompatible with EPI. Having in mind that EPI in HTSC at and near optimal doping is strong with 2 < λ ep < 3.5 (see below), we assume for the moment that the leading pairing mechanism in cuprates, which gives d-wave pairing, is due to some nonphononic mechanism. For instance, let us assume an exitonic mechanism due to the high-energy pairing boson (Ω nph ω ph ) and with the bare critical temperature T c0 and look for the effect of EPI on T c . If EPI is approximately isotropic, like in most LTSC materials, then it would be very detrimental for dwave pairing. In the case of dominating isotropic EPI in the normal state and the exitonic-like pairing, then near T c the linearized Eliashberg equations have an approximative form for a weak nonphonon interaction (with the large characteristic frequency Ω nph ) For pure d-wave pairing with the pairing potential and the equation for T c -see [3][4][5] ln Here Ψ is the di-gamma function. At temperatures near T c one has Γ ep ≈ 2πλ ep T c and the solution of (9) is approx- This means that for T max c ∼ 160 K and λ ep > 1 the bare T c0 due to the nonphononic interaction must be very large, that is, T c0 > 500 K.
Concerning other nonphononic mechanisms, such as the SFI one, the effect of EPI in the framework of Eliashberg equations was studied numerically in [76]. The latter is based on (A.1) in Appendix A with the kernels in the normal and superconducting channels λ Z kp (iν n ) and λ Δ kp , respectively. Usually, the spin-fluctuation kernel λ sf,kp (iν n ) is taken in the FLEX approximation [77]. The calculations [76] confirm the very detrimental effect of the isotropic (k-independent) EPI on d-wave pairing due to SFI. For the bare SFI critical temperature T c0 ∼ 100 K and for λ ep > 1 the calculations give very small (renormalized) critical temperature T c 100 K. These results tell us that a more realistic pairing interaction must be operative in cuprates and that EPI must be strongly momentum dependent and peaked at small transfer momenta [78][79][80]. Only in that case does strong EPI conform with dwave pairing, either as its main cause or as a supporter of a nonphononic mechanism. In Section 2 we will argue that the strongly momentum-dependent EPI is important scattering mechanism in cuprates providing the strength of the pairing mechanism, while the residual Coulomb interaction (by including weaker SFI) triggers it to d-wave pairing.

Experimental Evidence for Strong EPI.
In the following we discuss some important experiments which give evidence for strong electron-phonon interaction (EPI) in cuprates. However, before doing it, we will discuss some indicative inelastic magnetic neutron scattering (IMNS) measurements in cuprates whose results in fact seriously doubt in the effectiveness of the phenomenological SFI mechanism of pairing which is advocated in [12][13][14][15][16][17]81]. First, the experimental results related to the pronounced imaginary part of the susceptibility Im χ(k, k z , ω) in the normal state at and near the AF wave vector k = Q = (π, π) were interpreted in a number of papers as a support for the SFI mechanism for pairing [12][13][14][15][16][17]81]. Second, the existence of the so called magnetic resonance peak of Im χ(k, k z , ω) (at some energies ω < 2Δ) in the superconducting state was also interpreted in a number of papers either as the origin of superconductivity or as a mechanism strongly affecting superconducting gap at the antinodal point. Im χ(k, ω) we point out that in the (phenomenological) theories based on the spin-fluctuation interaction (SFI) the quasiparticle self-energy Σ sf (k, ω n ) (ω n is the Matsubara frequency and τ 0 is the Nambu matrix) in the normal and superconducting state and the effective (repulsive) pairing potential V sf (k, ω) (where iω n → ω + iη) are assumed in the form [12][13][14][15][16][17] where ω − nm ≡ ω n − ω m . Although the form of V sf cannot be justified theoretically, except in the weak coupling limit (g sf W b ) only, it is often used in the analysis of the quasiparticle properties in the normal and superconducting state of cuprates where the spin susceptibility (spectral function) Im χ(q, ω) is strongly peaked at and near the AF wave vector Q = (π/a, π/a).
Can the pairing mechanism in HTSC cuprates be explained by such a phenomenology and what is the prise for it is? The best answer is to look at the experimental results related to the inelastic magnetic neutron scattering (IMNS) which gives Im χ(q, ω). In that respect very indicative and impressive IMNS measurements on YBa 2 Cu 3 O 6+x , which are done by Bourges group [30], demonstrate that the normalstate susceptibility Im χ (odd) (q, ω) (the odd part of the spin susceptibility in the bilayer system) at q = Q = (π, π) is strongly dependent on the hole-doping as it is shown in Figure 2.
The most pronounced result for our discussion is that by varying doping there is a huge rearrangement of Im χ (odd) (Q, ω) in the normal state, especially in the energy (frequency) region which might be important for superconducting pairing, let us say 0 meV < ω < 60 meV. This is clearly seen in the last two curves in Figure 2 where this rearrangement is very pronounced, while at the same time there is only small variation of the critical temperature T c . It is seen in Figure 2 that in the underdoped YBa 2 Cu 3 O 6.92 crystal Im χ (odd) (Q, ω) and S(Q) = N(0)g 2 sf 60 0 dω Im χ (odd) (Q, ω) are much larger than that in the near optimally doped YBa 2 Cu 3 O 6.97 , that is, one has S 6.92 (Q) S 6.97 (Q), although the difference in the corresponding critical temperatures T c is very small, that is, T (6.92) c = 91 K (in YBa 2 Cu 3 O 6.92 ) and T (6.97) c = 92.5 K (in YBa 2 Cu 3 O 6.97 ). This pronounced rearrangement and suppression of Im χ (odd) (Q, ω) in the normal state of YBCO by doping (toward the optimal doping) but with the negligible change in T c is strong evidence that the SFI pairing mechanism is not the dominating one in HTSC cuprates. This insensitivity of T c , if interpreted in terms of the SFI coupling constant λ sf (∼ g 2 sf ), means that the latter is small, that is, λ (exp) sf 1. We stress that the explanation of high T c in cuprates by the SFI phenomenological theory [12][13][14][15][16][17] assumes very large SFI coupling energy with g (th) sf ≈ 0.7 eV while the frequency (energy) dependence of Im χ(Q, ω) is extracted from the fit of the NMR relaxation rate T −1 1 which gives T (NMR) c ≈ 100 K [12][13][14][15][16][17]. To this point, the NMR measurements (of T −1 1 ) give that there is an anticorrelation between the decrease of the NMR spectral function I Q = lim ω → 0 Im χ (NMR) (Q, ω)/ω and the increase of T c by increasing doping toward the optimal one-see [6] and references therein. The latter result additionally disfavors the SFI model of pairing [12][13][14][15][16][17] since the strength of pairing interaction is little affected by SFI. Note that if instead of taking Im χ(Q, ω) from NMR measurements one takes it from IMNS measurements, as it was done in [82], than for the same value g (th) sf one obtains much smaller T c . For instance, by taking the experimental Advances in Condensed Matter Physics 9 values for Im χ (IMNS) (Q, ω) in underdoped YBa 2 Cu 3 O 6.6 with T c ≈ 60 K one obtains T (IMNS) c < T (NMR) c /3 [82], while T (IMNS) c → 50 K for g (th) sf 1. The situation is even worse if one tries to fit the resistivity with Im χ (IMNS) (Q, ω) in YBa 2 Cu 3 O 6.6 since this fit gives T (IMNS) c < 7 K. These results point to a deficiency of the SFI phenomenology (at least that based on (10)) to describe pairing in HTSC cuprates.
Having in mind the results in [82], the recent theoretical interpretation in [81] of IMNS experiments [83,84] and ARPES measurements [85,86] on the underdoped YBa 2 Cu 3 O 6.6 in terms of the SFI phenomenology deserve to be commented. The IMNS experiments [83,84] give evidence for the "hourglass" spin excitation spectrum (in the superconducting state) for the momenta q at, near and far from Q, which is richer than the common spectrum with magnetic resonance peaks measured at Q. In [81] the self-energy of electrons due to their interaction with spin excitations is calculated by using (10) with g 2 sf = (3/2) U 2 and Im χ(q, ω) taken from [83,84]. However, in order to fit the ARPES self-energy and low-energy kinks (see discussion in Section 1.3.3) the authors of [81] use very large value U = 1.59 eV, that is, much larger than the one used in [82]. Such a large value of U has been obtained earlier within the Monte Carlo simulation of the Hubbard model [87]. In our opinion this value for U is unrealistically large in the case of strongly correlated systems where spin fluctuations are governed by the effective electron-exchange interaction J Cu-Cu 0.15 eV [88]. This implies that U 1 eV and T c 60 K. Note that this value for J Cu-Cu (∼0.15 eV) comes out also from the theory of strongly correlated electrons in the three-band Emery model which gives J Cu-Cu ≈ [4t 4 pd /(Δ dp + U pd ) 2 , (1/U d )+2/(U p +2Δ)]-for parameters see Section 2.3. We would like to emphasize here that an additional richness of the spin-fluctuations spectrum (the hourglass instead of the spin resonance) does not change the situation with the smallness of the exchange coupling constant U (and g sf ).
Concerning the problem related to the rearrangement of the SFI spectral function Im χ(Q, ω) in YB 2 Cu 3 O 6+x [30] we would like to stress that despite the fact that the latter results were obtained ten years ago they are not disputed by the new IMNS measurements [31] on high quality samples of the same compound (where much longer counting times were used in order to reduce statistical errors). In fact the results in [30] are confirmed in [31] where the magnetic intensity I(q, ω)(∼ Im χ(q, ω)) (for q at and in the broad range of Q) for the optimally doped YBa 2 Cu 3 O 6.95 (with T c = 93 K) is at least three times smaller than in the underdoped YBa 2 Cu 3 O 6.6 with T c = 60 K. This result is again very indicative sign of the weakness of SFI since such a huge reconstruction would decrease T c in the optimally doped YBa 2 Cu 3 O 6.95 if analyzed in the framework of the phenomenological SFI theory based on (10). It also implies that due to the suppression of Im χ(q, ω) by increasing doping toward the optimal one a straightforward extrapolation of the theoretical approach in [81] to the explanation of T c in the optimally doped YBa 2 Cu 3 O 6.95 would require an increase of U to the value even larger than 4 eV, which is highly improbable.
(b) Ineffectiveness of the Magnetic Resonance Peak. A less direct argument for smallness of the SFI coupling constant, that is, g exp sf ≤ 0.2 eV and g exp sf g sf , comes from other experiments related to the magnetic resonance peak in the superconducting state, and this will be discussed next. In the superconducting state of optimally doped YBCO and BSCO, Im χ(Q, ω) is significantly suppressed at low frequencies except near the resonance energy ω res ≈ 41 meV where a pronounced narrow peak appears-the magnetic resonance peak. We stress that there is no magnetic resonance peak in some families of HTSC cuprates, for instance, in LSCO, and consequently one can question the importance of the resonance peak in the scattering processes. The experiments tell us that the relative intensity of this peak (compared to the total one) is small, that is, I 0 ∼ (1-5)%-see Figure 3. In underdoped cuprates this peak is present also in the normal state as it is seen in Figure 2.
After the discovery of the resonance peak there were attempts to relate it, first, to the origin of the superconducting condensation energy and, second, to the kink in the energy dispersion or the peak-dimp structure in the ARPES spectral function. In order that the condensation energy is due to the magnetic resonance, it is necessary that the peak intensity I 0 is small [89]. I 0 is obtained approximately by equating the condensation energy E con ≈ N(0)Δ 2 /2 with the change of the magnetic energy E mag in the superconducting state, that is, δE mag ≈ 4I 0 · E mag : where S(k, ω) = (1/π)[1 + n(ω)] Im χ(k, ω) is the spin structure factor and n(ω) is the Bose distribution function. By taking Δ ≈ 2T c and the realistic value N(0) ∼ 1/(10J) ∼ 1 states/eV · spin, one obtains I 0 ∼ 10 −1 (T c /J) 2 ∼ 10 −3 . However, such a small intensity cannot be responsible for the anomalies in ARPES and optical spectra since it gives rise to small coupling constant λ sf,res for the interaction of holes with the resonance peak, that is, λ sf,res ≈ (2I 0 N(0)g 2 sf /ω res ) 1. Such a small coupling does not affect superconductivity at all. Moreover, by studying the width of the resonance peak one can extract an order of magnitude of the SFI coupling constant g sf . Since the magnetic resonance disappears in the normal state of the optimally doped YBCO, it can be qualitatively understood by assuming that its broadening scales with the resonance energy ω res , that is, γ res < ω res , where the line width is given by γ res = 4π(N(0)g sf ) 2 ω res [89]. This condition limits the SFI coupling to g sf < 0.2 eV. We stress that in such a way obtained g sf is much smaller (at least by factor three) than that assumed in the phenomenological spin-fluctuation theory [12][13][14][15][16][17]81] where g sf ∼ 0.6-0.7 eV and U ≈ 1.6 eV, but much larger than estimated in [89] (where g sf < 0.02 eV). The smallness of g sf comes out also from the analysis of the antiferromagnetic state in underdoped metals of LSCO and YBCO [90], where the small (ordered) magnetic moment μ(<0.1μ B ) points to an itinerant antiferromagnetism with small coupling constant g sf < 0.2 eV. The conclusion from this analysis is that in the optimally doped YBCO the sharp magnetic resonance is a consequence of the onset of superconductivity and not its cause. There is also one principal reason against the pairing due to the magnetic resonance peak at least in optimally doped cuprates. Since the intensity of the magnetic resonance near T c is vanishingly small, though not affecting pairing at the second-order phase transition at T c , then, if it would be solely the origin for superconductivity, the phase transition at T c would be first order, contrary to experiments. Recent ARPES experiments give evidence that the magnetic resonance cannot be related to the kinks in ARPES spectra [91,92]-see the discussion below.
Finally, we would like to point out that the recent magnetic neutron scattering measurements on optimally doped large-volume crystals Bi 2 Sr 2 CaCu 2 O 8+δ [93], where the absolute value of Im χ(q, ω) is measured, are questioning also the interpretation of the electronic magnetism in cuprates in terms of the itinerant magnetism. This experiment shows a lack of temperature dependence of the local spin susceptibility Im χ(ω) = q Im χ(q, ω) across the superconducting transition T c = 91 K, that is, there is only a minimal change in Im χ(ω) between 10 K and 100 K. Note that if the magnetic excitations were due to itinerant quasiparticles we should have seen dramatic changes of Im χ(ω) as a function of T over the whole energy range. This T-independence of Im χ(ω) strongly opposes the theoretical results in [24][25][26][27] which assume that the bosonic spectral function is proportional to Im χ(ω) and that the former can be extracted from optic measurements. Namely, the fitting procedure in [24][25][26][27] gives that Im χ(ω) is strongly Tdependent contrary to the experimental results in [93]-see more in Section 1.3.2 on optical conductivity.

Optical Conductivity and EPI.
Optical spectroscopy gives information on optical conductivity σ(ω) and on twoparticle excitations, from which one can indirectly extract the transport spectral function α 2 tr F(ω). Since this method probes bulk sample (on the skin depth), contrary to ARPES and tunnelling methods which probe tiny regions (10-15Å) near the sample surface, this method is indispensable. However, one should be careful not to overinterpret the experimental results since σ(ω) is not a directly measured quantity but it is derived from the reflectivity R(ω) = |( ε ii (ω) − 1)/( ε ii (ω) + 1)| 2 with the transversal dielectric tensor ε ii (ω) = ε ii,∞ + ε ii,latt + 4πiσ ii (ω)/ω. Here, ε ii,∞ is the high-frequency dielectric function, ε ii,latt describes the contribution of the lattice vibrations, and σ ii (ω) describes the optical (dynamical) conductivity of conduction carriers. Since R(ω) is usually measured in the limited-frequency interval ω min < ω < ω max , some physical modelling for R(ω) is needed in order to guess it outside this range-see more in reviews in [3][4][5][6]. This was the reason for numerous misinterpretations of optic measurements in cuprates, which will be uncovered below. An illustrative example for this claim is large dispersion in the reported value of ω pl -from 0.06 to 25 eV-that is, almost three orders of magnitude. However, it turns out that IR measurements of R(ω) in conjunction with elipsometric measurements of ε ii (ω) at high frequencies allow more reliable determination of σ(ω) [94].
(1) Transport and Quasiparticle Relaxation Rates. The widespread misconception in studying the quasiparticle scattering in cuprates was an ad hoc assumption that the transport relaxation rate γ tr (ω) is equal to the quasiparticle relaxation rate γ(ω), in spite of the well-known fact that the inequality γ tr (ω) / = γ(ω) holds in a broad-frequency (energy) region Allen. This (incorrect) assumption was one of the main arguments against the relevance of the EPI scattering mechanism in cuprates. Although we have discussed this problem several times before, we do it again due to the importance of this subject.

11
The dynamical conductivity σ(ω) consists of two parts, that is, σ(ω) = σ inter (ω) + σ intra (ω) where σ inter (ω) describes interband transitions which contribute at higher than intraband energies, while σ intra (ω) is due to intraband transitions which are relevant at low energies ω < (1-2) eV. (Note that in the IR measurements the frequency is usually given in cm −1 , where the following conversion holds: 1 cm −1 = 29.98 GHz = 0.123985 meV = 1.44 K.) The experimental data for σ(ω) = σ 1 + iσ 2 in cuprates are usually processed by the generalized (extended) Drude formula [32][33][34][35][36]95]: where m is the mass of the band electrons while the quantity ω tr (ω) is defined in (19). The expression (12) is a useful representation for systems with single-band electron-boson scattering which is justified in HTSC cuprates. However, this procedure is inadequate for interpreting optical data in multiband systems such as new high-temperature superconductors Fe-based pnictides since even in absence of the inelastic intra-and interband scattering the effective optic relaxation rate may be strongly frequency dependent [96].
(The usefulness of introducing the optic relaxation ω tr (ω) will be discussed below.) Here, i = a, b enumerates the plane axis; ω p , γ tr (ω, T), and m op (ω) are the electronic plasma frequency, the transport (optical) scattering rate, and the optical mass, respectively. Very frequently it is analyzed the quantity γ * tr (ω, T) given by [95] γ * tr (ω, In the weak coupling limit λ ep < 1, the formula for conductivity given in Appendix A, equations (A.20) and (A.21) can be written in the form of (12) where γ tr reads [33][34][35][36] γ tr (ω, T) = π l ∞ 0 dν α 2 tr,l F l (ν) Here n B (ω) is the Bose distribution function. For completeness we give also the explicit form of the transport mass m tr (ω), see [3][4][5][6][32][33][34][35][36]: with the Kernel K(x, y) = (i/ y)+{( In the presence of impurity scattering one should add γ imp,tr to γ tr . It turns out that (14) holds within a few percents also for large λ ep (>1). Note that α 2 tr,l F l (ν) / = α 2 l F l (ν) and the index l enumerates all scattering bosons-phonons-spin fluctuations, and so forth. For comparison, the quasiparticle scattering rate γ(ω, T) is given by where n F is the Fermi distribution function. For completeness we give also the expression for the quasiparticle effective mass m * (ω): The term γ imp is due to the impurity scattering. By comparing (14) and (16), it is seen that γ tr and γ are different quantities, that is, γ tr / = γ, where the former describes the relaxation of Bose particles (excited electron-hole pairs) while the latter one describes the relaxation of Fermi particles. This difference persists also at T = 0 K where one has (due to simplicity we omit in the following summation over l) [32] In the case of EPI with the constant electronic density of states, the above equations give that γ ep (ω) = const for ω > ω max ph while γ ep,tr (ω) (as well as γ * ep,tr ) is monotonic growing for ω > ω max ph , where ω max ph is the maximal phonon frequency. So, the growing of γ ep,tr (ω) (and γ * ep,tr ) for ω > ω max ph is ubiquitous and natural for the EPI scattering and has nothing to do with some exotic scattering mechanism. This behavior is clearly seen by comparing γ(ω, T), γ tr (ω, T), and γ * tr which are calculated for the EPI spectral function α 2 ep (ω)F ph (ω) extracted from tunnelling experiments in YBCO (with ω max ph ∼ 80 meV ) [42][43][44][45]-see Figure 4. The results shown in Figure 4 clearly demonstrate the physical difference between two scattering rates γ ep and γ ep,tr (or γ * tr ). It is also seen that γ * tr (ω, T) is even more linear function of ω than γ tr (ω, T). From these calculations one concludes that the quasilinearity of γ tr (ω, T) (and γ * tr ) is not in contradiction with the EPI scattering mechanism but it is in fact a natural consequence of EPI. We stress that such behavior of γ ep and γ ep,tr (and γ * ep,tr ), shown in Figure 4, is in fact not exceptional for HTSC cuprates but it is generic for many metallic systems, for instance, 3D metallic oxides, lowtemperature superconductors such as Al, Pb, and so forthsee more in [3][4][5][6] and references therein.
Let us discuss briefly the experimental results for R(ω) and γ * tr (ω, T) and compare these with theoretical predictions obtained by using a single-band model with α 2 ep (ω)F ph (ω) extracted from the tunnelling data with the EPI coupling constant λ ep 2 [42][43][44][45]. In the case of YBCO the  Figure 4: (a) Scattering rates γ(ω, T), γ tr (ω, T), and γ * tr -from top to bottom-for the Eliashberg function in (b). From [33][34][35]. (b) Eliashberg spectral function α 2 ep (ω)F ph (ω) obtained from tunnelling experiments on break junctions [42][43][44][45]. Inset shows γ * tr with (full line) and without (dashed line) interband transitions [3][4][5]. agreement between measured and calculated R(ω) is very good up to energies ω < 6000 cm −1 , which confirms the importance of EPI in scattering processes. For higher energies, where a mead-infrared peak appears, it is necessary to account for interband transitions [3][4][5]. In optimally doped Bi 2 Sr 2 CaCu 2 O 6 (Bi2212) [97,98] the experimental results for γ * tr (ω, T) are explained theoretically by assuming that the EPI spectral function α 2 ep (ω)F(ω) ∼ F ph (ω), where F ph (ω) is the phononic density of states in BSCO, with λ ep = 1.9 and γ imp ≈ 320 cm −1 -see Figure 5(a). At the same time the fit of γ * tr (ω, T) by the marginal Fermi liquid phenomenology fails as it is evident in Figure 5(b). Now we will comment the so called pronounced linear behavior of γ tr (ω, T) (and γ * tr (ω, T)) which was one of the main arguments for numerous inadequate conclusions regarding the scattering and pairing bosons and EPI. We stress again that the measured quantity is reflectivity R(ω) and derived ones are σ(ω), γ tr (ω, T), and m tr (ω), which are very sensitive to the value of the dielectric constant ε ∞ . This sensitivity is clearly demonstrated in Figure 6 for Bi-2212 where it is seen that γ tr (ω, T) (and γ * tr (ω, T)) for ε ∞ = 1 is linear up to much higher ω than in the case ε ∞ > 1.
Furthermore, we would like to comment two points related to σ, γ tr , and γ. First, the parametrization of σ(ω) with the generalized Drude formula in (12) and its relation to the transport scattering rate γ tr (ω, T) and the transport mass m tr (ω, T) is useful if we deal with electron-boson scattering in a single-band problem. In [36,96] it is shown that σ(ω) of a two-band model with only elastic impurity scattering can be represented by the generalized (extended) Drude formula with ω and T dependence of effective parameters γ eff tr (ω, T), m eff tr (ω, T) despite the fact that the inelastic electron-boson scattering is absent. To this end we stress that the singleband approach is justified for a number of HTSC cuprates such as LSCO, BSCO, and so forth. Second, at the beginning we said that γ tr (ω, T) and γ(ω, T) are physically different quantities and it holds that γ tr (ω, T) / = γ(ω, T). In order to give the physical picture and qualitative explanation for this difference we assume that α 2 tr F(ν) ≈ α 2 F(ν). In that case the renormalized quasiparticle frequency ω(ω) = Z(ω)ω = ω−Σ(ω) and the transport one ω tr (ω)-defined in (12)-are related and at T = 0 they are given by [32,36] (For the definition of Z(ω) see Appendix A.) It gives the relation between γ tr (ω) and γ(ω) as well as m tr (ω) and m * (ω), respectively: The physical meaning of (19) is the following: in optical measurements one photon with the energy ω is absorbed and two excited particles (electron and hole) are created above and below the Fermi surface. If the electron has energy ω and the hole ω − ω , then they relax as quasiparticles with the renormalized frequency ω. Since ω takes values 0 < ω < ω, then the optical relaxation ω tr (ω) is the energyaveraged ω(ω) according to (19). The factor 2 is due to the two quasiparticles-electron and hole. At finite T, the generalization reads [32,36] (2) Inversion of the Optical Data and α 2 tr (ω)F(ω). In principle, the transport spectral function α 2 tr (ω)F(ω) can be extracted from σ(ω) (or γ tr (ω)) only at T = 0 K, which follows from (18) as or equivalently as α 2 tr (ω)F(ω) = (1/2π)∂ 2 (ωγ tr (ω))/∂ω 2 . However, real measurements are performed at finite T (at T > T c which is rather high in HTSC cuprates) and the inversion procedure is an ill-posed problem since α 2 tr (ω)F(ω) is the deconvolution of the inhomogeneous Fredholm integral equation of the first kind with the temperaturedependent Kernel K 2 (ω, ν, T)-see (14). It is known that an ill-posed mathematical problem is very sensitive to input since experimental data contain less information than one needs. This procedure can cause, first, that the fine structure of α 2 tr (ω)F(ω) get blurred (most peaks are washed out) in the extraction procedures and, second, the extracted α 2 tr (ω)F(ω) be temperature dependent even when the true This artificial T dependence is especially pronounced in HTSC cuprates because T c (∼ 100 K) is very high. In the context of HTSC cuprates, this problem was first studied in [33][34][35][36] where this picture is confirmed by the following results: (1) the extracted shape of α 2 tr (ω)F(ω) in YBa 2 Cu 3 O 7−x as well as in other cuprates is not unique and it is temperature dependent, that is, at higher T > T c the peak structure is smeared and only a single peak (slightly shifted to higher ω) is present. For instance, the experimental data of R(ω) in YBCO were reproduced by two different spectral functions α 2 tr (ω)F(ω), one with single peak and the other one with three-peak structure as it is shown in Figure 7, where all spectral functions give almost identical R(ω). The similar situation is realized in optimally doped BSCO as it is seen in Figure 8 where again different functions α 2 (ω)F(ω) reproduce very well curves for R(ω) and σ(ω). However, it is important to stress that the obtained width of the extracted α 2 tr (ω)F(ω) in both compounds coincide with the width of the phonon density of states F ph (ω) [33][34][35][36]99].
These results demonstrate the importance of EPI in cuprates [33][34][35][36]. We point out that the width of α 2 tr (ω)F(ω) which is extracted from the optical measurements [33][34][35][36] coincides with the width of the quasiparticle spectral function α 2 (ω)F(ω) obtained in tunnelling and ARPES spectra (which we will discuss below), that is, both functions are spread over the energy interval 0 < ω < ω max ph ( 80 meV). Since in cuprates this interval coincides with the width in the phononic density of states F(ω) and since the maxima of 0 0.25 α 2 (ω)F(ω) and F(ω) almost coincide, this is further evidence for the importance of EPI.
To this end, we would like to comment two aspects which appear from time to time in the literature. First, in some reports [24][25][26][27] it is assumed that α 2 tr (ω)F(ω) of cuprates can be extracted also in the superconducting state by using (22). However, (22) holds exclusively in the normal state (at T = 0) since σ(ω) can be described by the generalized (extended) Drude formula in (12) only in the normal state. Such an approach does not hold in the superconducting state since the dynamical conductivity depends not only on the electron-boson scattering but also on coherence factors and on the momentum and energy dependent order parameter Δ(k, ω). Second, if R(ω)'s (and σ(ω)'s ) in cuprates are due to some other bosonic scattering which is pronounced up to much higher energies ω c ω max ph , this should be seen in the width of the extracted spectral function α 2 tr (ω)F(ω). In that respect in [25][26][27] it is assumed that SFI dominates in the quasiparticle scattering and that . This claim is based on reanalyzing of some IR measurements [25][26][27] and the transport spectral function α 2 tr (ω)F(ω) is extracted in [25] by using the maximum entropy method in solving the Fredholm equation. However, in order to exclude negative values in the extracted α 2 tr (ω)F(ω), which is an artefact and due to the chosen method, in [25] it is assumed that α 2 tr (ω)F(ω) has a rather large tail at large energies-up to 400 meV. It turns out that even such an assumption in extracting α 2 tr (ω)F(ω) does not reproduce the experimental curve Im χ(ω) [107] in some important aspects. First, the relative heights of the two peaks in the extracted spectral function α 2 tr (ω)F(ω) at lower temperatures are opposite to the experimental curve Im χ(ω) [107]-see [25, Figure 1]. Second, the strong temperature dependence of the extracted α 2 tr (ω)F(ω) found in [25][26][27] is not an intrinsic property of the spectral function but it is an artefact due to the high sensitivity of the extraction procedure on temperature. As it is already explained before, this is due to the illposed problem of solving the Fredholm integral equation of the first kind with strong T-dependent kernel. Third, the extracted spectral weight I B (ω) = α 2 tr (ω)F(ω) in [25] has much smaller values at larger frequencies (ω > 100 meV) than it is the case for the measured Im χ(ω), that is, (I B (ω > 100 meV)/I B (ω max )) Im χ(ω > 100 meV)/ Im χ(ω max )see [25, Figure 1]. Fourth, the recent magnetic neutron scattering measurements on optimally doped large-volume crystals Bi 2 Sr 2 CaCu 2 O 8+δ [93] (where the absolute value of Im χ(q, ω) is measured) are not only questioning the theoretical interpretation of magnetism in HTSC cuprates in terms of itinerant magnetism but also opposing the finding in [25][26][27]. Namely, this experiment shows that the local spin susceptibility Im χ(ω) = q Im χ(q, ω) is temperature independent across the superconducting transition T c = 91 K, that is, there is only a minimal change in Im χ(ω) between 10 K and 100 K. This T-independence of Im χ(ω) strongly opposes the (above discussed) results in [24][25][26][27], where the fit of optic measurements gives strong T dependence of Im χ(ω).
Fifth, the transport coupling constant λ tr extracted in [25] is too large, that is, λ tr > 3 contrary to the previous findings that λ tr 1.5 [33][34][35][36]99]. Since in HTSC one has λ > λ tr , this would probably give λ ≈ 6, which is not confirmed by other experiments. Sixth, the interpretation of α 2 tr (ω)F(ω) in LSCO and BSCO solely in terms of Im χ(ω) is in contradiction with the magnetic neutron scattering in the optimally doped and slightly underdoped YBCO [30]. The latter was discussed in Section 1.3.1, where it is shown that Im χ(Q, ω) is small in the normal state and its magnitude is even below the experimental noise. This means that if the assumption that α 2 tr (ω)F(ω) ≈ g 2 sf Im χ(ω) were correct then the contribution to Im χ(ω) from the momenta 0 < k Q would be dominant, which is detrimental for d-wave superconductivity.
Finally, we point out that very similar (to cuprates) properties, of σ(ω), R(ω) (and ρ(T) and electronic Raman spectra), were observed in 3D isotropic metallic oxides La 0.5 Sr 0.5 CoO 3 and Ca 0.5 Sr 0.5 RuO 3 which are nonsuperconducting [108] and in Ba 1−x K x BiO 3 which is superconducting below T c 30 K at x = 0.4. This means that in all of these materials the scattering mechanism might be of similar origin. Since in these compounds there are no signs of antiferromagnetic fluctuations (which are present in cuprates), then the EPI scattering plays important role also in other oxides.
(3) Restricted Optical Sum Rule. The restricted optical sum rule was studied intensively in HTSC cuprates. It shows some peculiarities not present in low-temperature superconductors. It turns out that the restricted spectral weight W(Ω c , T) is strongly temperature dependent in the normal and superconducting state, which was interpreted either to be due to EPI [39,40] or to some nonphononic mechanisms [109]. In the following we demonstrate that the temperature dependence of W(Ω c , T) = W(0) − βT 2 in the normal state can be explained in a natural way by the T dependence of the EPI transport relaxation rate γ ep tr (ω, T) [39,40]. Since the problem of the restricted sum rule has attracted much interest, it will be considered here in some details. In fact, there are two kinds of sum rules related to σ(ω). The first one is the total sum rule which in the normal state reads while in the superconducting state it is given by the Tinkham-Ferrell-Glover (TFG) sum rule Here, n is the total electron density, e is the electron charge, m is the bare electron mass, and λ L is the London penetration depth. The first (singular) term c 2 /8λ 2 L in (24) is due to the superconducting condensate which contributes σ S 1,cond (ω) = (c 2 /4λ 2 L )δ(ω). The total sum rule represents the fundamental property of matter-the conservation of the electron number. In order to calculate it one should use the total Hamiltonian H tot = T e + H int where all electrons, electronic bands, and their interactions H int (Coulomb, EPI, with impurities, etc.) are accounted for. Here, T e is the kinetic energy of bare electrons: The partial sum rule is related to the energetics solely in the conduction (valence) band which is described by the Hamiltonian of the conduction (valence) band electrons: H v contains the band energy with the dispersion p (ξ p = p − μ) and the effective Coulomb interaction of the valence electrons V v,c . In this case the partial sum rule in the normal state reads [110] (for a general form of p ) where the number operator n v,p = σ c † pσ c pσ ; 1/m p = ∂ 2 p /∂p 2 x is the momentum-dependent reciprocal mass and V is volume. In practice, the optic measurements are performed up to finite frequency and the integration over ω goes up to some cutoff frequency Ω c (of the order of the band plasma frequency). In this case the restricted sum rule has the form where K d is the diamagnetic Kernel given by (30) below and Π(ω) is the paramagnetic (current-current) response function. In the perturbation theory without vertex correction Π(iω n ) (at the Matsubara frequency ω n = πT(2n + 1)) is given by [39,40] where ω + nm = ω n + ω m and G(p, iω n ) = (iω n − ξ p − Σ(p, iω n )) −1 is the electron Green's function. In the case when the interband gap E g is the largest scale in the problem, that is, when W b < Ω c < E g , in this region one has approximately Im Π(ω) ≈ 0 and the limit Ω c → ∞ in (28) is justified. In that case one has Π(0) ≈ Ωc 0 (Im Π(ω)/ω)dω which gives the approximate formula for W(Ω c , T): where n p = n v,p is the quasiparticle distribution function in the interacting system. (30) does not depend on Ω c . So, one should be careful not to overinterpret the experimental results in cuprates by this formula. In that respect the best way is to calculate W(Ω c , T) by using the exact result in (28) which apparently depends on Ω c . However, (30) is useful for appropriately chosen Ω c , since it allows us to obtain semiquantitative results. In most papers related to the restricted sum rule in HTSC cuprates it was assumed, due to simplicity, the tight-binding model with nearest neighbors (n.n.) with the energy p = −2t(cos p x a + cos p y a) which gives 1/m p = −2ta 2 cos p x a. It is straightforward to show that in this case (30) is reduced to a simpler one: where T v Hv = p p n v Hv is the average kinetic energy of the band electrons, a is the Cu-Cu lattice distance, and V is the volume of the system. In this approximation W(Ω c , T) is a direct measure of the average band (kinetic) energy. In the superconducting state the partial band sum rule reads In order to introduce the reader to (the complexity of) the problem of the T dependence of W(Ω c , T), let us consider the electronic system in the normal state and in absence of the quasiparticle interaction. In that case one has n p = f p ( f p for Ω c ≈ 1.25 eV in two underdoped Bi2212 (with T c = 88 K and T c = 66 K). From [111].
is the Fermi distribution function) and W n (Ω c , T) increases with the decrease of the temperature, that is, To this end, let us mention in advance that the experimental value β exp is much larger than β b , that is, β exp β b , thus telling us that the simple Sommerfeld-like smearing of f p by the temperature effects cannot explain quantitatively the T dependence of W(Ω c , T). We stress that the smearing of f p by temperature lowers the spectral weight compared to that at T = 0 K, that is, W n (Ω c , T) < W n (Ω c , 0). In that respect it is not surprising that there is a lowering of W s (Ω c , T) in the BCS superconducting state, W BCS s (Ω c , T T c ) < W n (Ω c , T T c ) since at low temperatures f p is smeared mainly due to the superconducting gap, that is, Let us enumerate and discuss the main experimental results for W(Ω c , T) in HTSC cuprates. (1) In the normal state is increasing by decreasing T, even at T below T * -the temperature for the opening of the pseudogap. The increase of W n (Ω c , T) from the room temperature down to T c is no more than 5%. (2) In the superconducting state (T < T c ) of some underdoped and optimally doped Bi-2212 compounds [111,113,114] (and underdoped Bi-2212 films [115]) there is an effective increase of W s (Ω c , T) with respect to that in the normal state, that is, W s (Ω c , T) > W n (Ω c , T) for T < T c . This is a non-BCS behavior which is shown in Figure 9. Note that in the tight binding model the effective band (kinetic) energy T v is negative ( T v < 0) and in the standard BCS case (32) gives that W s (T < T c ) decreases due to the increase of T v . Therefore the experimental increase of W s (T < T c ) by decreasing T is called the non-BCS behavior. The latter means a lowering of the kinetic energy T v which is frequently interpreted to be due either to strong correlations or to a Bose-Einstein condensation (BEC) of the preformed tightly bound Cooper pair-bosons, for instance, bipolarons [116]. It is known that in the latter case the kinetic energy of bosons is decreased below the BEC critical temperature T c . In [117] it is speculated that the latter case might be realized in underdoped cuprates.
However, in some optimally doped and in most over- T)) which is the BCS-like behavior [112] as it is seen in Figure 10.
We stress that the non-BCS behavior of W s (Ω c , T) for underdoped (and in some optimally doped) systems was obtained by assuming that Ω c ≈ (1-1.2) eV. However, in [104] these results were questioned by claiming that the conventional BCS-like behavior was observed (W s (Ω c , T) < W n (Ω c , T)) in the optimally doped YBCO and slightly underdoped Bi-2212 by using larger cutoff energy Ω c = 1.5 eV. This discussion demonstrates how risky is to make definite conclusions on some fundamental physics based on the parameter-(such as the cutoff energy Ω c ) dependent analysis. Although the results obtained in [104] look very trustfully, it is fair to say that the issue of the reduced spectral weight in the superconducting state of the underdoped cuprates is still unsettled and under dispute. In overdoped Bi-2212 films, the BCS-like behavior The first question is the following. How to explain the strong temperature dependence of W(Ω c , T) in the normal state? In [39,40] W(T) is explained solely in the framework of the EPI physics where the EPI relaxation γ ep (T) plays the main role in the T dependence of W(Ω c , T). The main theoretical results of [39,40] are the following: the calculations of W(T) based on the exact (30) give that for Ω c Ω D (the Debye energy) the difference in spectral weights of the normal and superconducting states is small, that is, In the case of large Ω c the calculations based on (30) give In the case when EPI dominates one has ). It turns out that for α 2 (ω)F(ω), shown in Figure 4, one obtains (i) γ ep (T) ∼ T 2 in the temperature interval 100 K < T < 200 K as it is seen in Figure 11 for the T dependence of W(Ω c , T) [39,40]; (ii) the second term in (33) is much larger than the last one (the Sommerfeld-like term). For the EPI coupling constant λ ep,tr = 1.5 one obtains rather good agreement between the theory in [39,40] and experiments in [104,111,113,114]. At lower temperatures, γ ep (T) deviates from the T 2 behavior and this deviation depends on the structure of the spectrum in α 2 (ω)F(ω). It is seen in Figure 11 that, for a softer Einstein spectrum (with Ω E = 200 K), W(Ω c , T) lies above the curve with the T 2 asymptotic behavior, while the curve with a harder phononic spectrum (with Ω E = 400 K) lies below it. This result means that different behavior of W(Ω c , T) in the superconducting state of cuprates for different doping might be simply related to different contributions of lowand high-frequency phonons. We stress that such a behavior of W(Ω c , T) was observed in experiments in [104,111,113,114].
To summarize, the above analysis demonstrates that the theory based on EPI is able to explain in a satisfactory way the temperature behavior of W(Ω c , T) above and below T c in systems at and near the optimal doping.
(4) α 2 (ω)F(ω) and the In-Plane Resistivity ρ ab (T). The temperature dependence of the in-plane resistivity ρ ab (T) in cuprates is a direct consequence of the quasi-2D motion of quasiparticles and of the inelastic scattering which they experience. At present, there is no consensus on the origin of the linear temperature dependence of the in-plane resistivity ρ ab (T) in the normal state. Our intention is not to discuss this problem, but only to demonstrate that the EPI spectral function α 2 (ω)F(ω), which is obtained from tunnelling experiments in cuprates (see Section 1.3.4), is able to explain the temperature dependence of ρ ab (T) in the optimally doped YBCO. In the Boltzmann theory ρ ab (T) is given by where The residual resistivity ρ imp is due to the impurity scattering.
Since ρ(T) = 1/σ(ω = 0, T) and having in mind that the dynamical conductivity σ(ω, T) in YBCO (at and near the optimal doping) is satisfactory explained by the EPI scattering, then it is to expect that ρ ab (T) is also dominated by EPI in some temperature region T > T c . This is indeed confirmed in the optimally doped YBCO, where ρ imp is chosen appropriately and the spectral function α 2 tr (ω)F(ω) is taken from the tunnelling experiments in [42][43][44][45]. The very good agreement with the experimental results [118] is shown in Figure 12. We stress that in the case of EPI there is always a temperature region where γ tr (T) ∼ T for T > αΘ D , α < 1 depending on the shape of α 2 tr (ω)F(ω) (for the simple Debye spectrum α ≈ 0.2). In the linear regime one has ρ(T) [108] and ρ ≈ 0.6 in the oriented YBCO films and ρ ≈ 0.3-0.4 in single crystals of BSCO, one obtains λ tr ≈ 0.6-1.4. In case of YBCO single crystals, there is a pronounced anisotropy in ρ ab (T) [119] which gives ρ x (T) = 0.6 μΩcm/K and ρ y (T) = 0.25 μΩcm/K. The function λ tr (ω pl ) is shown in Figure 13 where the plasma frequency ω pl can be calculated by LDA-DFT and also extracted from the width (∼ ω * pl ) of the Drude peak at small frequencies, where ω pl = √ ε ∞ ω * pl . We stress that the rather good agreement of theoretical and experimental results for ρ ab (T), in some optimally doped HTSC cuprates such as YBCO, should not be overinterpreted in the sense that the above rather simple electron-phonon approach can explain the resistivity in other HTSC cuprates and for various doping. For instance, in highly underdoped systems ρ ab (T) is very different from the behavior in Figure 12 and the simple Migdal-Eliashberg theory based on the EPI spectral function is inadequate. In this case one should certainly take into account polaronic effects [8][9][10][11], strong correlations, and so forth. The above analysis on the resistivity in the optimally doped YBCO demonstrates only that in this case if in (35) one uses the EPI spectral function α 2 (ω)F(ω) obtained from the tunnelling experiments (and optics) one obtains the correct T dependence of ρ a,b (T). This result is an additional evidence for the importance of EPI.
Concerning the temperature dependence of the resistivity in other (than YBCO) families of the optimally doped HTSC cuprates we would like to point out that there is some evidence that the linear (in T) resistivity is observed in some of them even at temperatures T < 0.2Θ D [122,123]. This possibility is argued also theoretically in [124] where it is shown that in two-dimensional systems with a broad interval of phonon spectra the quasilinear behavior of ρ ab (T) is realized even at T < 0.2Θ D . The quasilinear behavior of the resistivity at T 0.2Θ D has been observed in Bi 2 (Sr 0.97 Pr 0.003 ) 2 CuO 6 [125], in LSCO, and in 1-layer Bi-2201 [122,123,126,127], where in all these systems the critical temperature is rather small, T c ≈ 10 K. In that respect all existing theories based on the electronboson scattering are plagued and having difficulties to explain this low-temperature behavior of ρ ab (T). To this point, we would like to emphasize here that some of these (experimental) observations are contradictory. For example, the results obtained by the Vedeneev group [127] show that some samples demonstrate the quasilinear behavior of the resistivity up to T = 10 K but some others with approximately the same T c have the usual Bloch-Grüneisentype behavior characteristic for the EPI scattering. In that respect it is very unlikely that the linear resistivity up to T = 10 K can be simply explained in the standard way by interactions of electrons with some known bosons either by phonons or spin fluctuations (magnons). The question why in some cuprates the linear resistivity is observed up to T = 10 K is still a mystery and its explanation is a challenge for all kinds of the electron-boson scattering, not only for EPI. In that respect it is interesting to mention that the existence of the forward scattering peak in EPI (with the width q c k F ), which is due to strong correlations, may give rise to the linear behavior of ρ(T) down to very low temperatures T ∼ Θ D /30 [6,128,129]-see more in Section 2.3.4, item (6). We will argue in Section 1.3.4 that if one interprets the tunnelling experiments in systems near optimal doping [42][43][44][45][46][47][48][49][50][51][52][53][54] in the framework of the Eliashberg theory one obtains the large EPI coupling constant λ ep ≈ 2-3.5 which implies that λ tr ∼ (λ/3). This means that EPI is reduced much more in transport properties than in the self-energy. We stress that such a large reduction of λ tr cannot be obtained within the LDA-DFT band-structure calculations, which means that λ ep and λ tr contain renormalization which do not enter in the LDA-DFT theory. In Section 2 we will argue that the strong suppression of λ tr may have its origin in strong electronic correlations [78][79][80]130] and in the long-range Madelung energy [3][4][5][6].
The femtosecond time-resolved optical spectroscopy (FTROS) has been developed in the last couple of years and applied to HTSC cuprates. In this method a femtosecond (1 fs = 10 −15 sec) laser pump excites in materials electron-hole pairs via interband transitions. These hot carriers release their energy via electron-electron (with the relaxation time τ ee ) and electron-phonon scattering reaching states near the Fermi energy within 10-100 fs-see [131]. The typical energy density of the laser pump pulses with the wavelength λ ≈ 810 nm (Dω = 1.5 eV) was around F ∼ 1 μJ/cm 2 (the excitation fluence F) which produces approximately 3 × 10 10 carriers per pulse (by assuming that each photon produces Dω/Δ carriers, Δ is the superconducting gap). By measuring photoinduced changes of the reflectivity in time, that is, ΔR(t)/R 0 , one can extract information on the relaxation dynamics of the low-laying electronic excitations.
Since ΔR(t) relax to equilibrium, the fit with exponential functions is used as where is the Heavyside function) describes the finite rise-time. The parameters A, B depend on the fluence F. This method was used in studying the superconducting phase of La 2−x Sr x CuO 4 , with x = 0.1 and 0.15 and T c = 30 K and 38 K, respectively [41]. In that case one has A / = 0 for T < T c and A = 0 for T > T c , while the signal B was present also at T > T c . It turns out that the signal A is related to the quasiparticle recombination across the superconducting gap Δ(T) and has a relaxation time of the order τ A > 10 ps at T = 4.5 K. At the so called threshold fluence (F T = 4.2 ± 1.7 μJ/cm 2 for x = 0.1 and F T = 5.8 ± 2.3 μJ/cm 2 for x = 0.15) the vaporization (destroying) of the superconducting phase occurs, where the parameter A saturates. This vaporization process takes place at the time scala τ r ≈ 0.8 ps. The external fluence is distributed in the sample over the excitation volume which is proportional to the optical penetration depth λ op (≈150 nm at λ ≈ 810 nm) of the pump. The energy densities stored in the excitation volume at the vaporization threshold for x = 0.1 and x = 0.15 are U p = F T /λ op = 2.0 ± 0.8 K/Cu and 2.6 ± 1.0 K/Cu, respectively. The important fact is that U p is much larger than the superconducting condensation energy which is U cond ≈ 0.12 K/Cu for x = 0.1 and U cond ≈ 0.3 K/Cu for x = 0.15, that is, U p U cond . This means that the energy difference U p − U cond must be stored elsewhere on the time scale τ r . The only present reservoir which can absorb the difference in energy is the bosonic baths of phonons and spin fluctuations. The energy required to heat the spin The measured specific heat C sf (T) in La 2 CuO 4 [41] gives very small value U sf ≈ 0.01 K. In the case of the phonon reservoir one obtains U ph = Tc T C ph (T)dT = 9 K/Cu for x = 0.1 and 28 K/Cu for x = 0.15, where C ph is the phonon specific heat. Since U sf U p − U cond , the spin reservoir cannot absorb the rest energy U p − U cond . The situation is opposite with phonons since U ph U p − U cond and phonon can absorb the rest energy in the excitation volume. The complete vaporization dynamics can be described in the framework of the Rothwarf-Taylor model which describes approaching of electrons and phonons to quasiequilibrium on the time scale of 1 ps [132]. We will not go into details but only summarize by quoting the conclusion in [132] that only phonon-mediated vaporization is consistent with the experiments, thus ruling out spin-mediated quasiparticle recombination and pairing in HTSC cuprates. The FTROS method tells us that at least for nonequilibrium processes EPI is more important than SFI. It gives also some opportunities for obtaining the strength of EPI but at present there is no reliable analysis.
In conclusion, optics and resistivity measurements in the normal state of cuprates give evidence that EPI is important while the spin-fluctuation scattering is weaker than it is believed. However, some important questions related to the transport properties remain to be answered. (i) What are the values of λ tr and ω pl ? (ii) What is the reason that λ tr λ is realized in cuprates? (iii) What is the role of the Coulomb scattering in σ(ω) and ρ(T)? Later on we will argue that ARPES measurements in cuprates give evidence for an appreciable contribution of the Coulomb scattering at higher frequencies, where γ(ω) ≈ γ 0 + λ c ω for ω > ω ph max with λ c ∼ 1. One should stress that despite the fact that EPI is suppressed in transport properties it is sufficiently strong in the quasiparticle self-energy, as it comes out from tunnelling measurements discussed below.

ARPES and the EPI Self-Energy.
The angle-resolved photoemission spectroscopy (ARPES) is nowadays one of leading spectroscopy methods in the solid-state physics [22,23]. In some favorable conditions it provides direct information on the one-electron removal spectrum in a complex many-body system. The method involves shining light (photons) with energies between E i = 5-1000 eV on samples and by detecting momentum (k)-and energy(ω)distribution of the outgoing electrons. The resolution of ARPES has been significantly increased in the last decade with the energy resolution of ΔE ≈ 1-2 meV (for photon energies ∼20 eV) and angular resolution of Δθ 0.2 • . On the other side the ARPES method is surfacesensitive technique, since the average escape depth (l esc ) of the outgoing electrons is of the order of l esc ∼ 10Å, depending on the energy of incoming photons. Therefore, very good surfaces are needed in order that the results be representative for bulk samples. The most reliable studies were done on the bilayer Bi 2 Sr 2 CaCu 2 O 8 (Bi2212) and its single-layer counterpart Bi 2 Sr 2 CuO 6 (Bi2201), since these materials contain weakly coupled BiO planes with the longest interplane separation in the cuprates. This results in a natural cleavage plane making these materials superior to others in ARPES experiments. After a drastic improvement of sample quality in other families of HTSC materials, the ARPES technique has became an important method in theoretical considerations. The ARPES can indirectly give information on the momentum and energy dependence of the pairing potential. Furthermore, the electronic spectrum of the (abovementioned) cuprates is highly quasi-2D which allows rather unambiguous determination of the initial state momentum from the measured final state momentum, since the component parallel to the surface is conserved in photoemission. In this case, the ARPES probes (under some favorable conditions) directly the single-particle spectral function A(k, ω). In the following we discuss mainly those ARPES experiments which give evidence for the importance of the EPI in cuprates-see more in [22,23].
ARPES measures a nonlinear response function of the electron system and it is usually analyzed in the so-called three-step model, where the total photoemission intensity I tot (k, ω) ≈ I 1 · I 2 · I 3 is the product of three independent terms: (1) I 1 that describes optical excitation of the electron in the bulk, (2) I 2 that describes the scattering probability of the travelling electrons, and (3) I 3 that describes the transmission probability through the surface potential barrier. The central quantity in the three-step model is I 1 (k, ω) and it turns out that for k = k it can be written in the form I 1 (k, ω) I 0 (k, υ) f (ω)A(k, ω) [22,23] with I 0 (k, υ) ∼ | ψ f |pA|ψ i | 2 and the quasiparticle spectral function A(k, ω) = − Im G(k, ω)/π: Here, ψ f |p · A|ψ i is the dipole matrix element which depends on k, polarization, and energy E i of the incoming photons. The knowledge of the matrix element is of a great importance and its calculation from first principles was done in [133]. f (ω) is the Fermi function; G and Σ = Re Σ + i Im Σ are the quasiparticle Green's function and the selfenergy, respectively. We summarize and comment here some important ARPES results which were obtained in the last several years and which confirm the existence of the Fermi surface and importance of EPI in the quasiparticle scattering [22,23].
ARPES in the Normal State. (N1) There is well-defined Fermi surface in the metallic state of optimally and near optimally doped cuprates with the topology predicted by the LDA-DFT. However, the bands are narrower than LDA-DFT predicts which points to a strong quasiparticle renormalization. (N2) The spectral lines are broad with |ImΣ(k, ω)| ∼ ω (or ∼ T for T > ω) which tells us that the quasiparticle liquid is a noncanonical Fermi liquid for larger values of T, ω. (N3) There is a bilayer band splitting in Bi2212 (at least in the overdoped state), which is also predicted by LDA-DFT. In the case when the coherent hopping t ⊥ between two layers in the bilayer dominates, then the antibonding and bonding bands ξ a,b k = ξ k ±t ⊥ k with t ⊥ k = [t ⊥ (cos 2 k x −cos 2 k y )+ · · · ] have been observed. It is worth to mention that the previous experiments did not show this splitting which was one of the reasons for various speculations on some exotic electronic scattering and non-Fermi liquid scenarios. (N4) In the underdoped cuprates and at temperatures T c < T < T * there is a d-wave-like pseudogap Δ pg (k) ∼ Δ pg,0 (cos k x − cos k y ) in the quasiparticle spectrum where Δ pg,0 increases by lowering doping. We stress that the pseudogap phenomenon is not well understood at present and since we are interested in systems near optimal doping where the pseudogap phenomena are absent or much less pronounced we will not discuss this problem here. Its origin can be due to a precursor superconductivity or due to a competing order, such as spin-or charge-density wave, strong correlations, and so forth. (N5) The ARPES self-energy gives evidence that EPI interaction is rather strong. The arguments for the latter statement are the following: (i) at T > T c there are kinks in the quasiparticle dispersion ω(ξ k ) in the nodal direction (along the (0, 0) − (π, π) line) at the characteristic phonon energy ω (70) ph ∼ (60-70) meV [91], see Figure 14 (top), and near the antinodal point (π, 0) at 40 meV [134]see Figure 14 (bottom).
(ii) The kink structure is observed in a variety of the hole-doped cuprates such as LSCO, Bi2212, Bi2201, Tl2201 (Tl 2 Ba 2 CuO 6 ), Na-CCOC (Ca 2−x Na x CuO 2 Cl 2 ). These kinks exist also above T c , which excludes the scenario with the magnetic resonance peak in Im χ s (Q, ω). Moreover, since the tunnelling and magnetic neutron scattering measurements give small SFI coupling constant g sf < 0.2 eV, then the kinks are not due to SFI. (iii) The position of the nodal kink is practically doping independent which points towards phonons as the scattering and pairing boson. (N6) The quasiparticles (holes) at and near the nodal-point k N couple practically to a rather broad spectrum of phonons since at least three groups of phonons were extracted in the bosonic spectral function α 2 F(k N , ω) from the ARPES effective selfenergy in La 2−x Sr x CuO 4 [135]- Figure 15.
The latter result is in a qualitative agreement with numerous tunnelling measurements [42][43][44][45][46][47][48][49][50][51][52][53][54] which apparently demonstrate that the very broad spectrum of phonons couples with holes without preferring any particular phononssee discussion below. (N7) Recent ARPES measurements in Bi2212 [92] show very different slope dω/dξ k of the quasiparticle energy ω(ξ k ) at small |ξ k | ω ph and at large energies |ξ k | ω ph -see Figure 16. The theoretical analysis [137] of these results gives the total coupling constant λ Z = λ Z ep + λ Z c ≈ 3, and for the EPI coupling λ Z ep ≈ 2, while the Coulomb coupling (SFI is a part of it) is λ Z c ≈ 1 [137]see Figure 16. (Note that the upper index Z in the coupling constants means the quasiparticle renormalization in the normal part of the self-energy.) To this end let us mention some confusion which is related to the value of the EPI coupling constant extracted from ARPES. Namely, [22,23,138,139] the EPI self-energy was obtained by subtracting the high-energy slope of the quasiparticle spectrum ω(ξ k ) at ω ∼ 0.3 eV. The latter is apparently due to the Coulomb interaction. Although the position of the low-energy kink is not affected by this procedure (if ω max ph ω c ), this subtraction procedure gives in fact an effective EPI selfenergy Σ ep eff (k, ω) and the effective coupling constant λ Z ep,eff (k) only. We demonstrate below that the λ Z ep,eff (k) is smaller than the real EPI coupling constant λ Z ep (k). The total selfenergy is Σ(k, ω) = Σ ep (k, ω) + Σ c (k, ω) where Σ c is the contribution due to the Coulomb interaction. At very low energies ω ω c one has usually Σ c (k, ω) = −λ Z c (k)ω, where ω c (∼1 eV) is the characteristic Coulomb energies and λ Z c is the Coulomb coupling constant. The quasiparticle is the bare bandstructure energy. At low energies ω < ω max ph ω c it can be rewritten in the form with ξ ren (k) Since at very low energies ω ω max ph one has Re Σ ep (k, ω) = −λ Z ep (k)ω and Re Σ ep eff (k, ω) = −λ Z ep,eff (k)ω, then the real coupling constant is related to the effective one by As a result one has λ Z ep (k) > λ Z ep,eff (k). At higher energies ω max ph < ω < ω c , where the EPI effects are suppressed and Σ ep (k, ω) stops growing, one has Re Σ(k, ω) ≈ Re Σ ep (k, ω) − λ Z c (k)ω. The measured Re Σ exp (k, ω) at T = 10 K near and slightly away from the nodal point in the optimally doped Bi-2212 with T c = 91 K [136] is shown in Figure 16.
It is seen that Re Σ exp (k, ω) has two kinks-the first one at low energy ω 1 ≈ ω high ph ≈ 50-70 meV which is (as we already argued) most probably of the phononic origin [22,23,138,139], while the second kink at higher energy ω 2 ≈ ω c ≈ 350 meV which is due to the Coulomb interaction. However, the important results in [136] are that the slopes of Re Σ exp (k, ω) at low (ω < ω high ph ) and high energies (ω high ph < ω < ω c ) are very different. The lowenergy and high-energy slope near the nodal point are shown in Figure 16 schematically (thin lines). From Figure 16    2.1 > 2λ Z ep,eff (k), that is, the EPI coupling is at least twice larger than the effective EPI coupling constant obtained in the previous analysis of ARPES results [22,23,138,139]. This estimation tells us that at (and near) the nodal point, the EPI interaction dominates in the quasiparticle scattering at low energies since λ Z ep (≈ 2.1) ≈ 2λ c z > 2λ Z sf , while at large energies (compared to ω ph ) the Coulomb interaction with λ Z c ≈ 1.1 dominates. We point out that EPI near the antinodal point can be even larger than in the nodal point, mostly due to the higher density of states near the antinodal point. (N8) Recent ARPES spectra in the optimally doped Bi2212 near the nodal and antinodal point [139] show a low-energy isotope effect in Re Σ exp (k, ω), which can be well described in the framework of the Migdal-Eliashberg theory for EPI [140]. At higher energies ω > ω ph obtained in [139] very pronounced isotope effect cannot be explained by the simple Migdal-Eliashberg theory [140]. However, there are controversies with the strength of the high-energy isotope effect since it was not confirmed in other measurements [141,142]-see the discussion in Section 1.3.6(2) related to the isotope effects in HTSC cuprates. (N9) The ARPES experiments in Ca 2 CuO 2 Cl 2 give strong evidence for the formation of small polarons in undoped cuprates which are due to phonons and strong EPI, while in the doped systems quasiparticles are formed and there are no small polarons [143]. Namely, in [143] a broad peak around −0.8 eV is observed at the top of the band (k = (π/2, π/2)) with the dispersion similar to that predicted by the t-J model-see Figure 17.
However, the peak in Figure 17(a) is of Gaussian shape and can be described only by coupling to bosons, that is, this peak is a boson side band-see more in [10,11] and references therein. The theory based on the t-J model (in the antiferromagnetic state of the undoped compound) by including coupling to several (half-breathing, apical oxygen, low-lying) phonons, which is given in [144][145][146], explains successfully this broad peak of the boson side band by the formation of small polarons due to the EPI coupling (λ ep ≈ 1.2). Note that this value of λ ep is for the polaron at the bottom of the band while in the case where the Fermi surface exists (in doped systems) this coupling is even larger due to the larger density of states at the Fermi surface [144][145][146]. In [144][145][146] it was stressed that even when the electronmagnon interaction is stronger than EPI the polarons in the undoped systems are formed due to EPI. The latter mechanism involves excitation of many phonons at the lattice site (where the hole is seating), while it is possible to excite only one magnon at the given site.   [148] show kink at energies 50-70 meV in the quasiparticle dispersion relation along both the nodal and antinodal, directions as it is shown in Figure 18. It is seen from this figure that the effective EPI coupling constant λ ep,eff (< λ ep ) is isotropic and λ ep,eff ≈ 0.8-1. It seems that the kink in the electron-doped cuprates is due solely to EPI and in that respect the situation is similar to the one in the hole-doped cuprates.
ARPES Results in the Superconducting State. (S1) There is an anisotropic superconducting gap in most HTSC compounds [22,23], which is predominately d-wave like, that is, Δ(k) ≈ Δ 0 (cos k x − cos k y ) with 2Δ 0 /T c ≈ 5-6 in the optimally doped systems. (S2) The particle-hole coherence in the superconducting state which is expected for the BCS-like theory of superconductivity has been observed first in [149] and confirmed with better resolution in [150], where the particle-hole mixing is clearly seen in the electron and hole quasiparticle dispersion. To remind the reader, the excited Bogoliubov-Valatin quasiparticles ( α k,± ) with energies E α± where the coherence factors u k , v k are given by |u k | 2 = 1 − |v k | 2 = (1 + ξ k /E k )/2. Note that |u k | 2 + |v k | 2 = 1, which is exactly observed, together with d-wave pairing Δ(k) = Δ 0 (cos k x − cos k y ), in experiments in [150]. This is very important result since it proves that the pairing in HTSC cuprates is of the BCS type and not exotic one as it was speculated long time after the discovery of HTSC cuprates. (S3) The kink at (60)(61)(62)(63)(64)(65)(66)(67)(68)(69)(70) meV in the quasiparticle energy around the nodal point is not shifted (in energy) while the antinodal kink at ω (40) ph ∼ 40 meV is shifted (in energy) in the superconducting state by Δ 0 (= (25-30) meV), that is, ω (40) ph → ω (40) ph + Δ 0 = (65-70) meV [22,23]. To remind the reader, in the standard Eliashberg theory the kink in the normal state at ω = ω ph should be shifted in the superconducting state to ω ph + Δ 0 at all k-points at the Fermi surface. This puzzling result (that the quasiparticle energy around the nodal point is not shifted in the superconducting state) might be a smoking gun result since it makes an additional constraint on the quasiparticle interaction in cuprates. Until now there is only one plausible explanation [151] of this nonshift puzzle which is based on an assumption of the forward scattering peak (FSP) in EPI-see more in Section 2. The FSP in EPI means that electrons scatter into a narrow region (q < q c k F ) around the initial point in the k-space, so that at the most part of the Fermi surface there is practically no mixing of states with different signs of the order parameter Δ(k). In that case the EPI bosonic spectral function (which is defined in Appendix A) α 2 F(k, k , Ω) ≈ α 2 F(ϕ, ϕ , Ω) (ϕ is the angle on the Fermi surface) has a pronounced forward scattering peak (at δϕ = ϕ − ϕ = 0) due to strong correlationssee Section 2. Its width δϕ c is narrow, that is, δϕ c 2π and the angle integration goes over the region δϕ c around the point ϕ. In that case the kink is shifted (approximately) by the local gap Δ(ϕ) = Δ max cos 2ϕ-for more details see [151]. As a consequence, the antinodal kink is shifted by the maximal gap, that is, |Δ(ϕ AN ≈ π/2)| = Δ max while the nodal gap is practically unshifted since |Δ(ϕ AN ≈ π/4)| ≈ 0. (S4) The recent ARPES spectra [152] in the undoped single crystalline 4-layered cuprate with the apical fluorine (F), Ba 2 Ca 3 Cu 4 O 8 F 2 (F0234) give rather convincing evidence against the SFI mechanism of pairing-see Figure 19.
First, F0234 is not a Mott insulator-as expected from valence charge counting which puts Cu valence as 2 + , but it is a superconductor with T c = 60 K. Moreover, the ARPES data [152] reveal at least two metallic Fermi-surface sheets with corresponding volumes equally below and above halffilling-see Figure 20.
Second, one of the Fermi surfaces is due to the electronlike (N) band (with 20 ± 6% electron-doping) and the other one due to the hole-like (P) band (with 20 ± 8% hole-doping) and their splitting along the nodal direction is significant and cannot be explained by the LDA-DFT calculations [153]. This electron and hole self-doping of inner and outer layers is in an appreciable contrast to other multilayered cuprates where there is only hole selfdoping. For instance, in HgBa 2 Ca n Cu n+1 O 2n+2 (n = 2, 3) and (Cu, C)Ba 2 Ca n Cu n+1 O 3n+2 (n = 2, 3, 4), the inner CuO 2 layers are less hole-doped than outer layers. It turns out, unexpectedly, that the superconducting gap on the N-band Fermi surface is significantly larger than on the P-one, where in Ba 2 Ca 3 Cu 4 O 8 F 2 the ratio is anomalous (Δ N /Δ P ) ≈ 2 and Δ N is an order of magnitude larger than in the electron-doped cuprate Nd 2−x Ce x CuO 4 . Third, the N-band Fermi surface is rather far from the antinodal point at (π, 0). This is very important result which means that the antiferromagnetic spin fluctuations with the AF wave-vector Q = (π, π), as well as the van Hove singularity, are not dominant in the pairing in the N-band. To remind the reader, the SFI scenario assumes that the pairing is due to spin fluctuations with the wave-vector Q (and near it) which connects two antinodal points which are near the van Hove singularity at the hole-surface (at (π, 0) and (0, π)) giving rise to large density of states. This is apparently not the case for the N-band Fermi surface-see Figure 20. The ARPES data give further that there is a kink at ∼85 meV in the quasiparticle dispersion of both bands, while the kink in the N-band is stronger than that in the P-band. This result, together with the anomalous ratio (Δ N /Δ P ) ≈ 2, disfavors SFI as a pairing mechanism. (S5) Despite the presence of significant elastic quasiparticle scattering in a number of samples of optimally doped Bi-2212, there are dramatic sharpenings of the spectral function near the antinodal point (π, 0) at T < T c (in the superconducting state) [154]. This effect can be explained by assuming that the small qscattering (the forward scattering peak) dominates in the elastic impurity scattering as it is pointed in [78-80, 130, 155, 156]. As a result, one finds that the impurity scattering rate in the superconducting state is almost zero, that is, γ imp (k, ω) = γ n (k, ω) + γ a (k, ω) ≈ 0 for |ω| < Δ 0 for any kind of pairing (s-, p-, d-wave, etc.) since the normal (γ n ) and the anomalous (γ a ) scattering rates compensate each other. This collapse of the elastic scattering rate is elaborated in details in [154] and it is a consequence of the Andersonlike theorem for unconventional superconductors which is due to the dominance of the small q-scattering [78-80, 130, 155, 156]. In such a case d-wave pairing is weakly unaffected by nonmagnetic impurities and as a consequence there is small reduction in T c [156,157]. The physics behind this result is rather simple. The small q-scattering (usually called forward scattering) means that electrons scatter into a small region in the k-space, so that at the most part of the Fermi surface there is no mixing of states with different signs of the order parameter Δ(k). In such a way the detrimental effect of nonmagnetic impurities on d-wave pairing is significantly reduced. This result points to the importance of strong correlations in the renormalization of the nonmagnetic impurity scattering too-see discussion in Section 2.
In conclusion, in order to explain the ARPES results in cuprates it is necessary to take into account (1) the electron-phonon interaction (EPI) since it dominates in the quasiparticle scattering in the energy region important for pairing, (2) the elastic nonmagnetic impurities with the forward scattering peak (FSP) due to strong correlations, and (3) the Coulomb interaction which dominates at higher energies ω > ω ph . In this respect, the presence of ARPES kinks and the knee-like shape of the T dependence of the spectral width are important constraints on the scattering and pairing mechanism in HTSC cuprates.

Tunnelling Spectroscopy and Spectral Function α 2 F(ω).
By measuring current-voltage I-V characteristics in NIS (normal metal-insulator-superconductor) tunnelling junctions with large tunnelling barrier one obtains from tunnelling conductance G NS (V ) = dI/dV the so called tunnelling density of states in superconductors N T (ω).  Moreover, by measuring of G NS (V ) at voltages eV > Δ it is possible to determine the Eliashberg spectral function α 2 F(ω) and finally to confirm the phonon mechanism of pairing in LTSC materials. Four tunnelling techniques were used in the study of HTSC cuprates: (1) vacuum tunnelling by using the STM technique-scanning tunnelling microscope; (2) point-contact tunnelling; (3) break-junction tunnelling; (4) planar-junction tunnelling. Each of these techniques has some advantages although in principle the most potential one is the STM technique since it measures superconducting properties locally [158]. Since tunnelling measurements probe a surface region on the scale of the superconducting coherence length ξ 0 , then this kind of measurements in HTSC materials with small coherence length ξ 0 (ξ ab ∼ 20Å in the a − b plane and ξ c ∼ 1-3Å along the c-axis) depends strongly on the surface quality and sample preparation. Nowadays, many of the material problems in HTSC cuprates are understood and as a result consistent picture of tunnelling features is starting to emerge.
From tunnelling experiments one obtains the (energydependent) gap function Δ(ω) in the superconducting state. Since we have already discussed this problem in [6], we will only briefly mention some important result. For instance, in most systems G NS (V ) has V -shape in all families of HTSC hole-and electron-doped cuprates. The V -shape is characteristic for d-wave pairing with gapless spectrum, which is also confirmed in the interference experiments on hole-and electron-doped cuprates [75]. Some experiments give a U-shape of G NS (V ) which resembles s-wave pairing. This controversy is explained to be the property of the tunnelling matrix element which filters out states with the maximal gap.
Here we are interested in the bosonic spectral function α 2 F(ω) of HTSC cuprates near optimal doping which can be extracted by using tunnelling spectroscopy. We inform the reader in advance that the shape and the energy width of α 2 F(ω), which are extracted from the second derivative d 2 I/dV 2 at voltages above the superconducting gap, in most HTSC cuprates resemble the phonon density of states F ph (ω). This result is strong evidence for the importance of EPI in the pairing potential of HTSC cuprates. For instance, plenty of break junctions made from Bi2212 single crystals [42][43][44][45] show that the peaks (and shoulders) in −d 2 I/dV 2 (or dips-negative peaks in d 2 I/dV 2 ) coincide with the peaks (and shoulders) in the phonon density of states F ph (ω) measured by neutron scattering-see Figure 21.
The tunnelling spectra in Bi-2212 break junctions [42][43][44][45], which are shown in Figure 21 indicates that the spectral function α 2 F(ω) is independent of magnetic field, which is in contradiction with the theoretical prediction based on the SFI pairing mechanism where this function should be sensitive to magnetic field. The reported broadening of the peaks in α 2 F(ω) is partly due to the gapless spectrum of dwave pairing in HTSC cuprates. Additionally, the tunnelling density of states N T (ω) at very low T and for ω > Δ shows a pronounced gap structure. It was found that 2Δ/T c = 6.2-6.5, where T c = 74-85 K and Δ is some average value of the gap. In order to obtain α 2 F(ω) the inverse procedure was used by assuming s-wave superconductivity and the effective Coulomb parameter μ * ≈ 0.1 [42][43][44][45]. The obtained α 2 F(ω) gives large EPI coupling constant λ ep ≈ 2.3. Although this analysis [42][43][44][45] was done in terms of s-wave pairing, it mimics qualitatively the case of d-wave pairing, since one expects that d-wave pairing does not change significantly the global structure of d 2 I/dV 2 at eV > Δ albeit introducing a broadening in it-see the physical meaning in Appendix A. We point out that the results obtained in [42][43][44][45] were reproducible on more than 30 junctions. In that respect very important results on slightly overdoped Bi2212-GaAs and on Bi2212-Au planar tunnelling junctions are obtained in [46,47]-see Figure 22.
These results show very similar features to those obtained in [42][43][44][45] on break junctions. It is worth mentioning that several groups [48][49][50][51][52] have obtained similar results for the shape of the spectral function α 2 F(ω) from the I-V measurements on various HTSC cuprates-see the comparison in Figure 23. These facts leave no much doubts about the importance of the EPI in pairing mechanism of HTSC cuprates.
In that respect, the tunnelling measurements on slightly overdoped Bi 2 Sr 2 CaCu 2 O 8 [46,47,53,54] give impressive results. The Eliashberg spectral function α 2 F(ω) of this compound was extracted from the measurements of d 2 I/dV 2 and by solving the inverse problem-see Appendix A. The extracted α 2 F(ω) has several peaks in broad energy region up to 80 meV as it is seen in Figures 22 and 23, which coincide rather well with the peaks in the phonon density of states F ph (ω)-more precisely the generalized phonon density of states GPDS(ω) defined in Appendix A. In [53,54] numerous peaks, from P1-P13, in α 2 F(ω) are discerned as shown in Figure 24, which correspond to various groups of phonon modes-laying in (and around) these peaks. Moreover, in [46,47,53,54] the coupling constants for these modes are extracted as well as their contribution (ΔT c ) to T c as it is seen in Table 1. Note that due to the nonlinearity of the problem the sum of (ΔT c ) i , i = 1, 2, . . . , 13, due to various modes is not equal to T c . The next remarkable result is that the extracted EPI coupling constant is very large, that is, λ ep (= 2 dω α 2 F(ω)/ω) = i λ i ≈ 3.5-see Table 1. It is obvious from Figure 24 and Table 1 that almost all phonon modes contribute to λ ep and T c , which means that on the average each particular phonon mode is not too strongly coupled to electrons since λ i < 1.3, i = 1, 2, . . . , 13, thus keeping the lattice stable.
Let us discuss the content of Table 1 in more details where it is shown the strength of the EPI coupling and the relative contribution of different phononic modes to T c . In Table 1 it is seen that lower-frequency modes from P1-P3, corresponding to Cu, Sr, and Ca vibrations, are rather strongly coupled to electrons (with λ κ ∼ 1) which give appreciable contributions to T c . It is also seen in Table 1 that the coupling constants λ i of the high-energy phonons (P9-P13 with ω ≥ 70 meV) have λ i 1 and give moderate contribution to T c -around 10%. These results give solid evidence for the importance of the lowenergy modes related to the change of the Madelung energy in the ionic-metallic structure of HTSC cuprates-the idea advocated in [3][4][5][6] and discussed in Section 2. If confirmed in other HTSC families, these results are in favor of the moderate oxygen isotope effect in cuprates near the optimal doping since the oxygen modes are higher-energy modes and give smaller contribution to T c . We stress that each peak P1-P13 in α 2 F(ω) corresponds to many modes. For a better understanding of the EPI coupling in these systems we show in Figure 25 the total and partial density of phononic states. It Table 1: Phonon frequency ω, EPI coupling constant λ i of the peaks P1-P13, and contribution ΔT c to T c of each peak in α 2 F(ω)-as shown in Figure 24-obtained from the tunnelling conductance of Bi 2 Sr 2 CaCu 2 O 8 . ΔT c is the decrease in T c when the peak in α 2 F(ω) is eliminated. From [53,54]. is seen that the low-energy phonons are due to the vibrations of the Ca, Sr, and Cu ions which correspond to the peaks P1-P2 in Figures 23 and 24. In order to obtain information on the structure of vibrations which are strongly involved in pairing, we show in Figures 26 and 27  these vibrations at special points in the Brillouin zone. It is seen in Figure 26 that the low-frequency phonons P1-P2 are dominated by Cu, Sr, Ca vibrations.
Further, based on Table 1 one concludes that the P3 modes are strongerly coupled to electrons than the P4 ones, although the density of state for the P4 modes is larger. The reason for such an anomalous behavior might be due to symmetries of the corresponding phonons as it is seen in Figure 27. Namely, to the P3 peak contribute axial vibrations of O(1) in the CuO 2 plane which are odd under inversion, while in the P4 peak these modes are even. The in-plane modes of Ca and O(1) are present in P3 which are in-phase and out-of-phase modes, while in P4 they are all out-of-phase modes. For more information on other modes, P5-P13, see [53,54]. We stress that the Eliashberg equations based on the extracted α 2 F(ω) of the slightly overdoped Bi 2 Sr 2 CaCu 2 O 8 with the ratio (2Δ/T c ) ≈ 6.5 describe rather well numerous optical, transport, and thermodynamic properties [53,54]. However, in underdoped systems with (2Δ/T c ) ≈ 10, where the pseudogap phenomena are pronounced, there are serious disagreements between experiments and the Migdal-Eliashberg theory [53,54]. We would like to stress that the contribution of the highfrequency modes (mostly the oxygen modes) to α 2 F(ω) may be underestimated in tunnelling measurements due to their sensitivity to the surface contamination and defects. Namely, the tunnelling current probes a superconductor to a depth of order of the quasiparticle mean-free path l(ω) = v F γ −1 (ω). Since the relaxation time γ −1 (ω) decreases with increasing ω, the mean-free path can be rather small and the effects of the high-energy phonons are sensitive to the surface contamination.
Similar conclusion regarding the structure of the EPI spectral function α 2 F(ω) in HTSC cuprates comes out from tunnelling measurements on Andreev junctions (the BTK parameter Z 1-low barrier) and Giaver junctions (Z 1-high barrier) in La 2−x Sr x CuO 4 and YBCO compounds [160], where the extracted α 2 F(ω) is in good accordance with the phonon density of states F ph (ω)-see Figure 28.
Note that the BTK parameter Z is related to the transmission and reflection coefficients for the normal metal (1 + Z 2 ) −1 and Z 2 (1 + Z 2 ) −1 , respectively.
Although most of the peaks in α 2 F(ω) of HTSC cuprates coincide with the peaks in the phonon density of states, it is legitimate to put the following question. Can the magnetic resonance in the superconducting state give significant contribution to α 2 F(ω)? In that respect the inelastic magnetic neutron scattering measurements of the magnetic resonance as a function of doping [161] give that the resonance energy E r scales with T c , that is, E r = (5-6)T c as shown in Figure 29. This means that if one of the peaks in α 2 F(ω) is due to the magnetic resonance at ω = E r , then it must shift strongly with doping as it is observed in [161]. This is contrary to phonon peaks (energies) whose positions are practically doping independent. To this end, recent tunnelling experiments on Bi-2212 [55] show clear doping independence of α 2 F(ω) as it is seen in Figure 30. This remarkable result is an additional evidence in favor of EPI and against the SFI mechanism of pairing in HTSC cuprates which is based on the magnetic resonance peak in the superconducting state. In that respect the analysis in [162] of the tunneling spectra of the electron-doped cuprate Pr 0.88 Ce 0.12 CuO 4 with T c = 24 K shows the existence of the bosonic mode at ω B = 16 meV which is significantly larger than the magneticresonance mode with ω r = (10-11) meV. This result excludes the magnetic-resonance mode as an important factor which modifies superconductivity.
The presence of pronounced phononic structures (and the importance of EPI) in the I(V ) characteristics was quite recently demonstrated by the tunnelling measurements on the very good La 1.85 Sr 0.15 CuO 4 films prepared by the molecular beam epitaxy on the [001]-symmetric SrTiO 3 bicrystal substrates [56]. They give unique evidence for eleven peaks in the (negative) second derivative, that is, −d 2 I/dV 2 . Furthermore, these peaks coincide with the peaks in the intensities of the phonon Raman scattering data measured at 30 K in single crystals of LSCO with 20% of Sr [57]. These results are shown in Figure 31. In spite of the  [48,49], and Shimada et al. [46,47]. The generalized density of states GPDS for Bi2212 is at the bottom. From [46,47].
lack of a quantitative analysis of the data in the framework of the Eliashberg equations, the results in [56] are important evidence that phonons are relevant pairing bosons in HTSC cuprates. It is interesting that in the c-axis vacuum tunnelling STM measurements [163] the fine structure in d 2 I/dV 2 at eV > Δ was not seen below T c , while the pseudogap structure is observed at temperatures near and above T c . This result could mean that the STM tunnelling is likely dominated by the nontrivial structure of the tunnelling matrix element (along the c-axis), which is derived from the band-structure calculations [164]. However, recent STM experiments on Bi2212 [61][62][63] give information on the possible nature of the bosonic mode which couples with electrons. In [61][62][63] the local conductance dI/dV (r, E) is measured where it is found that d 2 I/dV 2 (r, E) has peak at E(r) = Δ(r) + Ω(r) where dI/dV (r, E) has the maximal slope-see Figure 32(a).
It turns out that the corresponding average phonon energy Ω depends on the oxygen mass, that is, Ω ∼ M −1/2 O , with Ω 16 = 52 meV and Ω 18 ≈ 48 meV-as it is seen in Figure 32(b). This result is interpreted in [61][62][63] as an evidence that the oxygen phonons are strongly involved in  [46,47]. The averaged one is divided into 13 components. The origin of the ordinate is 2, 1, 0, and −0.5 from the top down. From [46,47,53,54]. the quasiparticle scattering. A possible explanation is put forward in [61][62][63] by assuming that this isotope effect is due to the B 1g phonon which interacts with the antinodal quasiparticles. However, this result requires a reanalysis since the energy of the bosonic mode in fact coincides with the dip and not with the peak of d 2 I/dV 2 (r, E)-as reported in [61][62][63].
The important message of numerous tunnelling experiments in HTSC cuprates near and at the optimal doping is that there is strong evidence for the importance of EPI in the quasiparticle scattering and that no particular phonon mode can be singled out in the spectral function α 2 F(ω) as being the only one which dominates in pairing mechanism. This important result means that the high T c is not attributable to a particular phonon mode in the EPI mechanism but all phonon modes contribute to λ ep . Having in mind that the phonon spectrum in HTSC cuprates is very broad (up to 80 meV), then the large EPI constant (λ ep 2) obtained in the tunnelling experiments is not surprising at all. Note that similar conclusion holds for some other oxide superconductors such as Ba 9.6 K 0.4 BiO 3 with T c = 30 K where the peaks in the bosonic spectral function/extracted from tunnelling measurements coincide with the peaks in the phononic density of states [165][166][167].

Phonon Spectra and EPI.
Although experiments related to phonon spectra and their renormalization by EPI, such as inelastic neutron, inelastic X-ray, and Raman  scattering, do not give the spectral function α 2 F(ω), they nevertheless can give useful, but indirect, information on the strength of EPI for some particular phonons. We stress in advance that the interpretation of the experimental results in HTSC cuprates by the theory of EPI for weakly correlated electrons is inadequate since in strongly correlated systems, such as HTSC cuprates, the phonon renormalization due to EPI is different than in weakly correlated metals [168]. Since these questions are reviewed in [168], we will briefly enumerate the main points: (1) in strongly correlated systems the EPI coupling for a number of phononic modes can be significantly larger than the LDA-DFT and Hartree-Fock methods predict. This is due to many-body effects not contained in LDA-DFT [168,169]. The lack of the LDA-DFT calculations in obtaining phonon line-widths is clearly demonstrated, for instance, in experiments on L 2−x Sr x CuO 4 -see review in [170] and references therein, where the bond-stretching phonons at q = (0.3, 0, 0) are softer and much broader than the LDA-DFT calculations predict. (Note the wave vector q is in units (2π/a, 2π/b, 2π/c)-for instance, in these units (π, π) corresponds to (0.5, 0.5).) (2) The calculation of phonon spectra is in principle very difficult problem since besides the complexity of structural properties in a given material one should take into account appropriately the long-range Coulomb interaction of electrons as well as strong short-range repulsion. Our intention is not to discuss this complexity here-for that see, for instance, [  we only stress some important points which will help to understand problems with which is confronted the theory of phonons in cuprates. The phonon Green's function D(q, ω) depends on the phonon self-energy Π(q, ω) which takes into account all the enumerated properties (note that D −1 (q, ω) = D −1 0 (q, ω) − Π(q, ω)). In cases when the EPI coupling constant g ep (k, k ) is a function on the transfer momentum q = k − k only, then Π(q) (q = (q, iω n )) depends on the quasiparticle charge susceptibility χ c (q) = P(q)/ε e (q):  From [46,47,53,54]. and P(q) is the irreducible electronic polarization given by The screening due to the long-range Coulomb interaction is contained in the electronic dielectric function ε e (q) while the "screening" due to (strong) correlations is described by the charge vertex function Γ c (p, q). Due to complexity of the physics of strong correlations the phonon dynamics was studied in the t-J model but without the long-range Coulomb interaction [168,169,171], in which case one has ε e = 1 and χ c (q) = P(q). However, in studying the phonon Advances in Condensed Matter Physics  Figure 28: (a) d 2 I/dV 2 of a Giaver-like contact in La 2−x Sr x CuO 4note the large structure below 50 meV; (b) d 2 I/dV 2 of an Andreevand Giaver-like contact compared to the peaks in the phonon density of states. From [160]. spectra in HTSC cuprates it is believed that this deficiency might be partly compensated by choosing the bare phonon frequency ω 0 (q) (contained in D −1 0 (q, ω)) to correspond to the undoped compounds [168,171]. It is a matter of future investigations to incorporate all relevant interactions in order to obtain a fully microscopic and reliable theory of phonons in cuprates. Additionally, the electron-phonon interaction (with the bare coupling constant g ep (q)) is dominated by the change of the energy of the Zhang-Rice singlet-see more in Section 2.3-and (41) for Π(q) is adequate one [6,168,169]. Since the charge fluctuations in HTSC cuprates are strongly suppressed (no doubly occupancy of the Cu 3d 9 state) due to strong correlations, and since the suppressed value of χ c (q) cannot be obtained by the band-structure calculations, this means that LDA-DFT underestimates the EPI coupling constant significantly. In the following we discuss this important result briefly.

k B T c
Underdoped Overdoped Figure 29: Doping dependence of the energy E r of the magnetic resonance peak at Q = (π, π) in YBCO and Bi2212 measured at low temperatures by inelastic neutron scattering. From [161].
(1) Inelastic Neutron and X-Ray Scattering-The Phonon Softening and the Line-Width due to EPI. The appreciable softening and broadening of numerous phonon modes has been observed in the normal state of HTSC cuprates, thus giving evidence for pronounced EPI effects and for inadequacy of the LDA-DFT calculations in treating strong correlations and suppression of the charge susceptibility [6,10,11,168,171].
There are several relevant reviews on this subject [10,11,168,170,172] [173][174][175] as it is seen in Figure 33. While the softening can be partly described by the LDA-DFT method [176], the latter theory predicts an order of magnitude smaller broadening than the experimental one. This failure of LDA-DFT is due to the incorrect treatment of the effects of strong correlations on the charge susceptibility χ c (q) and due to the absence of many-body effects which can increase the coupling constant g ep (q)-see more in Section 2.
The neutron scattering measurements in La 1.85 Sr 0.15 CuO 4 give evidence for large (30%) softening of the O Z Z with Λ 1 symmetry with the energy ω ≈ 60 meV, which is theoretically predicted in [177], and for the large line-width about 17 meV which also suggests strong EPI. These apex modes are favorable for d-wave pairing since their coupling constants are peaked at small momentum q [10,11]. Having in mind the above results, then it is not surprising that  the recent calculations of the EPI coupling constant λ ep in the framework of LDA-DFT give very small EPI coupling constant λ ep ≈ 0.3 [28,29]. The critical analysis of the LDA-DFT results in HTSC cuprates is done in [6] and additionally argued in [10,11,178] by pointing their disagreement with the inelastic neutron and X-ray scattering measurements-as it is shown in Figure 33.
In Section 2 we will discuss some theoretical approaches related to EPI in strongly correlated systems but without discussing the phonon renormalization. The latter problem was studied in more details in the review articles in [10,11,170]. Here, we point out only three (for our purposes) relevant results. First, there is an appreciable difference in the phonon renormalization in strongly and weakly correlated systems. Namely, the change of the phonon frequencies in the presence of the conduction electrons is proportional to the squared coupling constant |g q | and charge susceptibility χ c , that is, δω(q) ∼ |g ep (q)| 2 Re χ c , while the line-width is given by Γ ω(q) ∼ |g ep (q)| 2 | Im χ c |. All these quantities can be calculated in LDA-DFT and as we discussed above, where for some modes one obtains that Γ (LDA) ω(q) . However, it turns out that in strongly correlated systems doped by holes (with the concentration δ 1) the charge fluctuations are suppressed in which case the following sum rule holds [10,11,171]: while in the LDA-DFT method one has The inequality Γ (LDA) ω(q) (for some phonon modes) together with (43)-(44) means that for low doping δ 1 the LDA calculations strongly underestimate the EPI coupling constant in the large portion of the Brillouin zone, that is, one has |g (LDA) ep (q)| |g (exp) ep (q)|. The large softening and the large line-width of the half-breathing mode at q = (0.5, 0), but very moderate effects for the breathing mode at q = (0.5, 0.5), are explained in the framework of the one slave-boson (SB) theory (for U = ∞) in [171], where χ c (q, ω) (i.e., Γ c (p, q) = Γ c (p, q)) is calculated in leading O(1/N) order. We stress that there is another method for studying strong correlations-the X-method-where the controllable 1/N expansion is performed in terms of the Hubbard operators and where the charge vertex Γ c (p, q) is calculated [6, 78-80, 130, 179, 180]. It turns out that in the adiabatic limit (ω = 0) the vertex functions Γ c (p F , q) in these two methods have important differences. For instance, Γ (X) c (p F , q) (in the X-method) is peaked at q = 0-the so called forward scattering peak (FSP)-while Γ (SB) c (p F , q) has maximum at finite |q| / = 0 [181]-see Section 2.3.5. The enumerated properties of Γ (X) c (p F , q) are confirmed by the numerical Monte Carlo calculations in the finite-U Hubbard model [182], where it is found that FSP exists for all U, but it is especially pronounced in the limit U t. These results are also confirmed in [183] where the calculations are performed in the four-slave-boson technique-see more in Section 2.3.5. Having in mind this difference it would be useful to have calculations of χ c (q, ω) in the framework of the X-method which are unfortunately not done yet. Second, the many-body theory gives that for coupling to some modes the coupling constant |g ep (q)| in HTSC cuprates can be significantly larger than the LDA-DFT calculations predict [10,11], which is due to some many-body effects not present in the latter [169]. In Section 2 it will be argued that for some phonon modes one has |g ep (q)| 2 |g (LDA) ep (q)| 2 . For instance, for the half-breathing mode, one has |g ep (q)| 2 ≈ 3|g (LDA) ep (q)| 2 [10,11,169]-see Section 2. These two results point to an inadequacy of LDA-DFT in calculations of EPI effects in HTSC cuprates. Third, the phonon self-energy (Π(q)) and quasiparticle self-energy Σ(k) are differently renormalized by strong correlations [6, 10, 11, 78-80, 130, 179, 180], which is the reason that Π(q) is much more suppressed than Σ(k)-see Section 2. The effects of the charge vertex on Π(q) and Σ(k) are differently manifested. Namely, the vertex function enters quadratically in Σ(k) and the presence of the forward scattering peak in the charge vertex strongly affects the EPI coupling constant g ep (q) in Σ(k): where g(k)(≡ G(k)/Q) is the quasiparticle Green's function, γ c (k, q) = Γ c (k, q)/Q is the quasiparticle vertex, and Q(∼ δ) is the Hubbard quasiparticle spectral weight-see Section 2.3.
In the adiabatic limit |q| > q ω = ω ph /v F one has γ c (k, q) ≈ γ c (k, q) and for q q c (≈ δ · π/a) the charge vertex is strongly suppressed (γ c (k, q) 1) making the effective EPI coupling (which also enters the pairing potential) small at large (transfer) momenta q. This has strong repercussion on the pairing due to EPI since for small doping it makes the d-wave pairing coupling constant to be of the order of the s-one (λ d ≈ λ s ). Then in the presence of the residual Coulomb interaction EPI gives rise to d-wave pairing. On the other side the charge vertex Γ c (k, q) enters Π(q) linearly and it is additionally integrated over the quasiparticle momentum k-see (42). Therefore, one expects that the effects of the forward scattering peak on Π(q) are less pronounced than on Σ(k). Nevertheless, the peak of Γ c (k, q) at q = 0 may be (partly) responsible that the maximal experimental softening and broadening of the stretching (half-breathing) mode in La 1.85 Sr 0.15 CuO 4 and YBa 2 Cu 3 O 7 is at q (exp) hb = (0.3, 0, 0) [170] and not at q hb = (0.5, 0) for which g ep (q hb ) reaches maximum. This means that the charge vertex function pushes the maximum of the renormalized EPI coupling constant to smaller momenta q. It would be very interesting to have calculations for other phonons by including the vertex function obtained by the X-method-see Section 2.3.
(O dominated mode in the CuO 2 planes) show pronounced asymmetry which is absent in YBa 2 Cu 3 O 6 . This asymmetry means that there is an appreciable interaction of Raman active phonons with continuum states (quasiparticles).
(ii) The phonon frequencies for some A 1g and B 1g are strongly renormalized in the superconducting state, between (6-10)%, pointing again to the importance of EPI [188]see also [6,37,38]. To this point we mention that there is a Advances in Condensed Matter Physics 37 remarkable correlation between the electronic Raman crosssection S exp (ω) and the optical conductivity in the a − b plane σ ab (ω), that is, S exp (ω) ∼ σ ab (ω) [6]. In previous subsections it is argued that EPI with the very broad spectral function α 2 F(ω) (0 < ω 80 meV) explains in a natural way the ω and T dependence of σ ab (ω). This means that the electronic Raman spectra in cuprates can be explained by EPI in conjunction with strong correlations. This conclusion is supported by the calculations of the Raman cross-section [189] which take into account EPI with α 2 F(ω) extracted from the tunnelling measurements in YBa 2 Cu 3 O 6+x and Bi 2 Sr 2 CaCu 2 O 8+x [6,[42][43][44][45][46][47][48][49][50][51][52][53][54]. Quite similar properties (to cuprates) of the electronic Raman scattering, as well as of σ(ω), R(ω), and ρ(T), were observed in experiments [108] on isotropic 3D metallic oxides La 0.5 Sr 0.5 CoO 3 and Ca 0.5 Sr 0.5 RuO 3 where there are no signs of antiferromagnetic fluctuations. This means that low dimensionality and antiferromagnetic spin fluctuations cannot be a prerequisite for anomalous scattering of quasiparticles and EPI must be inevitably taken into account since it is present in all these compounds. Σ(k, ω). The isotope effect α TC in the critical temperature T c was one of the very important proofs for the EPI pairing mechanism in low-temperature superconductors (LTSCs). As a curiosity the isotope effect in LTSC systems was measured almost exclusively in monoatomic systems and in few polyatomic systems: the hydrogen isotope effect in PdH, the Mo and Se isotope shift of T c in Mo 6 Se 8 , and the isotope effect in Nb 3 Sn and MgB 2 . We point out that very small (α TC ≈ 0 in Zr and Ru) and even negative (in PdH) isotope effects in some polyatomic systems of LTSC materials are compatible with the EPI pairing mechanism but in the presence of substantial Coulomb interaction or lattice anharmonicity. The isotope effect α TC cannot be considered as the smoking gun effect since it is sensitive to numerous influences. For instance, in MgB 2 it is with certainty proved that the pairing is due to EPI and strongly dominated by the boron vibrations, but the boron isotope effect is significantly reduced, that is, α TC ≈ 0.3 and the origin for this smaller value is still unexplained. The situation in HTSC cuprates is much more complicated because they are strongly correlated systems and contain many atoms in unit cell. Additionally, the situation is complicated with the presence of intrinsic and extrinsic inhomogeneities, low dimensionality which can mask the isotope effects. On the other hand new techniques such as ARPES, STM, and μSR allow studies of the isotope effects in quasiparticle self-energies, that is, α Σ , which will be discussed below.

Isotope Effect in T c and
(1) Isotope Effect α TC in T c . This problem will be discussed only briefly since more extensive discussion can be found in [6]. It is well known that in the pure EPI pairing mechanism the total isotope coefficient α is given by α TC = i,p α is the mass of the ith element in the pth crystallographic position. We stress that the total isotope effect is not measured in HTSC cuprates but only some partial ones. Note that, in the case when the screened Coulomb interaction is negligible, that is, μ * c = 0, the theory predicts α TC = 1/2. From this formula one can deduce that the relative change of T c , δT c /T c , for heavier elements should be rather small-for instance, it is 0.02 for 135 Ba → 138 Ba, 0.03 for 63 Cu → 65 Cu, and 0.07 for 138 La → 139 La. This means that the measurements of α i for heavier elements are confronted with the ability of the present experimental techniques. Therefore most isotope effect measurements were done by substituting light atoms 16 there is a Cu-isotope effect which is of the order of the oxygen one, that is, α Cu ≈ α O giving α Cu + α O ≈ 0.25-0.35 for optimally doped systems (x = 0.15). In case when [190,191]. The appreciable copper isotope effect in La 2−x M x CuO 4 tells us that vibrations other than oxygen ions are important in giving high T c . In that sense one should have in mind the tunnelling experiments discussed above, which tell us that all phonons contribute to the Eliashberg pairing function α 2 F(k, ω) and according to these results the oxygen modes give moderate contribution to T c [53,54]. Hence the small oxygen isotope effect α (O) Tc in optimally doped cuprates, if it is an intrinsic property at all (due to pronounced local inhomogeneities of samples and quasi-two-dimensionality of the system), does not exclude the EPI mechanism of pairing.
(2) Isotope Effect α Σ in the Self-Energy. The fine structure of the quasiparticle self-energy Σ(k, ω), such as kinks and slopes, can be resolved in ARPES measurements and in some respect in STM measurements. It turns out that there is isotope effect in the self-energy in the optimally doped Bi2212 samples [139,141,142]. In the first paper on this subject [139] it is reported a red shift δω k,70 ∼ − (10)(11)(12)(13)(14)(15) meV of the nodal kink at ω k, 70 70 meV for the 16 O → 18 O substitution. In [139] it is reported that the isotope shift of the self-energy δΣ = Σ 16 − Σ 18 ∼ 10 meV is very pronounced at large energies ω = 100-300 meV. Concerning the latter result, there is a dispute since it is not confirmed in other experiments [141,142]. However, the isotope effect in Re Σ(k, ω) at low energies [141,142] is well described in the framework of the Migdal-Eliashberg theory for EPI [140] which is in accordance with the recent ARPES measurements with low-energy photons ∼7 eV [192]. The Effective Im Σ determined from MDC full widths. An impurity term is subtracted at ω = 0. From [192].
latter allowed very good precision in measuring the isotope effect in the nodal point of Bi-2212 with T 16 c = 92.1 K and T 18 c = 91.1 K [192]. They observed a shift in the maximum of Re Σ(k N , ω)-at ω k,70 ≈ 70 meV (it corresponds to the half-breathing or to the breathing phonon)-by δω k,70 ≈ 3.4 ± 0.5 meV as shown in Figure 34.
By analyzing the shift in ImΣ(k N , ω)-shown in Figure 34-one finds similar result for δω k,70 ≈ 3.2 ± 0.6 meV. The similar shift was obtained in STM measurements [61][62][63] which is shown in Figure 32(b) and can have its origin in different phonons. We would like to stress two points: (i) in compounds with T c ∼ 100 K the oxygen isotope effect in T c is moderate, that is, α (O) Tc < 0.1 [192]. If we consider this value to be intrinsic, then even in this case it is not in conflict with the tunnelling experiments [53,54] since the latter give evidence that vibrations of heavier ions contribute significantly to T c -see the discussion in Subsection 1.3.4 on the tunnelling spectroscopy. (ii) In ARPES measurements of [192] the effective EPI coupling constant λ ep,eff 0.6 is extracted, while the theory in Subsection 1.3.3 gives that the real coupling constant is larger, that is, λ ep > 1.2. This value is significantly larger than the LDA-DFT theory predicts λ ep,LDA < 0.3 [28,29]. This again points that the LDA-DFT method does not pick up the many-body effects due to strong correlations-see Section 2.
1.4. Summary of Section 1. The analysis of experimental data in HTSC cuprates which are related to optics, tunnelling, and ARPES measurements near and at the optimal doping gives evidence for the large electron-phonon interaction (EPI) with the coupling constant 1 < λ ep < 3.5. We stress that this analysis is done in the framework of the Migdal-Eliashberg theory for EPI which is a reliable approach for systems near the optimal doping. The spectral function α 2 F(ω), averaged over the Fermi surface, is extracted from various tunnelling measurements on bulk materials and tin films. It contains peaks at the same energies as the phonon density of states F ph (ω). So obtained spectral function when inserted in the Eliashberg equations provides sufficient strength for obtaining high critical temperature T c ∼ 100 K. These facts are a solid proof for the important role of EPI in the normal-state scattering and pairing mechanism of cuprates. Such a large (experimental) value of the EPI coupling constant and the robustness of the d-wave superconductivity in the presence of impurities imply that the EPI potential and the impurity scattering amplitude must be strongly momentum dependent. The IR optical reflectivity data provide additional but indirect support for the importance of EPI since by using the spectral function (extracted from tunnelling measurements) one can quantitatively explain frequency dependence of the dynamical conductivity, optical relaxation rate, and optical mass. These findings related to EPI are additionally supported by ARPES measurements on BSCO compounds. The ARPES kinks, the phononic features and the isotope effect in the quasiparticle self-energy in the nodal and antinodal points at low energies (ω ω c ) persist in the normal and superconducting state. They are much more in favor of EPI than for the spin fluctuation (SFI) scattering mechanism. The transport EPI coupling constant in HTSC cuprates is much smaller than λ ep , that is, λ tr ∼ λ ep /3, which points to some peculiar scattering mechanism not met in low-temperature superconductors. The different renormalization of the quasiparticle and transport selfenergies by the Coulomb interaction (strong correlations) hints to the importance of the small-momentum scattering in EPI. This will be discussed in Section 2.
The ineffectiveness of SFI to solely provide pairing mechanism in cuprates comes out also from the magnetic neutron scattering on YBCO and BSCO. As a result, the imaginary part of the susceptibility is drastically reduced in the low-energy region by going from slightly underdoped toward optimally doped systems, while T c is practically unchanged. This implies that the real SFI coupling constant λ sf (∼ g 2 sf ) is small since the experimental value g Advances in Condensed Matter Physics 39 is much smaller than the assumed theoretical value g (th) sf ≈ (0.7-1.5) eV. Inelastic neutron and X-ray scattering measurements in HTSC cuprates show that the broadening of some phonon lines is by an order of magnitude larger than the LDA-DFA methods predict. Since the phonon line-widths depend on the EPI coupling and the charge susceptibility, it is evident that calculations of both quantities are beyond the range of applicability of LDA-DFT. As a consequence, the LDA-DFT calculations overestimate the electronic screening and thus underestimate the EPI coupling, since many-body effects due to strong correlations are not contained in this mean-field theory. However, in spite of the promising and encouraging experimental results about the dominance of EPI in cuprates, the theory is still confronted with difficulties in explaining sufficiently large coupling constant in the d-channel. At present there is not such a satisfactory microscopic theory although some concepts, such as the the dominant EPI scattering at small transfer momenta, are understood at least qualitatively. This set of problems and questions will be discussed in Section 2.

Theory of EPI in HTSC
The experimental results in Section 1 give evidence that the electron-phonon interaction (EPI) in HTSC cuprates is strong and in order to be conform with d-wave pairing EPI must be peaked at small transfer momenta. A number of other experiments in HTSC cuprates give evidence that these are strongly correlated systems with large on-site Coulomb repulsion of electrons on the Cu-ions. However, at present there is no satisfactory microscopic theory of pairing in HTSC cuprates which is able to calculate T c and the order parameter. This is due to mathematical difficulties in obtaining a solution of the formally exact ab initio many-body equations which take into account two important ingredients-EPI and strong correlations [6]. In Section 2.1 we discuss first the ab initio many-body theory of superconductivity in order to point places which are most difficult to be solved. Since the superconductivity is low energy phenomenon (also in HTSC cuprates), one can simplify the structure of the ab initio equations in the low-energy sector (the Migdal-Eliashberg theory), where the highenergy processes are incorporated in the (so called) ideal band-structure (nonlocal) potential V IBS (x, y) and the vertex function Γ. This program of calculations of V IBS (x, y), Γ, and the EPI coupling (matrix elements) g ep (x, y) is not realized in HTSC superconductors due to its complexity. However, one pragmatical way out is to calculate g ep in the framework of the LDA-DFT method which is at present stage unable to treat strong correlations in a satisfactory manner. Some achievements and results of the LDA-DFT methods which are related to HTSC cuprates are discussed in Section 2.2.
In the case of very complicated systems, such as the HTSC cuprates, the standard (pragmatical) procedure in physics is to formulate a minimal theoretical model-sometimes called toy model-which includes minimal set of important ingredients necessary for qualitative and semiquantitative study of a phenomenon. As a consequence of the experimental results, the minimal theoretical model must comprise two important ingredients: (1) EPI and (2) strong correlations. In Section 2.3 we will formulate such a minimal theoretical model-called the t-J model which includes EPI too. In the framework of this model we will discuss the renormalization of EPI by strong correlations. In recent years the interest in these problems is increased and numerous numerical calculations were done mostly on small clusters with n × n atoms (n < 8). We will not discuss this subject which is fortunately covered in the recent comprehensive review in [10,11]. The analytical approaches in studying the renormalization of EPI by strong correlations, which are based on a controllable and systematic theory, are rather scarce. We will discuss such a systematic and controllable theory in the framework of the t-J model with EPI, which is formulated and solved in terms of Hubbard operators. The theory of this (toy) model predicts some interesting effects which might be important for understanding the physics of HTSC cuprates. It predicts that the high-energy processes (due to the suppression of doubly occupancy for U W b ) give rise to a nonlocal contribution to the band-structure potential (self-energy Σ(x, y, ω = 0)) as well as to EPI. This nonlocality in EPI is responsible for the peak in the effective pairing potential (V ep,eff (q, ω)) at small transfer momenta q( q c k F ) [6,[78][79][80]130]. The latter property allows that the (strong) EPI is conform with d-wave pairing in HTSC cuprates. Furthermore, the peculiar structural properties of HTSC cuprates and corresponding electronic quasi-twodimensionality give an additional nonlocality in EPI. The latter is due to the change of the weakly screened Madelung energy which is involved in most of the lattice vibrations along the c-axis. Since at present there is no quantitative theory for the latter effect, we tackle this problem here only briefly. The next task for the future studies of the physics of HTSC cuprates is to incorporate these structural properties in the minimal theoretical t-J model.
Finally, by writing this chapter our intention is not to overview the theoretical studies of EPI in HTSC cuprateswhich is an impossible task-but first to elucidate the descending way from the (old) well-defined ab initio microscopic theory of superconductivity to the one of the minimal model which treats the interplay of EPI and strong correlations. Next, we would like to encourage the reader to further develop the theory of HTSC cuprates.

Ab Iniitio Many-Body
Theory. The many-body theory of superconductivity is based on the fully microscopic electron-ion Hamiltonian for electrons and ions in the crystal-see, for instance, [193,194]. It comprises mutually interacting electrons which interact also with the periodic lattice and with the lattice vibrations. In order to pass continually to the problem of the interplay of EPI and strong correlations and also to explain why the LDA-DFT method is inadequate for HTSC cuprates, we discuss this problem here with restricted details-more extended discussion can be found in [6,194]. In order to describe superconductivity the Nambu-spinor ψ † (r) = ( ψ † ↑ (r) ψ ↓ (r)) is introduced which operates in the electron-hole space ( ψ(r) = ( ψ † (r)) † ) where ψ ↑ (r), ψ † ↑ (r) are annihilation and creation operators for spin up, respectively, and so forth. The microscopic Hamiltonian of the system under consideration contains three parts: H = H e + H i + H e-i . The electronic Hamiltonian H e , which describes the kinetic energy and the Coulomb interactions of electrons, is given by where 0 ( p) = p 2 /2m is the kinetic energy of electron and V c (r − r ) = e 2 /|r − r | is the electron-electron Coulomb interaction. Note that in the electron-hole space the pseudospin (Nambu) matrices τ i , i = 0, 1, 2, 3 are Pauli matrices. Since we will discuss only the electronic properties, the explicit form of the lattice Hamiltonian H i [6,194] is omitted here. The electron-ion Hamiltonian describes the interaction of electrons with the equilibrium lattice and with its vibrations, respectively: Here, V e-i (r − R 0 n ) is the electron-ion potential and its form depends on the level of description of the electronic subsystem. For instance, in the all-electron calculations one has V e-i (r − R 0 n ) = −Ze 2 /|r − R 0 n | where Ze is the ionic charge. The second term which is proportional to the lattice distortion operator Φ(r) = − n,α u αn ∇ α V e-i (r − R 0 n ) + Φ anh (r) (because of convenience it includes also the EPI coupling ∇ α V e-i ) describes the interaction of electrons with harmonic (∼ u αn ) (or anharmonic ∼ Φ anh (r)) lattice vibrations.
Dyson's equations for the electron and phonon Green's functions G (1,2) is the bare inverse electronic Green's function. Here, 1 = (r 1 , τ 1 ), where τ 1 is the imaginary time in the Matsubara technique, and the effective one-body potential u eff (1) = V e-i (1) + V H + Φ(1) , where V H is the Hartree potential. The electron and phonon selfenergies Σ(1, 2) and Π(1, 2) take into account many-body dynamics of the interacting system. The electronic self-energy Σ(1, 2) = Σ c (1, 2) + Σ ep (1,2) is obtained in the form where integration (summation) over the bar indices is understood. The effective retarded potential V eff (1, 1) in (48) contains the screened (by the electron dielectric function ε e (1, 2)) Coulomb and EPI interactions: The inverse electronic dielectric permeability ε −1 (48) is the solution of the complicated (and practically unsolvable) integro-differential functional equation Note that the effective vertex function Γ eff (1, 2; 3), which takes into account all renormalizations going beyond the simple Coulomb (RPA) screening, is the functional of both the electronic and phononic Green's functions G and D, thus making at present the ab initio microscopic equations practically unsolvable.

Low-Energy Migdal-Eliashberg Theory.
If the vertex function Γ eff would be known, we would have a closed set of equations for Green's functions which describe dynamics of the interacting electrons and lattice vibrations (phonons) in the normal and superconducting state. However, this is a formidable task and at present far from any practical realization. Fortunately, we are mostly interested in lowenergy phenomena (with energies |ω n |, ξ ω c and for momenta k = k F + δk in the shell δk δk c near the Fermi momentum k F ; ω c and δk c are some cutoffs), which allows us further simplification of equations [1,2]. Therefore, the strategy is to integrate high-energy processes-see more in [194]. Here, we sketch this procedure briefly. Namely, Green's function G(k, ω n ) = [iω n − (k 2 /2m − μ) τ 3 − Σ(k, ω n )] −1 can be formally written in the form where is the low-energy Green's function and G high (k, ω n ) = G(k, ω n )Θ(|ω n | − ω c )Θ(δk − δk c ) is the high-energy one and analogously D = D low (k, ω n ) + D high (k, ω n ). By introducing the small parameter of the theory s ∼ (ω/ω c ) ∼ (δk/δk c ) 1 one has in leading order G low (k, ω n ) ∼ s −1 , G high (k, ω n ) 1 and D low (k, ω n ) ∼ s 0 , D high (k, ω n ) ∼ s 2 . Note that the coupling constants (V ei , ∇V ei , V ii , etc.) are of the order s 0 = 1.
The procedure of separating low-energy and highenergy processes lies also behind the adiabatic approximation since in most materials the characteristic phonon (Debye) energy ω D of lattice vibrations is much smaller than the characteristic electronic Fermi energy E F (ω D E F ). In the Advances in Condensed Matter Physics 41 small s( 1) limit the Migdal theory [1,2] keeps in the total self-energy Σ linear terms in the phonon propagator D (D) only. In that case the effective vertex function can be written in the form Γ eff (1, 2; 3) ∼ = Γ c (1, 2; 3) + δ Γ ep (1, 2; 3) [1,2], where the Coulomb charge vertex Γ c (1, 2; 3) (3) contains correlations due to the Coulomb interaction only but does not contain EPI and phonon propagator D explicitly. The part δ Γ ep (1, 2; 3) = δ Σ ep (1, 2)/δu eff (3) contains all linear terms with respect to EPI. Note that in these diagrams enters the dressed Green's function which contains implicitly EPI up to infinite order. By careful inspection of all (explicit) contributions to δ Γ ep (1, 2; 3) which is linear in D one can express the selfenergy in terms of the charge (Coulomb) vertex Γ c (1, 2; 3) only. As a result of this approximation, the part of the selfenergy due to Coulomb interaction is given by where V sc c (1, 2) = V c (1, 2)ε −1 e (2, 2) is the screened Coulomb interaction. The part which is due to EPI has the following form: 2) is the screened EPI potential. Note that Σ ep (1, 2) depends now quadratically on the charge vertex Γ c , which is due to the adiabatic theorem.
It is well known that the Coulomb self-energy Σ c (1, 2) is the most complicating part of the electronic dynamics, but since we are interested in low-energy physics when s 1, then the term Σ c (1, 2) can be further simplified by separating it in two parts: The term Σ (h) c (1, 2) is due to high-energy processes contained in the product G high (1, 2) Γ high c (2, 2; 1) (e.g., due to the large Hubbard U in strongly correlated systems) and Σ (l) c (1, 2) is due to low-energy processes. The leading part of For further purposes we define the quantity V 0 as where V e-i , V H are also of order s 0 . After the Fourier transform with respect to time (and for small |ω n | ω c ) Σ (h) c is given by As we said, Σ (h) c0 ∼ s 0 while ( Σ (h) c0 ) · ω n ∼ s 1 because ω n ∼ s 1 . From (52) it is seen that the part Σ (l) c (1, 2) contains the lowenergy Green's function G low (1, 2) and this skeleton diagram is of order s 1 . The similar analysis based on (53) for Σ ep (1,2) gives that the leading order is s 1 which describes the lowenergy part of EPI. After the separations of terms (of s 0 and s 1 orders) the Dyson equation in the low-energy region has the form where x means integration d 3 x over the crystal volume. The Coulomb renormalization function Z c (x, y) = δ(x − y) − (Σ (h) 0c ) (x, y, 0) and the single-particle Hamiltonian H 0 (x, y) collect formally all high-energy processes which are unaffected by superconductivity (which is low-energy process) where with One can further absorb Z c (x, y) into the renormalized Green's function G r x, y, ω n = Z 1/2 c (x, x) G low x, y, ω n Z 1/2 c y, y , Here, is the ideal band-structure potential (sometimes called the excitation potential) and apparently nonlocal quantity, which is contrary to the standard local potential V g (x) in the LDA-DFT theories-see Section 2.2. The static potential V IBS (x, y) is of order s 0 and includes high-energy processes. Finally, we obtain the matrix Dyson equation for the renormalized Green's function G r (x, y, ω n ) which is the basis for the (strong-coupling) Migdal-Eliashberg theory in the low-energy region

42
Advances in Condensed Matter Physics where Σ (l) c,r and Σ ep,r have the same form as (52)-(53) but with the renormalized Green's and vertex functions G r , Γ r instead of G, Γ. We stress that (64) holds in the low-energy region only. In the superconducting state the set of Eliashberg equations in (64) are written explicitly in Appendix A, where it is seen that the superconducting properties depend on the Eliashberg spectral function α 2 kp F(ω). The latter function is defined also in Appendix A, (A.4), and it depends on material properties of the system.
The important ingredients of the low-energy Migdal-Eliashberg theory are the ideal band-structure Hamiltonian h 0 (x, y)-given by (62) which contains many-body (excitation) ideal band-structure nonlocal periodic crystal potential V IBS (x, y). The Hamiltonian h 0 (x, y) determines the ideal energy spectrum (k) of the conduction electrons and the wave function ψ i,p (x) through where μ is the chemical potential. We stress that the Hamiltonian h 0 (x, y) also governs transport properties of metals in low-energy region.
After solving (65) the next step is to expand all renormalized Green's function, self-energies, vertices, and the renormalized EPI matrix element (written symbolically as g ep,r = g ep,0 Γ ren ε −1 e ) in the basis of ψ i,p (x) and to write down the Eliashberg equations in this basis. We will not elaborate further this program and refer the reader to the relevant literature in [193,194]. We point out that even such simplified program of the low-energy Migdal-Eliashberg theory was never fully realized in low-temperature superconductors, because the nonlocal potential V IBS (x, y) (enters the ideal band-structure Hamiltonian h 0 (x, y)) and the renormalized vertex function (entering the EPI coupling constant g ep,r ) which include electronic correlations are difficult to calculate especially in strongly correlated metals. Therefore, it is not surprising at all that the situation is even more difficult in HTS materials which are strongly correlated systems with complex structural and material properties. Due to these difficulties the calculations of the electronic band structure and the EPI coupling are usually done in the framework of LDA-DFT where the manybody excitation potential V IBS (x, y) is replaced by some (usually local) potential V LDA (x) which in fact determines the ground-state properties of the crystal. In the next section we briefly describe (i) the LDA-DFT procedure in calculating the EPI coupling constant and (ii) some results of the LDA-DFT calculations related to HTSC cuprates. We will also discuss why this approximation is inappropriate when applied to HTS materials.

LDA-DFT Calculations of the EPI Matrix Elements.
We point out again two results which are important for the future microscopic theory of pairing in HTSC cuprates. First, numerous experiments (discussed in Part I) give evidence that the EPI coupling constant which enters the normal part of the quasiparticle self-energy λ Z ep = λ s + λ d + · · · is rather large, that is, 1 < λ Z ep < 3.5. In order to be conform with d-wave pairing the effective EPI potential must be nonlocal (and peaked at small transfer momenta q), which implies that the s-wave and d-wave coupling constants are of the same order, that is, λ d ≈ λ s . Second, the theory based on the minimal t-J model, which will be discussed in Section 2.3, gives that strong electronic correlations produce a peak at small transfer momenta in the effective EPI pairing potential thus giving rise to λ d ≈ λ s . This is a striking property which allows that EPI is conform with d-wave pairing. However, the theory is seriously confronted with the problem of calculation of the coupling constants λ Z ep . It turns out that at present it is an illusory task to calculate λ Z ep and λ d since it is extremely difficult (if possible at all) to incorporate the peculiar structural properties of HTSC cuprates (layered structure, ionic-metallic system, etc.) and strong correlations effects in a consistent and reliable microscopic theory which is described in Section 2.1. As it is stressed several times, the LDA-DFT methods miss some important many-body effects (especially in the band-structure potential) and therefore fail to describe correctly screening properties of HTSC cuprates and the strength of EPI. However, the LDA-DFT methods are able to incorporate diverse structural properties of HTSC cuprates much better than the simplified minimal t-J (toy) model. Here, we discuss briefly some achievements of the advanced LDA-DFT calculations which are able to take partially into account some nonlocal effects in the EPI. The latter are mainly due to the almost ionic structure along the c-axis which is reflected in the very small c-axis plasma frequency (ω c ω ab ). The main task of the LDA-DFT theory in obtaining the EPI matrix elements is to calculate the change of the groundstate (self-consistent) potential δV g (r)/δR α and the EPI coupling constant (matrix element) g LDA α (k, k ) (see its definition below), which is the most difficult part of calculations. Since in the LDA-DFT method the EPI scattering cannot be formulated, then the recipe is that the calculated g LDA α (k, k ) is inserted into the many-body Eliashberg equations. By knowing g LDA α (k, k ) one can define the total (λ) and partial (λ q,ν ) EPI coupling constants for the νth mode, respectively [195], as where p = 3κ is the number of phonon branches (κ is the number of atoms in the unit cell) and N(0) is the density of states at the Fermi energy (per spin and unit cell). The phonon line-width γ q,ν is defined in the Migdal-Eliashberg theory by Here, e α ν (q) is the phonon polarization vectors; n F is the Fermi function. Since the ideal energy spectrum from (65) ξ l,k = E l,k − μ and the corresponding eigenfunctions ψ kl are unknown, then instead of these one sets in (67)  α,ll is defined by the change of the ground-state potential δV g (r)/δR α : The index n means summation over the lattice sites; α = x, y, z and the wave function ψ (LDA) kl are the solutions of the Kohn-Sham equation-see [6]. In the past various approximations within the LDA-DFT method have been used in calculating δV g (r)/δR α and λ while here we comment some of them only. (i) In most calculations in LTS systems and in HTSC cuprates the rigid-ion (RI) approximation was used as well as its further simplifications which inevitably (due to its shortcomings and obtained small λ) deserves to be commented. The RI approximation is based on the very specific assumption that the ground-state (crystal) potential V g (r) can be considered as a sum of ionic potentials V g (r) = n V g (r − R n ) where the ion potential V g (r − R n ) and the electron density ρ e (r) are carried rigidly with the ion at R n during the ion displacement (R n = R 0 n + u αn ). In the RI approximation the change of V g (r) is given by which means that RI does not take into account changes of the electron density during the ion displacements. In numerous calculations applied to HTSC cuprates the rigidion model is even further simplified by using the rigid muffintin approximation (RMTA) (or similar version with the rigidatomic sphere)-see discussions in [195][196][197][198]. The RMTA assumes that the ground-state potential and the electron density follow ion displacements rigidly inside the Wigner-Seitz cell while outside it V g (r) is not changed because of the assumed very good metallic screening (e.g., in simple metals): This means that the dominant EPI scattering is due to the nearby atoms only and that the scattering potential is isotropic. All nonlocal effects related to the interaction of electrons with ions far away are neglected in the RMTA.
In this case g LDA α,n (k, k ) is calculated by the wave function centered at the given ion R 0 n which can be expanded inside the muffin-tin sphere (outside it the potential is assumed to be constant) in the angular momentum basis {l, m}, that is, (the angles φ, θ are related to the vector r = (r − R 0 n )/|r − R 0 n |). The radial function ρ l (|r − R 0 n |) is zero outside the muffin-tin sphere. In that case the EPI matrix element is given by g RMTA α,n (k, k ) ∼ Y lm | r|Y l m and because r is vector the selection rule implies that only terms with Δl ≡ l − l = ±1 contribute to the EPI coupling constant in the RMTA. This result is an immediate consequence of the assumed locality of the EPI potential in RMTA. However, since nonlocal effects, such as the long-range Madelunglike interaction, are important in HTSC cuprates, then additional terms contribute also to the coupling constant g α,n , that is, g α,n (k, k ) = g RMTA α,n (k, k ) + g nonloc α,n (k, k ), where a part (δg nonloc α,n ) of the nonlocal contribution to g nonloc α,n is represented schematically: From (72) comes out the selection rule Δl = l − l = 0 for the nonlocal part of the E − P interaction. We stress that the Δl = 0 (nonlocal) terms are omitted in the RMTA approach and therefore it is not surprising that this approximation works satisfactorily in elemental (isotropic) metals only. The latter are characterized by the large density of states at the Fermi surface which makes electronic screening very efficient. This gives rise to a local EPI where an electron feels potential changes of the nearby atom only. One can claim with certainty that the RMTA method is not suitable for HTSC cuprates which are highly anisotropic systems with pronounced ionic character of binding and pronounced strong electronic correlations. The RMTA method applied to HTSC cuprates misses just this important part-the longrange part EPI due to the change of the long-range Madelung energy in the almost ionic structure of HTSC cuprates. For instance, the first calculations done in [199] which are based on the RMTA give very small EPI coupling constant λ RMTA ∼ 0.1 in YBCO, which is in apparent contradiction with the experimental finding that λ ep is large-see Section 1. However, these nonlocal effects are taken into account in [195] by using the frozen-in phonon (FIP) method in evaluating of λ ep in La 2−x M x CuO 4 . In this method some symmetric phonons are considered and the band structure is calculated for the system with the super-cell which is determined by the periodicity of the phonon displacement. By comparing the unperturbed and perturbed energies the corresponding EPI coupling λ ν (for the considered phonon νth mode) is found. More precisely speaking, in this approach the matrix elements of δV g (r)/δR κ 0,α are determined from the finite difference of the ground-state potential ΔV g,q,ν (r) = V g (R κ 0,L + Δτ κ q,ν (L)) − V g (R κ 0,L ) = L,κ Δτ κ q,ν (L)∂V g (R κ 0,L )/∂R κ 0,L , where L, κ enumerate elementary lattice cells and atoms in the unit cell, respectively. The frozen-in atomic displacements of the phonon Δτ κ q,ν (L) of the νth mode are given by Δτ κ q,ν (L) = Δu q,ν (D/2M κ ω q,ν ) 1/2 Re[e κ,ν (q)e iq·R ] where Δu q,ν is the dimensionless phonon amplitude and the phonon polarization (eigen)vector e κ,ν (q) fulfills the condition κ e * κ,ν (q) · e κ,ν (q) = δ ν,ν . Based on this approach various symmetric A g (and some B 3g ) modes of La 2−x M x CuO 4 were studied [195], where it was found that the large matrix elements are due to unusually long-range Madelung-like, especial for the c-axis phonon modes. The obtained large λ ep ≈ 1.37 is the consequence of the following three main 44 Advances in Condensed Matter Physics facts. (i) The electronic spectrum in HTSC cuprates is highly anisotropic, that is, it is quasi-two-dimensional. This is an important fact for pairing because if the conduction electrons would be uniformly spread over the whole unit cell then due to the rather low electron density (n ∼ 10 21 cm −3 ) the density of states on the Cu and O in-plane atoms would be an order of magnitude smaller than the real value. This would further give an order of magnitude smaller EPI coupling constant λ ep . Note that the calculated density of states on the (heavy) Cu and (light) O in-plane atoms, N Cu (0) ∼ = 0.54 states/eV and N Oxy (0) ∼ = 0.35 states/eV, is of same order of magnitude as in some LTS materials. For instance, in NbC where T c ≈ 11 K one has on (the heavy) Nb atom N Nb (0) ∼ = 0.58 states/eV and on (the light) C atom N C (0) ∼ = 0.25 states/eV. So, the quasi-two-dimensional character of the spectrum is crucial in obtaining appreciable density of states on the light O atoms in the CuO 2 planes. (ii) In HTSC cuprates there is strong Cu-O hybridization leading to good in-plane metallic properties. This large covalency in the plane is due to the (fortunately) small energy separation of the electron levels on Cu and O xy atoms which comes out from the band-structure calculations [200], that is, Δ = | Cu − Oxy | ≈ 3 eV. The latter value gives rise to strong covalent mixing (the hybridization parameter t pd ) of the Cu d x 2 −y 2 and O px,y states, that is, t pd = −1.85 eV. It is interesting that the small value of Δ is not due to the ionic structure (crystal field effect) of the system but it is mainly due to the natural falling of the Cu d x 2 −y 2 states across the transitionmetal series. So, the natural closeness of the atomic energy levels of the Cu d x 2 −y 2 and O px, py states is this distinctive feature of HTSC cuprates which basically allows achievement of high T c . (iii) The ionic structure of HTSC cuprates which is very pronounced along the c-axis is responsible for the weak electronic screening along this axis and according to that for the significant contribution of the nonlocal (longrange) Madelung-like interaction to EPI. It turns out that because of the ionicity of the structure the La and O z axial modes are strongly coupled with charge carriers in the CuO 2 planes despite the fact that the local density of states on these atoms is very small [195], that is, N La (0) = 0.13 states/eV and N Oz (0) = 0.016 states/eV. (For comparison, on planar atoms Cu and O xy one has N Cu (0) = 0.54 states/eV and N Oxy (0) = 0.35 states/eV.) These calculations show that the lanthanum mode (with ω q,ν = 202 cm −1 ) at the q = (0, 0.2π/c) zone boundary (fully symmetric Z-point) has ten times larger coupling constant λ La q,ν (FIP) = 4.8 than it is predicted in the RMT approximation λ La q,ν (RMT) = 0.48. The similar increase holds for the average coupling constant, where λ La ν,average (FIP) = 1.0 but λ La ν,average (RMT) = 0.1. Note that for the q ≈ 0 La-mode one obtains λ La ν (FIP) = 4.54 compared to λ La ν (RMT) = 0.12. Similar results hold for the axial apexoxygen q = (0, 0.2π/c) mode (O z ) with ω q,ν = 396 cm −1 where the large (compared to the RMT method) coupling constant is obtained: λ Oz q,ν = 14 and λ Oz ν,average = 4.7, while for q ≈ 0 axial apex-oxygen modes with ω q,ν = 415 cm −1 one has λ Oz ν,average = 11.7. After averaging over all calculated modes it was estimated that λ = 1.37 and ω log ≈ 400 K. By assuming that μ * = 0.1 one obtains T c = 49 K by using Allen-Dynes formula for T c ≈ 0.83ω log exp{−1.04(1+λ)/[λ− μ * (1+0.62λ)]} with ω log = 2 dω dω α 2 (ω)F(ω) ln ω/λω. For μ * = 0.15 and 0.2 one obtains T c = 41 and 32 K, respectively. We stress that the rather large λ ep (and T c ) is due to the nonlocal (long range) effects of the metallic-ionic structure of HTSC cuprates and non-muffin-tin corrections in EPI, as was first proposed in [201,202]. However, we would like to stress that the optimistic results for λ ep obtained in [195] are in fact based on the calculation of the EPI coupling for some wave vectors q with symmetric vibration patterns and in fact the obtained λ ep is an extrapolated value. The allq calculations of λ ep,q which take into account long-range effects are a real challenge for the LDA-DFT calculations and are still awaiting.
Finally, it is worth to mention important calculations of the EPI coupling constant in the framework of the linear-response full-potential linear-muffin-tin-orbital method (LRFP-LMTO) invented in [203,204] and applied to the doped HTSC cuprate (Ca 1−x Sr x ) 1−y CuO 2 for x ∼ 0.7 and y ∼ 0.1 with T c = 110 K [205]. Namely, these calculations give strong evidence that the structural properties of HTSC cuprates already alone make the dominance of small-q scattering in EPI, whose effect is additionally increased by strong correlations. In order to analyze this compound in [205] the calculations are performed for CaCuO 2 doped by holes in a uniform, neutralizing back-ground charge. The momentum (q = (q , q ⊥ )) dependent EPI coupling constant (summed over all phonon branches ν) in different L channels (s, p, d.) is calculated by using a standard expression Here, ξ k is the quasiparticle energy, g k,q,ν is the EPI coupling constant (matrix element) with the νth branch, Y L (k) is the L-channel wave function, and the normalization factor M ∝ N −1 L (0) with the partial density of states is N L (0) ∝ k Y 2 L (k)δ(ξ k ). The total coupling constant in the L-channel is an average of λ L (q) over the whole 2D Brillouin zone (over q ), that is, λ L (q ⊥ ) = λ L (q ) BZ . We stress three important results of [205]. First, the sand d-coupling constants, λ s (q), λ d (q), are peaked at small transfer momenta q ∼ (π/3a, 0, 0) as it is shown in [205, Figure 3]. This result is mainly caused by the nesting properties of the Fermi surface shown in [205, Figure 1]. Second, the q-dependence of the integrated EPI matrix elements |g L,q | 2 = λ L (q)/χ L (q) (with χ L (q) ∝ k Y L (k + q)Y L (k)δ(ξ k+q )δ(ξ k )) for L = s, d is similar to that of λ L (q), that is, these are peaked at small transfer momenta q 2k F . Both of these results mean that the structural properties of HTSC cuprates imply the dominance of smallq EPI scattering. Third, the calculations give similar values for λ s (q ⊥ = 0) and λ d (q ⊥ = 0), that is, λ s = 0.47 for swave and λ d = 0.36 for d-wave pairing [205]. The result that λ d ≈ λ s is due to the dominance of the small q-scattering in EPI, which means that the nonlocal effects (long-range forces) in EPI of HTSC cuprates are very important. This result together with the finding of the dominance of the small-q scattering in EPI due to strong correlations [78-80, 130, 179, 180] mean that strong correlations and the i ( − → σ ) σ1σ2 X 0σ2 i where summation over the bar indices is understood. The operator σ X σσ i has the meaning of the hole number on the ith site. It is useful to introduce the operator X 00 i = X 0σ i X σ0 i at the ith lattice site which is the number of Zhang-Rice singlets on the ith site. For X 00 i |0 = 1|0 the ith site is occupied by the Zhang-Rice singlet, while for X 00 i |1 = 0|1 there is no Zhang-Rice singlet on the ith site (i.e., this site is occupied only by one 3d 9 hole on the Cu site). This property of X 00 i is due to the local constraint which forbids doubly occupancy of the ith site by holes. By projecting out doubly occupied (high-energy) states the t-J model reads The first term (∼ 0 i ) describes an effective local energy of the hole (or the Zhang-Rice singlet), the second one (∼ t i j ) describes hopping of the holes, and the third one (∼ J i j ) is the Heisenberg-like exchange energy between two holes. The theory [212] predicts that | 0 This property is very important in the study of EPI. H 3 contains three-site term which is usually omitted believing that it is not important. For charge fluctuation processes it is plausible to omit it, while for spin-fluctuation processes it is questionable approximation. If one introduces the enumeration α, β, γ, λ = 0, ↑, ↓, then the Hubbard operators satisfy the following algebra: where δ i j is the Kronecker symbol. Note that the Hubbard operators possess the projection properties with X αβ i X γλ i = δ βγ X αλ i . The (anti)commutation relations in (78) are more complicated than the canonical Fermi and Bose (anti)commutation relations, which complicates the mathematical structure of the theory. To escape these complications some novel techniques have been used, such as the one slave boson-technique. In this technique X 0σ are represented in terms of the fermion (spinon) operator f iσ which annihilates the spin on the ith and the boson (holon) operator b † i which creates the Zhang-Rice singlet.
In the minimal theoretical model the electron-phonon interaction (EPI) contains in principle two leading terms: which are the "ionic" one ( H ion ep ) and the "covalent" one ( H cov ep ). The "ionic" term describes the change of the energy of the hole (or the Zhang-Rice singlet) at the ith site due to lattice vibrations and it reads [6,[78][79][80]130] H ion where the "displacement" operator (which as in Section 2.1 includes the bare coupling constant) describes the change of the hole (or Zhang-Rice singlet) energy 0 a,i by displacing atoms in the lattice by the vector u Lκ . In the harmonic approximation the EPI potential is given where b q,λ and b † q,λ are the annihilation and creation operator of phonons with the polarization λ, respectively. This term describes in principle the following processes: (1) the change of the Ohole and Cu-hole bare energies 0 ph , 0 dh in the three-band model due to lattice vibrations, (2) the change of the longrange Madelung energy (which is due to the ionicity of the structure) by lattice vibrations along the c-axis, and (3) the change of the Cu-O hopping parameter t pd in the presence of vibrations, and so forth. Here, L and κ enumerate unit lattice vectors and atoms in the unit cell, respectively. Usually, the EPI scattering is studied in the harmonic approximation where the phonon operator Φ i is calculated in the harmonic approximation ( Φ ∼ u) for the EPI interaction of holes with some specific phononic modes, such as the breathing and half-breathing ones [10,11,169]. The theory which includes also all other (than oxygen) vibrations in Φ i is still awaiting.
It is interesting to make comparison of the EPI coupling constants in the t-J model and in the Hartree-Fock (HF) approximation (which is the analogous of the LDA-DFT method) of the three-band Emery (p-d) model in (74)-(75) when the problem is projected on the single band. For instance, the coupling constant with the half-breathing mode at the zone boundary in the HF approximation (which mimics the LDA-DFT approach) is given by while the coupling constant in the t-J model g t-J hb (= ∂ 0 /∂R Cu-O ) is given by (83) where p = 0.96-see [10,11,169] and references therein. It is obvious that in the t-J model the electron-phonon coupling is different from the HF one, since the former contains an additional term coming from the many-body effects, which are not comprised by the HF (LDA-DFT) calculations. The first term in (83) describes the hopping of a 3D hole into the O 2p-states and this term exists also in the LDA-DFT coupling constant-see (82). However, the second term in (83), which is due to many-body effects, describes the hopping of an O 2p-hole into the (already) single occupied Cu 3D state and it does not exist in the LDA-DFT approach.
Since the corresponding dimensionless coupling constant λ hb is proportional to |g hb | 2 , one obtains that the bare t-J coupling constant is almost three times larger than the LDA-DFT one: This example demonstrates clearly that the LDA-DFT method is inadequate for calculating the EPI coupling constant in HTSC cuprates. Note that there is also a covalent contribution to EPI which comes from the change of the effective hopping (t) in of the t-J model (77) and the exchange energy (J) in the presence of atomic displacements: Here, we will not go into details but only stress that since | 0 i | |t i j | then the covalent term in the effective t-J model is much smaller than the ionic term-see more in [6,10,11,169] and references therein-and in the following only the term H ion ep will be considered [6,[78][79][80]130].

Controllable X-Method for the Quasiparticle Dynamics.
The minimal model Hamiltonian for strongly correlated holes with EPI (discussed above) is expressed via the Hubbard operators which obey "ugly" noncanonical commutation relations. The latter property is rather unpleasant for making a controllable theory in terms of Feynmann diagrams (for these "ugly" operators) and some other approaches are required. A possible way out is to express the Hubbard operators in terms of fermions and bosons (which must be confined) as, for instance, in the slave boson (SB) method. However, in real calculations which are based on some approximations the SB method is confronted with some subtle constraints whose fulfillments require very sophisticated mathematical treatment. Fortunately, there is a mathematically controllable approach for treating the problem directly with Hubbard operators and without using slave-boson (or fermion) techniques. This method-we call it the Xmethod-is based on the general Baym-Kadanoff technique which allows to treat the problem by the well-defined and controllable 1/N expansion for the Green's functions in terms of Hubbard operators. This approach is formulated in [214] while the important refinement of the method is done in [78][79][80]130]. In the paramagnetic and homogeneous state (with finite doping) the Green's function with the Hubbard spectral weight Q = X 00 + X σσ . The function g(1 − 2) plays the role of the quasiparticle Green's function-see more in [6, 78-80, 130, 179, 180]. It turns out that in order to have a controllable theory (1/N expansion) one way is to increase the number of spin components from two to N by changing the constraint (76) into the new one In order to reach the convergence of physical quantities in the limit N → ∞ the hopping and exchange energy are also rescaled, that is, t i j = t 0,i j /N and J i j = J 0,i j /N. In order to eliminate possible misunderstandings we stress that in the case N > 2 the constraint in (87) spoils some projection properties of the Hubbard operators. Fortunately, these (lost) projection properties are not used at all in the refined theory. As a result one obtains the functional integral equation for G(1, 2), thus allowing unambiguous mathematical and physical treatment of the problem. In [78-80, 130, 179, 180] it is developed a systematic 1/N expansion for the quasiparticle Green's function g(1 − 2)(= g 0 + g 1 /N + · · · ), Q(= Nq 0 + q 1 + · · · ) (also for G(1 − 2)) and the self-energy. For large Here, δ is the concentration of the oxygen holes (that is, of the Zhang-Rice singlets) which is related to the chemical potential by the The quasiparticle Green's function g 0 (k, ω) and the quasiparticle spectrum 0 (k) in the leading order are given by The level shift is c = 0 + 2 p t(p)n F (p) and t(p) is the Fourier transform of the hopping integral t i j -see more in [6]. Let us summarize the main results of the X-method in leading O(1)-order for the quasiparticle properties in the t-J model [6, 78-80, 130, 179, 180]. (i) The Green's function g 0 (k, ω) describes the coherent motion of quasiparticles whose contribution to the total spectral weight of the Green's function G 0 (k, ω) is Q 0 = Nδ/2. The coherent motion of quasiparticles is described in leading order by G 0 (k, ω) = Q 0 g 0 (k, ω) and the quasiparticle residuum Q 0 disappears in the undoped Mott insulating state (δ = 0). This result is physically plausible since in the Mott insulating state the coherent motion of quasiparticles, which is responsible for finite conductivity, vanishes. (ii) The quasiparticle spectrum 0 (k) plays the same role as the eigenvalues of the ideal bandstructure Hamiltonian h 0 (x, y) (it contains the excitation potential V IBS (x, y) which is due to high-energy processes of the Coulomb interaction). So, if we would consider tb (k) = −t(k) as the tight-binding parametrization of the LDA-DFT band-structure spectrum which takes int account only weak correlations (with the local potential V xc (x)δ(x − y)), then one can define a nonlocal excitation potential V tJ IBS (x, y) = V tJ IBS (x, y) + V xc (x)δ(x − y) which mimics strong correlations in the t-J model Here, V 0 = 2 p t(p)n F (p) and t(x − y) is the Fourier transform of t(k) while J(x − y) is the Fourier transform of the third term in (89). The relative excitation potential V tJ IBS (x, y) is due to strong correlations (suppression of doubly occupancy on each lattice site) and as we will see below it is responsible for the short-range screening of EPI in such a way that the forward scattering peak appears in the effective EPI interaction-see discussion below. (iii) For the very low doping 0 (k) is dominated by the exchange parameter if J 0 > δ · t 0 . However, in the case when J 0 δ · t 0 there is a band narrowing by lowering the holedoping δ, where the band width is proportional to the holeconcentration δ, that is, W b = z · δ · t 0 . (iv) The O(1)-order quasiparticle Green's function g 0 (k, ω) and the quasiparticle spectrum 0 (k) in the X-method have similar form as the spinon Green's function g 0, f (k, ω) = − T f σ f † σ k,ω and the spinon energy s (k) in the SB method. However, in the SB method there is a broken gauge symmetry in the metallic state (with δ / = 0) which is characterized by b i / = 0. This broken local gauge symmetry in the slave-boson method in O(1) order, which is due to the local decoupling of spinon and holon, is in fact forbidden by Elitzur's theorem. On the other side the local gauge invariance is not broken in the Xmethod where Green's function G 0 (k, ω) describes motion of the composite object, that is, simultaneous creation of the hole and annihilation of the spin at a given lattice site, while in the SB theory there is a spin-charge separation because of the broken symmetry ( b i / = 0). The assumption of the broken symmetry b i / = 0 gives qualitative satisfactory results for the quasiparticle energy for the case N = ∞ in D > 2 dimensions. However, the analysis of response functions and of higher-order 1/N corrections to the self-energies very delicate in the SB theory and special techniques must be implemented in order to restore the gauge invariance of the theory. On the other side the X-method is intrinsically gauge invariant and free of spurious effects in all orders of the 1/N expansion. Therefore, one expects that these two methods may deliver different results in O(1) and higher order in response functions. This difference is already manifested in the calculation of EPI where the charge vertex in these two methods is peaked at different wave vectors q, that is, at q = 0 in the X-method and |q| / = 0 in the SB methodsee Section 2.3.5. (v) In [215,216] it is shown that in the superconducting state the anomalous self-energies (which are of O(1/N)-order in the 1/N expansion) of the Xand SB-methods differ substantially. As a consequence, the SB method [217] predicts false superconductivity in the t-J model (for J = 0) with large T c (due to the kinematical interaction), while the X-method gives extremely small T c (≈ 0) [215,216]. So, although the two approaches yield some similar results in leading O(1)-order they, are different at least in next to leading O(1/N)-order.

EPI Effective Potential in the t-J
Model. The theory of EPI in the minimal t-J model based on the X-method predicts that the leading term in the EPI self-energy Σ ep is given by the expression [6,[78][79][80]130] Σ ep (1, 2) = −V ep 1 − 2 γ c 1, 3; 1 g 0 3 − 4 γ c 4, 2; 2 , (91) where the screened (by the dielectric constant) EPI potential and V 0 ep (1−2) = − T Φ(1) Φ(2) is the "phonon" propagator which may also describe an anharmonic EPI. It is obvious that (91) is equivalent to (53) in spite the fact that the theory is formulated in terms of the Hubbard operators. The charge vertex γ c (1, 2; 3) = −δg −1 0 (1, 2)/δu eff (3) corresponds to the the renormalized vertex Γ c,r in (53) and it describes the screening by strong correlations. It depends on the relative excitation potential V tJ IBS (x, y). The electronic dielectric function ε e (1−2) describes the screening of EPI by the long-range part of the Coulomb interaction. Note that in the harmonic approximation Φ(1) contains the bare EPI coupling constant g 0 ep and lattice displacement u, that is, Φ ∼ g 0 ep u-see more in [6]. (Note that in the above equations summation and integration over bar indices are understood.) The self-energy Σ ep (k, ω) due to EPI reads with R(ω, ν) = −2πi(n B (ν) + 1/2) + ψ(1/2 + i) − ψ(1/2 − i(ν + ω)/2πT) where n B (ν) is the Bose distribution function and ψ is di-gamma function. The Eliashberg spectral function is given by where g ν (k, p) is the EPI coupling constant for the νth mode, where the renormalization by long-range Coulomb interaction is included, that is, g ν (k, p) = g 0 ep,ν (k, p)/ε e (p). · · · k denotes Fermi-surface average with respect to the momentum k and N(0) is the density of states renormalized by strong correlations. The effect of strong correlations in the adiabatic limit is stipulated in the charge vertex function γ c (k, k − k ) which, as we will see in Section 2.3.4, changes the properties of V ep (q, ν) drastically compared to weakly correlated systems. In fact the charge vertex depends on frequency ω but in the adiabatic limit (ω ph W) and for qv F > ω ph it is practically frequency independent, that is, γ (ad) c (k, q, ω) ≈ γ c (k, q, ω = 0) where the latter is real quantity. For Depending on the symmetry of the superconducting order parameter Δ(k, ω) (s-and d-wave pairing) various projected averages (over the Fermi surface) of α 2 F(k, k , ω) enter the Eliashberg equations. Assuming that the superconducting order parameter transforms according to the representation Γ i of the point group C 4v of the square lattice (in the CuO 2 planes) the appropriate symmetry-projected spectral function is given by where k and k are momenta on the Fermi line in the irreducible Brillouin zone (1/8 of the total Brillouin zone). T j , j = 1, . . . , 8 denotes the eight point-group transformations forming the symmetry group of the square lattice. This group has five irreducible representations which we distinguish by the label i = 1, 2, . . . , 5. In the following we discuss the representations i = 1 and i = 3, which correspond to the sand d-wave symmetry of the full rotation group, respectively. D i ( j) is the representation matrix of the jth transformation for the representation i. Assuming that the superconducting order parameter Δ(k, ω) does not vary much in the irreducible Brillouin zone, one can average over k and k in the Brillouin zone. For each symmetry one obtains the corresponding pairing spectral function α 2 F i (ω): which governs the transition temperature for the order parameter with the symmetry Γ i . For instance, α 2 F 3 (ω) is the pairing spectral function in the d-channel and it gives the coupling for d-wave superconductivity (the irreducible representation Γ 3 -sometimes labelled as B 1g ). Performing similar calculations for the phonon-limited resistivity, one finds that the resistivity is related to the transport spectral function α 2 F tr (ω): The effect of strong correlations on EPI was discussed in [130] within the model where g ν (k, p) and the phonon frequencies ω ν ( k − k ) are weakly momentum dependent. In order to elucidate the main effect of strong correlations on EPI and α 2 F i (ω) we consider the latter functions for a simple model with Einstein phonon, where these functions are proportional to the (so called) relative coupling constant Λ i : Similarly, the resistivity ρ(T)(∼ λ tr ∼ Λ tr ) is renormalized by the correlation effects where the transport coupling constant Λ tr is given by As we see, all projected spectral functions α 2 i F(ω) depend on the charge vertex function γ c (k, q) which describes the screening (renormalization) of EPI due to strong correlations (suppression of doubly occupancy) [78][79][80]130]. This important ingredient (which respects also the Ward identities) is a decisive step beyond the MFA renormalization of EPI in strongly correlated systems which was previously studied in connection with heavy fermions-see review in [218].

Charge Vertex and the EPI Coupling.
The charge vertex function γ c (k, q) (in the adiabatic approximation) has been calculated in [78-80, 130, 179, 180] in the framework of the 1/N expansion in the X-method-see also [6]-and here we discuss only the main results. Note that γ c (k, q) renormalizes all charge fluctuation processes, such as the EPI interaction, the long-range Coulomb interaction, the nonmagnetic impurity scattering, and so forth. In fact γ c (k, q) describes specific screening due to the vanishing of doubly occupancy in strongly correlated systems. Note that the latter constraint is at present impossible to incorporate into the LDA-DFT band-structure calculations, thus making the latter method unreliable in highly correlated systems. In [78-80, 130, 179, 180] γ c (k, q, ω) was calculated as a function of the model parameters t, t , δ, J in leading O(1) order of the t-J model: where χ αβ (q) = p G α (p, q)F β (p), F α (k) = [t(k), 1, 2J 0 cos k x , 2J 0 sin k x , 2J 0 cos k y , 2J 0 sin k y ], and G α (p, q) = [1, t(p + q), cos p x , sin p x , cos p y , sin p y ]Π(p, q). Here, Π(k, q) = −g(k)g(k + q) and q = (q, iq n ), q n = 2πnT, p = (p, ip m ), p m = πT(2m + 1). The physical meaning of the vertex function γ c (k, q) is following: in the presence of an external (or internal) charge perturbation there is screening due to the change of the excitation potential V tJ IBS (x, y), that is, of the change of the bandwidth, as well as of the local chemical potential. The central result is that for momenta k lying at (and near) the Fermi surface the vertex function γ c (k, q, ω = 0) has very pronounced forward scattering peak (at q = 0) especially at very low doping concentration δ( 1), while the backward scattering is substantially suppressed, as it is seen in Figure 35 where γ c (k F , q, ω = 0) is shown. The peak at q = 0 is very narrow at very small doping since its width q c is proportional to the doping δ, that is, q c ∼ δ(π/a) where a is the lattice constant. It is interesting that γ c (k, q), as well as the dynamics of charge fluctuations, depend only weakly on the exchange energy J and are mainly dominated by the constraint of having no doubly occupancy of sites, as it is shown in [78-80, 130, 179, 180]. The existence of the forward scattering peak in γ c (k, q) at q = 0 is confirmed by numerical calculations in the Hubbard model, which show that this peak is very pronounced at large U [182]. This is important result since it proves that the 1/N expansion in the X-method is reliable method in studying charge fluctuation processes in strongly correlated systems. The strong suppression of γ c (k, q) at large q(∼ k F ) means that at small distances the charge fluctuations are strongly suppressed (correlated). Such a behavior of the vertex function means that a quasiparticle moving in the strongly correlated medium digs up a giant correlation hole with the radius ξ ch (∼ π/q c ) ≈ a/δ, where a is the lattice constant. As a consequence of this effect the renormalized EPI becomes long ranged which is contrary to the weakly correlated systems where it is short ranged.
By knowing γ c (k, q) one can calculate the relative coupling constants Λ 1 ≡ Λ s , Λ 3 ≡ Λ d , Λ tr , and so forth. In the absence of correlations and for an isotropic band one has Λ 1 = Λ tr = 1, Λ i = 0 for i > 1. The averages in Λ s , Λ d , and Λ tr were performed numerically in [130] by using the realistic anisotropic band dispersion in the t-t -J model and the results are shown in Figure 36. For convenience, the three curves are multiplied with a common factor so that Λ s approaches 1 in the empty-band limit δ → 1, when strong correlations are absent. Note that the superconducting critical temperature T c in the weak coupling limit and in the ith channel scales like T (i) c ∼ exp(−1/(λ 0 Λ i − μ * i ) where λ 0 is some effective coupling constant which depends on microscopic details. The parameter μ * i is the effective residual Coulomb repulsion in the ith superconducting channel. We  Figure 36: Normalized s-wave Λ s , d-wave Λ d , and transport Λ tr · δ coupling constants as a function of doping δ for t = 0 and J = 0. From [179,180]. stress here several interesting results which come out from the above theory and which are partially presented in Figures  35 and 36.
(1) In principle the bare EPI coupling constant g 0 λ (k, q) depends on the quasiparticle momentum k and the transfer momentum q. In the t-J model the EPI coupling is dominated by the ionic coupling H ion ep (see (80)) and corresponding EPI depends only on the momentum transfer q, that is, g 0 λ (k, q) ≈ g 0 λ (q) while for the much smaller covalent coupling H cov ep depends on both k and q [6,10,11]. However, the EPI couplings for most phonon modes are renormalized by the charge vertex and since the latter is peaked at small momentum transfer q = |k − k | then the maxima of the corresponding effective potentials are pushed toward smaller values of q. The further consequence of the vertex renormalization is that in the absence of strong correlations the bare EPI coupling |g 0 (k, q)| 2 for some phonon modes (which enters in the effective t-J model) is detrimental for d-wave pairing; it can be less detrimental or even supports it in the presence of strong correlations (since the maximum is pushed toward smaller q). To illustrate this let us consider the in-plane oxygen breathing mode with the frequency ω br which is supposed to be important in HTSC cuprates. The bare coupling constant (squared) for this mode is approximately given by |g 0 br (k, q)| The advantage in this method is that the explicit knowledge of μ * is not required [229,230]. However, the integral equation for α 2 F(ω) is linear Fredholm equation of the first kind which is ill defined-see the discussion in Section 1.3.2 item (2) b. Phonon Effects in N T (ω). We briefly discuss the physical origin for the phonon effects in N T (ω) by considering a model with only one peak, at ω 0 , in the phonon density of states F(ω) by assuming for simplicity μ * = 0 and neglecting the weak structure in N T (ω) at nω 0 + Δ 0 , which is due to the nonlinear structure of the Eliashberg equations [232]. In Figure 37 it is seen that the real part of the gap function Δ R (ω) reaches a maximum at ω 0 + Δ 0 then decreases and becomes negative and zero, while Δ I (ω) is peaked slightly beyond ω 0 + Δ 0 that is the consequence of the effective electron-electron interaction via phonons. It follows that for ω < ω 0 + Δ 0 most phonons have higher energies than the energy ω of electronic charge fluctuations and there is overscreening of this charge by the ions giving rise to attraction. For ω ≈ ω 0 + Δ 0 the charge fluctuations are in resonance with ion vibrations giving rise to the peak in Δ R (ω). For ω 0 + Δ 0 < ω the ions move out of phase with respect to the charge fluctuations giving rise to repulsion and negative Δ R (ω). This is shown in Figure 37(b). The structure in Δ(ω) is reflected on N T (ω) as shown in Figure 37(c) which can be reconstructed from the approximate formula for N T (ω) expanded in powers of Δ/ω: As Δ R (ω) increases above Δ 0 , this gives N T (ω) > N BCS (ω), while for ω ω 0 + Δ 0 the real value Δ R (ω) decreases while Δ I (ω) rises and N T (ω) decreases giving rise for N T (ω) < N BCS (ω).

A.2. Transport Spectral Function α 2
tr F(ω). The spectral function α 2 tr F(ω) enters the dynamical conductivity σ i j (ω) (i, j = a, b, c axis in HTS systems) which generally speaking is a tensor quantity given by the formula where q = (q, ν) and k = (k = 0, ω) and the bare current vertex γ i (q, k + q; k = 0) is related to the Fermi velocity v F,i , that is, γ i (q, k + q; k = 0) = v F,i . The vertex function Γ j (q, k + q) takes into account the renormalization due to all scattering processes responsible for finite conductivity [233]. In the following we study only the in-plane conductivity at k = 0. The latter case is realized due to the fact that the long penetration depth in HTSC cuprates and the skin depth in the normal state are very large. In the EPI theory, Γ j (q, k + q) ≡ Γ j (q, iω n , iω n + iω m ) is the solution of an approximative integral equation written in the symbolic form [118] Γ j = v j + V eff GGΓ j . The effective potential V eff (due to EPI) is given by V eff = κ |g ren κ | 2 D κ , where D κ is the phonon Green's