Abstract

In this paper we discuss experimental evidence related to the structure and origin of the bosonic spectral function 𝛼2𝐹(𝜔) in high-temperature superconducting (HTSC) cuprates at and near optimal doping. Global properties of 𝛼2𝐹(𝜔), such as number and positions of peaks, are extracted by combining optics, neutron scattering, ARPES and tunnelling measurements. These methods give evidence for strong electron-phonon interaction (EPI) with 1<𝜆𝑒𝑝3.5 in cuprates near optimal doping. We clarify how these results are in favor of the modified Migdal-Eliashberg (ME) theory for HTSC cuprates near optimal doping. In Section 2 we discuss theoretical ingredients—such as strong EPI, strong correlations—which are necessary to explain the mechanism of d-wave pairing in optimally doped cuprates. These comprise the ME theory for EPI in strongly correlated systems which give rise to the forward scattering peak. The latter is supported by the long-range part of EPI due to the weakly screened Madelung interaction in the ionic-metallic structure of layered HTSC cuprates. In this approach EPI is responsible for the strength of pairing while the residual Coulomb interaction and spin fluctuations trigger the d-wave pairing.

1. 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 𝑇𝑐 in cuprates is high, with the maximum 𝑇𝑐max160K in the Hg-1223 compounds; (2) the pairing in cuprates is 𝑑-wave like, that is, Δ(𝐤,𝜔)Δ𝑑(𝜔)(cos𝑘𝑥cos𝑘𝑦). 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 electron-boson 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 𝐴𝛼𝜆(𝜔𝐵/𝑊𝑏) fulfills the condition 𝐴1, where 𝜆 is the electron-boson coupling constant, 𝜔𝐵 is the characteristic bosonic energy, 𝑊𝑏 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 𝛼2𝐹(𝐤,𝐤,𝜔) (or its average 𝛼2𝐹(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 [311], where mediating bosons are phonons and where the average spectral function 𝛼2𝐹(𝜔) is similar to the phonon density of states 𝐹ph(𝜔). Note that 𝛼2𝐹(𝜔) is not the product of two functions although sometimes one defines the function 𝛼2(𝜔)=𝛼2𝐹(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<𝜆𝑒𝑝3, which will be discussed in detail below. In the EPI approach 𝛼2𝐹ph(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 𝑇𝑐 be realized in the nonadiabatic limit 𝐴1?

The second approach [1217] assumes that EPI is too weak to be responsible for high 𝑇𝑐 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 Im𝜒(𝐤𝐤,𝜔), that is, 𝛼2𝐹(𝐤,𝐤,𝜔)𝑔2sfIm𝜒(𝐤𝐤,𝜔) where 𝑔sf 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 𝑇𝑐  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 (𝑇𝑐100K)? 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 𝑇𝑐, 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 high-temperature expansion for the pairing susceptibility [21] give evidence that there is no superconductivity at temperatures characteristic for cuprates and if it exists 𝑇𝑐 must be rather low—few Kelvins. These numerical results tell us that the lack of high 𝑇𝑐 (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 𝑇𝑐  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 𝜆𝑒𝑝>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 [2427] 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, 𝜆𝑒𝑝<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 𝛼2𝐹(𝜔) as well as to the transport spectral function 𝛼2tr𝐹(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 quasiparticles—although 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 [811]. 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 𝛼2𝐹(𝐤,𝐤,𝜔) and 𝛼2tr𝐹(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 X-ray 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 HTSC cuprates. We argue that any nonphononic mechanism of pairing should have very large bare critical temperature 𝑇𝑐0𝑇𝑐 in the presence of the large EPI coupling constant, 𝜆𝑒𝑝1, if the EPI spectral function is weakly momentum dependent, that is, if 𝛼2𝐹(𝐤,𝐤,𝜔)𝛼2𝐹(𝜔) like in low-temperature 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 [1217] related to the Eliashberg spectral function, that is, 𝛼2𝐹sf(𝐤,𝐤,𝜔)𝑔2sfIm𝜒(𝐪=𝐤𝐤,𝜔). We stress that such an approach can be theoretically justified only in the weak coupling limit, 𝑔sf𝑊𝑏, where 𝑊𝑏 is the band width and 𝑔sf 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 𝑇𝑐=92.5KIm𝜒(𝐐,𝜔) is drastically suppressed compared to that in slightly underdoped ones with 𝑇𝑐=91K. This fact implies that the SFI coupling constant 𝑔sf must be small.

(b) Optical Conductivity Measurements. From these measurements one can extract the transport relaxation rate 𝛾tr(𝜔) and indirectly an approximative shape of the transport spectral function 𝛼2tr𝐹(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 𝛾tr(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 [36, 3238]. (ii) Second are methods of extraction of the transport spectral function 𝛼2tr𝐹(𝜔). Although these methods give at finite temperature 𝑇 a blurred 𝛼2tr𝐹(𝜔) which is (due to the ill-defined methods) temperature dependent, it turns out that the width and the shape of the extracted 𝛼2tr𝐹(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 𝑇-dependence 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 La2𝑥Sr𝑥CuO4 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 𝛼2𝐹(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 (𝜔𝜔ph) and high energies (𝜔𝜔ph ) 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 𝛼2𝐹(𝜔) from tunnelling conductance. In this part we discuss the following items: (i) the extracted Eliashberg spectral function 𝛼2𝐹(𝜔) with the coupling constant 𝜆(tun)=2-3.5 from the tunnelling conductance of break-junctions in optimally doped YBCO and Bi-2212 [4255] which gives that the maxima of 𝛼2𝐹(𝜔) coincide with the maxima in the phonon density of states 𝐹ph(𝜔); (ii) the existence of eleven peaks in 𝑑2𝐼/𝑑𝑉2 in superconducting La1.84Sr0.16CuO4 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 𝐹ph(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 𝐵1𝑔 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 many-body 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 𝛼2𝐹(𝐤,𝐤,𝜔) and the transport spectral function 𝛼2tr𝐹(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 everything—which 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 for the coherence peak in the microwave response 𝜎𝑠(𝑇<𝑇𝑐,𝜔=const) at 𝜔=17GHz 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 17GHz in Nb qualitatively but not quantitatively. However, the measurements at higher frequency 𝜔60GHz 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 MgB2 (𝑇𝑐40K) which is smaller than the theoretical value, that is, 𝛼expB0.3<𝛼Bth=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/𝑇𝑐 which is on optimally doped YBCO and BSCO 5 and 7, respectively, while in underdoped BSCO one has even (2Δ/𝑇𝑐)20; (ii) peculiarities of the coherence peak in the microwave response 𝜎(𝑇) in HTSC cuprates, which is peaked at 𝑇 much smaller than 𝑇𝑐, contrary to the case of LTSC where it occurs near 𝑇𝑐; (iii) the dependence of 𝑇𝑐 on the number of CuO2 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 many-body theory of superconductivity which is based on the fundamental (microscopic) Hamiltonian and the many-body technique. This theory can in principle calculate measurable properties of materials such as the critical temperature 𝑇𝑐, 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,,𝑚𝑒,𝑀ion,𝑘𝐵 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 low-energy phenomenon characterized by the very small energy parameters (𝑇𝑐/𝐸𝐹,Δ/𝐸𝐹,𝜔ph/𝐸𝐹)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) strong-coupling 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) 𝑉IBS(𝐫,𝐫) (IBS—the ideal band structure) which includes the static self-energy (Σ()𝑐0(𝐫,𝐫,𝜔=0)) due to high-energy electronic processes, that is, 𝑉IBS(𝐫,𝐫)=[𝑉𝑒-𝑖(𝐫)+𝑉𝐻(𝐫)]𝛿(𝐫𝐫)+Σ()𝑐0(𝐫,𝐫,𝜔=0), with 𝑉𝑒-𝑖 and 𝑉𝐻 being the electron-ion and Hartree potential, respectively; (2) solving the low-energy Eliashberg equations. However, the calculation of the (excited) potential 𝑉IBS(𝐫,𝐫) and the real EPI coupling 𝑔𝑒𝑝(𝐫,𝐫)=𝛿𝑉IBS(𝐫,𝐫)/𝛿𝐑𝑛, 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 𝑉LDA(𝐫)—the ground-state potential—and the real EPI coupling by the “local” LDA one 𝑔𝑒𝑝(𝐫)=𝛿𝑉LDA(𝐫)/𝛿𝐑𝑛. Since the exchange-correlation effects enter 𝑉LDA(𝐫)=𝑉𝑒-𝑖(𝐫)+𝑉𝐻(𝐫)+𝑉XC(𝐫) via the local exchange-correlation potential 𝑉XC(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 𝑉IBS(𝐫,𝐫) and the nonlocal EPI coupling 𝑔𝑒𝑝(𝐫,𝐫). As a result one obtains the momentum-dependent EPI coupling 𝑔𝑒𝑝(𝐤𝐹,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 (𝜆tr) rather smaller than the pairing one 𝜆, that is, 𝜆tr𝜆/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 𝑇𝑐. 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 [6163], giving rise to a pronounced inhomogeneity of the superconducting order parameter Δ(𝐤,𝐑), where 𝐤 is the relative momentum of the Cooper 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 (10–20)% in HTSC cuprates which is many orders of magnitude larger than in standard semiconductors (1021 versus 1015 carrier concentration). There are some claims that high 𝑇𝑐 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 𝑇𝑐 solely within the inhomogeneous Hubbard or 𝑡-𝐽 model. Here we will not discuss this interesting problem but mention only that the concept of 𝑇𝑐 increase by inhomogeneity is not well-defined, since the increase of 𝑇𝑐 is defined with respect to the average value 𝑇𝑐. However, 𝑇𝑐 is experimentally not well defined quantity and the hypothesis of an increase of 𝑇𝑐 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.

1.2. EPI versus Nonphononic Mechanisms

Concerning the high 𝑇𝑐 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 𝜉𝐤(=𝜖𝐤𝜇) which are much larger than their inverse life-times, that is, 𝜉𝐤1/𝜏𝐤𝜉2𝐤/𝑊𝑏. Since the level width Γ=2/𝜏𝐤 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=0K there is an energy level 𝜉𝐤𝐹=0 which defines the Fermi surface on which the Fermi quasiparticle distribution function 𝑛𝐹(𝜉𝐤) has finite jump at 𝑘𝐹; (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 𝑘𝐵Θ𝐷(𝜔𝐷) (for the Debye spectrum ~Θ𝐷/5), that is, for 𝜉𝐤Θ𝐷, one has 𝜏𝐤1𝜉𝐤 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 𝜉𝐤Θ𝐷 one has 𝜏𝐤1𝜉𝐤 and they are well described by the Landau theory. However, in (the noncanonical) cases where for higher energies 𝜉𝐤Θ𝐷 one has 𝜏𝐤1𝜉𝐤, 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 𝛼2𝐹sf(𝐤,𝐤,𝜔) and its higher momenta. Since in HTSC cuprates the electron-boson (phonon) coupling is strong and 𝑇𝑐 is large, then it is natural that in the normal state (at 𝑇>𝑇𝑐) 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 𝑇𝑐 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 𝑉e(𝐤,𝜔) in the leading approximation looks like as for two external charges (𝑒) embedded in the medium with the total longitudinal dielectric function 𝜀tot(𝐤,𝜔) (𝐤 is the momentum and 𝜔 is the frequency) [68, 69], that is, 𝑉e𝑉(𝐤,𝜔)=ext(𝐤)𝜀tot=(𝐤,𝜔)4𝜋𝑒2𝑘2𝜀tot.(𝐤,𝜔)(1) In case of strong interaction between quasiparticles, the state of embedded quasiparticles changes significantly due to interaction with other quasiparticles, giving rise to 𝑉e(𝐤,𝜔)4𝜋𝑒2/𝑘2𝜀tot(𝐤,𝜔). In that case 𝑉e depends on other (than 𝜀tot(𝐤,𝜔)) response functions. However, in the case when (1) holds, that is, when the weak-coupling limit is realized, 𝑇𝑐 is given by 𝑇𝑐𝜔exp(1/(𝜆𝑒𝑝𝜇)) [6870]. Here, 𝜆𝑒𝑝 is the EPI coupling constant, 𝜔 is an average phonon frequency, and 𝜇 is the Coulomb pseudopotential, 𝜇=𝜇/(1+𝜇ln𝐸𝐹/𝜔) (𝐸𝐹 is the Fermi energy). The couplings 𝜆𝑒𝑝 and 𝜇 are expressed by 𝜀tot(𝐤,𝜔=0): 𝜇𝜆𝑒𝑝=𝑁(0)𝑉e(𝐤,𝜔=0)=𝑁(0)2𝑘𝐹0𝑘𝑑𝑘2𝑘2𝐹4𝜋𝑒2𝑘2𝜀tot,(𝐤,𝜔=0)(2) where 𝑁(0) is the density of states at the Fermi surface and 𝑘𝐹 is the Fermi momentum—see more in [35]. In [65, 66] it was claimed that the lattice stability of the system with respect to the charge density wave formation implies the condition 𝜀tot(𝐤,𝜔=0)>1 for all 𝐤. If this were correct, then from (2) it would follow that 𝜇>𝜆𝑒𝑝, which limits the maximal value of 𝑇𝑐 to the value 𝑇𝑐max𝐸𝐹exp(43/𝜆𝑒𝑝). In typical metals 𝐸𝐹<(1-10)eV, and if one accepts the statement in [65, 66] that 𝜆𝑒𝑝𝜇(0.5), one obtains 𝑇𝑐(1-10)K. The latter result, if it would be correct, means that EPI is ineffective in producing not only high-𝑇𝑐 superconductivity but also low-temperature superconductivity (LTS with 𝑇𝑐20K). However, this result is in conflict first of all with experimental results in LTSC, where in numerous systems one has 𝜇𝜆𝑒𝑝 and 𝜆𝑒𝑝>1. For instance, 𝜆𝑒𝑝2.6 is realized in PbBi alloy which is definitely much higher than 𝜇(<1), and so forth.

Moreover, the basic theory tells us that 𝜀tot(𝐤0,𝜔) is not the response function [68, 69] (contrary to the assumption in [65, 66]). Namely, if a small external potential 𝛿𝑉ext(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 𝛿𝑉tot(𝐤,𝜔)=𝛿𝑉ext(𝐤,𝜔)/𝜀tot(𝐤,𝜔), which means that 1/𝜀tot(𝐤,𝜔) is the response function. The latter obeys the Kramers-Kronig dispersion relation which implies the following stability condition [68, 69]: 1𝜀tot(𝐤,𝜔=0)<1,𝐤0,(3) that is, either 𝜀tot(𝐤0,𝜔=0)>1,(4) or 𝜀tot(𝐤0,𝜔=0)<0.(5) This important theorem invalidates the restriction on the maximal value of 𝑇𝑐 in the EPI mechanism given in [65, 66]. We stress that the condition 𝜀tot(𝐤0,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, ̂𝜀tot(𝐤+𝐐,𝐤+𝐐,𝜔), and 𝜀tot(𝐤,𝜔) is defined by 𝜀1tot(𝐤,𝜔)=̂𝜀1tot(𝐤+𝟎,𝐤+𝟎,𝜔). In dense metallic systems with one ion per cell (such as metallic hydrogen) and with the electronic dielectric function 𝜀el(𝐤,0), the macroscopic total dielectric function 𝜀tot(𝐤,0) is given by [7173] 𝜀tot𝜀(𝐤,0)=el(𝐤,0)11/𝜀el(𝐤,0)𝐺𝑒𝑝.(𝐤)(6) At the same time the energy of the longitudinal phonon 𝜔𝑙(𝐤) is given by 𝜔2𝑙Ω(𝐤)=2𝑝𝜀el(𝐤,0)1𝜀el(𝐤,0)𝐺𝑒𝑝,(𝐤)(7) where Ω2𝑝 is the ionic plasma frequency, and 𝐺𝑒𝑝 is the local (electric) field correction—see [7173]. The right condition for lattice stability requires that 𝜔2𝑙(𝐤)>0, which implies that for 𝜀el(𝐤,0)>0 one has 𝜀el(𝐤,0)𝐺𝑒𝑝(𝐤)<1. The latter condition gives automatically 𝜀tot(𝐤,0)<0. Furthermore, the calculations [7173] show that in the metallic hydrogen (H) crystal 𝜀tot(𝐤,0)<0 for all 𝐤𝟎. Note that in metallic H the EPI coupling constant is very large, that is, 𝜆𝑒𝑝7 and 𝑇𝑐 may reach very large value 𝑇𝑐600K [74]. Moreover, the analyses of crystals with more ions per unit cell [7173] give that 𝜀tot(𝐤𝟎,0)<0 is more a rule than an exception—see Figure 1. The physical reason for 𝜀tot(𝐤𝟎,0)<0 is local field effects described by 𝐺𝑒𝑝(𝐤). Whenever the local electric field 𝐄loc acting on electrons (and ions) is different from the average electric field 𝐄, that is, 𝐄loc𝐄, there are corrections to 𝜀tot(𝐤,0) which may lead to 𝜀tot(𝐤,0)<0.

The above analysis tells us that in real crystals 𝜀tot(𝐤,0)  can be negative in the large portion of the Brillouin zone thus giving rise to 𝜆𝑒𝑝𝜇>0 in (2). This means that analytic properties of the dielectric function 𝜀tot(𝐤,𝜔)  do not limit 𝑇𝑐 in the phonon mechanism of pairing. This result does not mean that there is no limit on 𝑇𝑐 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 𝜆tr (entering resistivity) rather smaller than the pairing EPI coupling constant 𝜆𝑒𝑝(>1) (entering 𝑇𝑐), that is, why one has 𝜆tr(0.6–1.4)𝜆𝑒𝑝(2–3.5)? (3) If EPI is ineffective for pairing in HTSC oxides, in spite of 𝜆𝑒𝑝>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 [1217]. This belief is mainly based, first, on the above discussed incorrect lattice stability criterion related to the sign of 𝜀tot(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<𝜆𝑒𝑝<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 (Ωnph𝜔ph) and with the bare critical temperature 𝑇𝑐0 and look for the effect of EPI on 𝑇𝑐. If EPI is approximately isotropic, like in most LTSC materials, then it would be very detrimental for 𝑑-wave pairing. In the case of dominating isotropic EPI in the normal state and the exitonic-like pairing, then near 𝑇𝑐 the linearized Eliashberg equations have an approximative form for a weak nonphonon interaction (with the large characteristic frequency Ωnph) 𝑍𝜔𝑛Δ𝑛(𝐤)𝜋𝑇𝑐Ωnph𝑚𝐪𝑉nphΔ(𝐤,𝐪,𝑛,𝑚)𝑚(𝐪)||𝜔𝑚||,𝑍𝜔𝑛Γ1+𝑒𝑝𝜔𝑛.(8) For pure 𝑑-wave pairing with the pairing potential 𝑉nph=𝑉nph(𝜃𝐤,𝜃𝐪)Θ(Ωnph|𝜔𝑛|)Θ(Ωnph|𝜔𝑛|) with 𝑉nph(𝐤,𝐪)=𝑉0𝑌𝑑(𝜃𝐤)𝑌𝑑(𝜃𝐪) and 𝑌𝑑(𝜃𝐤)=𝜋1/2cos2𝜃𝐤, one obtains Δ𝑛(𝐤)=Δ𝑑Θ(Ωnph|𝜔𝑛|)𝑌𝑑(𝜃𝐤) and the equation for 𝑇𝑐—see [35] 𝑇ln𝑐𝑇𝑐01Ψ21Ψ2+Γ𝑒𝑝2𝜋𝑇𝑐.(9) Here Ψ is the di-gamma function. At temperatures near 𝑇𝑐 one has Γ𝑒𝑝2𝜋𝜆𝑒𝑝𝑇𝑐 and the solution of (9) is approximately 𝑇𝑐𝑇𝑐0exp{𝜆𝑒𝑝} with 𝑇𝑐0Ωnphexp{𝜆nph}, 𝜆nph=𝑁(0)𝑉0. This means that for 𝑇𝑐max160K and 𝜆𝑒𝑝>1 the bare 𝑇𝑐0 due to the nonphononic interaction must be very large, that is, 𝑇𝑐0>500K.

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 𝜆𝑍𝐤𝐩(𝑖𝜈𝑛) and 𝜆Δ𝐤𝐩, respectively. Usually, the spin-fluctuation kernel 𝜆sf,𝐤𝐩(𝑖𝜈𝑛) 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 𝑇𝑐0100K and for 𝜆𝑒𝑝>1 the calculations give very small (renormalized) critical temperature 𝑇𝑐100K. 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 [7880]. Only in that case does strong EPI conform with 𝑑-wave 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.

1.3. 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 [1217, 81]. First, the experimental results related to the pronounced imaginary part of the susceptibility Im𝜒(𝐤,𝑘𝑧,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 [1217, 81]. Second, the existence of the so called magnetic resonance peak of Im𝜒(𝐤,𝑘𝑧,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.

1.3.1. Magnetic Neutron Scattering and the Spin-Fluctuation Spectral Function

(a) Huge Rearrangement of the SFI Spectral Function and Small Change of 𝑇𝑐
Before discussing experimental results in cuprates on the imaginary part of the spin susceptibility Im𝜒(𝐤,𝜔) we point out that in the (phenomenological) theories based on the spin-fluctuation interaction (SFI) the quasiparticle self-energy Σsf(𝐤,𝜔𝑛) (𝜔𝑛 is the Matsubara frequency and ̂𝜏0 is the Nambu matrix) in the normal and superconducting state and the effective (repulsive) pairing potential 𝑉sf(𝐤,𝜔) (where 𝑖𝜔𝑛𝜔+𝑖𝜂) are assumed in the form [1217] Σsf𝐤,𝜔𝑛=𝑇𝑁𝐤,𝑚𝑉sf𝐤𝐤,𝜔𝑛𝑚̂𝜏0𝐺𝐤,𝜔𝑚̂𝜏0,𝑉sf𝐤,𝜔𝑛𝑚=𝑔2sf𝑑𝜈𝜋Im𝜒𝐪,𝜈+𝑖0+𝜈𝑖𝜔𝑛𝑚,(10) where 𝜔𝑛𝑚𝜔𝑛𝜔𝑚. Although the form of 𝑉sf cannot be justified theoretically, except in the weak coupling limit (𝑔sf𝑊𝑏) 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) Im𝜒(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 Im𝜒(𝐪,𝜔). In that respect very indicative and impressive IMNS measurements on YBa2Cu3O6+𝑥, which are done by Bourges group [30], demonstrate that the normal-state susceptibility Im𝜒(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 Im𝜒(odd)(𝐐,𝜔) in the normal state, especially in the energy (frequency) region which might be important for superconducting pairing, let us say 0meV<𝜔<60meV. 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 𝑇𝑐. It is seen in Figure 2 that in the underdoped YBa2Cu3O6.92 crystal Im𝜒(odd)(𝐐,𝜔) and 𝑆(𝐐)=𝑁(0)𝑔2sf060𝑑𝜔Im𝜒(odd)(𝐐,𝜔) are much larger than that in the near optimally doped YBa2Cu3O6.97, that is, one has 𝑆6.92(𝐐)𝑆6.97(𝐐), although the difference in the corresponding critical temperatures 𝑇𝑐 is very small, that is, 𝑇𝑐(6.92)=91K (in YBa2Cu3O6.92) and 𝑇𝑐(6.97)=92.5K (in YBa2Cu3O6.97). This pronounced rearrangement and suppression of Im𝜒(odd)(𝐐,𝜔) in the normal state of YBCO by doping (toward the optimal doping) but with the negligible change in 𝑇𝑐 is strong evidence that the SFI pairing mechanism is not the dominating one in HTSC cuprates. This insensitivity of 𝑇𝑐, if interpreted in terms of the SFI coupling constant 𝜆sf(~𝑔2sf), means that the latter is small, that is, 𝜆(exp)sf1. We stress that the explanation of high 𝑇𝑐 in cuprates by the SFI phenomenological theory [1217] assumes very large SFI coupling energy with 𝑔(th)sf0.7eV while the frequency (energy) dependence of Im𝜒(𝐐,𝜔) is extracted from the fit of the NMR relaxation rate 𝑇11 which gives 𝑇(NMR)𝑐100K [1217]. To this point, the NMR measurements (of 𝑇11) give that there is an anticorrelation between the decrease of the NMR spectral function 𝐼𝐐=lim𝜔0Im𝜒(NMR)(𝐐,𝜔)/𝜔 and the increase of 𝑇𝑐 by increasing doping toward the optimal one—see [6] and references therein. The latter result additionally disfavors the SFI model of pairing [1217] since the strength of pairing interaction is little affected by SFI. Note that if instead of taking Im𝜒(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 𝑇𝑐. For instance, by taking the experimental values for Im𝜒(IMNS)(𝐐,𝜔) in underdoped YBa2Cu3O6.6 with 𝑇𝑐60K one obtains 𝑇(IMNS)𝑐<𝑇(NMR)𝑐/3 [82], while 𝑇(IMNS)𝑐50K for 𝑔(th)sf1. The situation is even worse if one tries to fit the resistivity with Im𝜒(IMNS)(𝐐,𝜔) in YBa2Cu3O6.6 since this fit gives 𝑇(IMNS)𝑐<7K. 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 YBa2Cu3O6.6 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 𝑔2sf𝑈=(3/2)2 and Im𝜒(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=1.59eV, 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 𝐽Cu-Cu0.15eV [88]. This implies that 𝑈1eV and 𝑇𝑐60K. Note that this value for 𝐽Cu-Cu(~0.15 eV) comes out also from the theory of strongly correlated electrons in the three-band Emery model which gives 𝐽Cu-Cu[4𝑡4𝑝𝑑/(Δ𝑑𝑝+𝑈𝑝𝑑)2,(1/𝑈𝑑)+2/(𝑈𝑝+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 𝑔sf).

Concerning the problem related to the rearrangement of the SFI spectral function Im𝜒(𝐐,𝜔) in YB2Cu3O6+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)(~Im𝜒(𝐪,𝜔)) (for 𝐪 at and in the broad range of 𝐐) for the optimally doped YBa2Cu3O6.95 (with 𝑇𝑐=93K) is at least three times smaller than in the underdoped YBa2Cu3O6.6 with 𝑇𝑐=60K. This result is again very indicative sign of the weakness of SFI since such a huge reconstruction would decrease 𝑇𝑐 in the optimally doped YBa2Cu3O6.95 if analyzed in the framework of the phenomenological SFI theory based on (10). It also implies that due to the suppression of Im𝜒(𝐪,𝜔) by increasing doping toward the optimal one a straightforward extrapolation of the theoretical approach in [81] to the explanation of 𝑇𝑐 in the optimally doped YBa2Cu3O6.95 would require an increase of 𝑈 to the value even larger than 4eV, which is highly improbable.

(b) Ineffectiveness of the Magnetic Resonance Peak
A less direct argument for smallness of the SFI coupling constant, that is, 𝑔expsf0.2eV and 𝑔expsf𝑔sf, 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, Im𝜒(𝐐,𝜔) is significantly suppressed at low frequencies except near the resonance energy 𝜔res41meV 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, 𝐼0(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 𝐼0 is small [89]. 𝐼0 is obtained approximately by equating the condensation energy 𝐸con𝑁(0)Δ2/2 with the change of the magnetic energy 𝐸mag in the superconducting state, that is, 𝛿𝐸mag4𝐼0𝐸mag: 𝐸mag=𝐽𝑑𝜔𝑑2𝑘(2𝜋)31cos𝑘𝑥cos𝑘𝑦𝑆(𝐤,𝜔),(11) where 𝑆(𝐤,𝜔)=(1/𝜋)[1+𝑛(𝜔)]Im𝜒(𝐤,𝜔) is the spin structure factor and 𝑛(𝜔) is the Bose distribution function. By taking Δ2𝑇𝑐 and the realistic value 𝑁(0)1/(10𝐽)1states/eVspin, one obtains 𝐼0101(𝑇𝑐/𝐽)2103. However, such a small intensity cannot be responsible for the anomalies in ARPES and optical spectra since it gives rise to small coupling constant 𝜆sf,res for the interaction of holes with the resonance peak, that is, 𝜆sf,res(2𝐼0𝑁(0)𝑔2sf/𝜔res)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 𝑔sf. 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 𝜔res, that is, 𝛾res<𝜔res, where the line width is given by 𝛾res=4𝜋(𝑁(0)𝑔sf)2𝜔res [89]. This condition limits the SFI coupling to 𝑔sf<0.2eV. We stress that in such a way obtained 𝑔sf is much smaller (at least by factor three) than that assumed in the phenomenological spin-fluctuation theory [1217, 81] where 𝑔sf0.6-0.7eV and 𝑈1.6eV, but much larger than estimated in [89] (where 𝑔sf<0.02eV). The smallness of 𝑔sf 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(<0.1𝜇𝐵) points to an itinerant antiferromagnetism with small coupling constant 𝑔sf<0.2eV. 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 𝑇𝑐 is vanishingly small, though not affecting pairing at the second-order phase transition at 𝑇𝑐, then, if it would be solely the origin for superconductivity, the phase transition at 𝑇𝑐 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 Bi2Sr2CaCu2O8+𝛿 [93], where the absolute value of Im𝜒(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 Im𝜒(𝜔)=𝑞Im𝜒(𝐪,𝜔) across the superconducting transition 𝑇𝑐=91K, that is, there is only a minimal change in Im𝜒(𝜔) between 10K and 100K. Note that if the magnetic excitations were due to itinerant quasiparticles we should have seen dramatic changes of Im𝜒(𝜔) as a function of 𝑇 over the whole energy range. This 𝑇-independence of Im𝜒(𝜔) strongly opposes the theoretical results in [2427] which assume that the bosonic spectral function is proportional to Im𝜒(𝜔) and that the former can be extracted from optic measurements. Namely, the fitting procedure in [2427] gives that Im𝜒(𝜔) is strongly 𝑇-dependent contrary to the experimental results in [93]—see more in Section 1.3.2 on optical conductivity.

1.3.2. Optical Conductivity and EPI

Optical spectroscopy gives information on optical conductivity 𝜎(𝜔) and on two-particle excitations, from which one can indirectly extract the transport spectral function 𝛼2tr𝐹(𝜔). Since this method probes bulk sample (on the skin depth), contrary to ARPES and tunnelling methods which probe tiny regions (10-15 Å) 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)=|(𝜀𝑖𝑖(𝜔)1)/(𝜀𝑖𝑖(𝜔)+1)|2 with the transversal dielectric tensor 𝜀𝑖𝑖(𝜔)=𝜀𝑖𝑖,+𝜀𝑖𝑖,latt+4𝜋𝑖𝜎𝑖𝑖(𝜔)/𝜔. Here, 𝜀𝑖𝑖, is the high-frequency dielectric function, 𝜀𝑖𝑖,latt describes the contribution of the lattice vibrations, and 𝜎𝑖𝑖(𝜔) describes the optical (dynamical) conductivity of conduction carriers. Since 𝑅(𝜔) is usually measured in the limited-frequency interval 𝜔min<𝜔<𝜔max, some physical modelling for 𝑅(𝜔) is needed in order to guess it outside this range—see more in reviews in [36]. 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 𝜔pl—from 0.06 to 25eV—that is, almost three orders of magnitude. However, it turns out that IR measurements of 𝑅(𝜔) in conjunction with elipsometric measurements of 𝜀𝑖𝑖(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 𝛾tr(𝜔) is equal to the quasiparticle relaxation rate 𝛾(𝜔), in spite of the well-known fact that the inequality 𝛾tr(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.

The dynamical conductivity 𝜎(𝜔) consists of two parts, that is, 𝜎(𝜔)=𝜎inter(𝜔)+𝜎intra(𝜔) where 𝜎inter(𝜔) describes interband transitions which contribute at higher than intraband energies, while 𝜎intra(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 cm1, where the following conversion holds: 1cm1=29.98GHz=0.123985meV=1.44K.) The experimental data for 𝜎(𝜔)=𝜎1+𝑖𝜎2 in cuprates are usually processed by the generalized (extended) Drude formula [3236, 95]: 𝜔𝜎(𝜔)=2𝑝14𝜋𝛾tr(𝜔)𝑖𝜔𝑚tr(𝜔)/𝑚𝑖𝜔2𝑝4𝜋𝜔tr,(𝜔)(12) where 𝑚 is the mass of the band electrons while the quantity 𝜔tr(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 𝜔tr(𝜔) will be discussed below.) Here, 𝑖=𝑎,𝑏 enumerates the plane axis; 𝜔𝑝,𝛾tr(𝜔,𝑇), and 𝑚op(𝜔) are the electronic plasma frequency, the transport (optical) scattering rate, and the optical mass, respectively. Very frequently it is analyzed the quantity 𝛾tr(𝜔,𝑇) given by [95] 𝛾tr𝑚(𝜔,𝑇)=𝑚tr(𝛾𝜔)tr(𝜔,𝑇)=𝜔Im𝜎(𝜔).Re𝜎(𝜔)(13) In the weak coupling limit 𝜆𝑒𝑝<1, the formula for conductivity given in Appendix A, equations (A.20) and (A.21) can be written in the form of (12) where 𝛾tr reads [3336] 𝛾tr(𝜔,𝑇)=𝜋𝑙0𝑑𝜈𝛼2tr,𝑙𝐹𝑙×2(𝜈)1+2𝑛𝐵𝜈(𝜈)2𝜔𝜔+𝜈𝜔𝑛𝐵+(𝜔+𝜈)𝜔𝜈𝜔𝑛𝐵.(𝜔𝜈)(14) Here 𝑛𝐵(𝜔) is the Bose distribution function. For completeness we give also the explicit form of the transport mass 𝑚tr(𝜔), see [36, 3236]: 𝑚tr(𝜔)𝑚2=1+𝜔𝑙0𝑥0200𝑑0𝑑𝜈𝛼2tr,𝑙𝐹𝑙𝜔(𝜈)Re𝐾,𝜈2𝜋𝑇,2𝜋𝑇(15) with the Kernel 𝐾(𝑥,𝑦)=(𝑖/𝑦)+{((𝑦𝑥)/𝑥)[𝜓(1𝑖𝑥+𝑖𝑦)𝜓(1+𝑖𝑦)]}{𝑦𝑦} where 𝜓 is the di-gamma function. In the presence of impurity scattering one should add 𝛾imp,tr to 𝛾tr. It turns out that (14) holds within a few percents also for large 𝜆𝑒𝑝(>1). Note that 𝛼2tr,𝑙𝐹𝑙(𝜈)𝛼2𝑙𝐹𝑙(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 𝛾(𝜔,𝑇)=2𝜋0𝑑𝜈𝛼2×𝐹(𝜈)2𝑛𝐵(𝜈)+𝑛𝐹(𝜈+𝜔)+𝑛𝐹(𝜈𝜔)+𝛾imp,(16) where 𝑛𝐹 is the Fermi distribution function. For completeness we give also the expression for the quasiparticle effective mass 𝑚(𝜔): 𝑚(𝜔)𝑚1=1+𝜔𝑙0𝑑𝜈𝛼2𝑙𝐹𝑙𝜓1(𝜈)×Re2+𝑖𝜔+𝜈12𝜋𝑇𝜓2𝑖𝜔𝜈.2𝜋𝑇(17) The term 𝛾imp is due to the impurity scattering. By comparing (14) and (16), it is seen that 𝛾tr and 𝛾 are different quantities, that is, 𝛾tr𝛾, 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=0K where one has (due to simplicity we omit in the following summation over 𝑙) [32] 𝛾tr(𝜔)=2𝜋𝜔𝜔0𝑑𝜈(𝜔𝜈)𝛼2tr(𝜈)𝐹(𝜈),𝛾(𝜔)=2𝜋𝜔0𝑑𝜈𝛼2(𝜈)𝐹(𝜈).(18) In the case of EPI with the constant electronic density of states, the above equations give that 𝛾𝑒𝑝(𝜔)=const for 𝜔>𝜔maxph while 𝛾𝑒𝑝,tr(𝜔) (as well as 𝛾𝑒𝑝,tr) is monotonic growing for 𝜔>𝜔maxph, where 𝜔maxph is the maximal phonon frequency. So, the growing of 𝛾𝑒𝑝,tr(𝜔) (and 𝛾𝑒𝑝,tr) for 𝜔>𝜔maxph 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),𝛾tr(𝜔,𝑇), and 𝛾tr which are calculated for the EPI spectral function 𝛼2𝑒𝑝(𝜔)𝐹ph(𝜔) extracted from tunnelling experiments in YBCO (with 𝜔maxph80meV ) [4245]—see Figure 4.

The results shown in Figure 4 clearly demonstrate the physical difference between two scattering rates 𝛾𝑒𝑝 and 𝛾𝑒𝑝,tr (or 𝛾tr). It is also seen that 𝛾tr(𝜔,𝑇) is even more linear function of 𝜔 than 𝛾tr(𝜔,𝑇). From these calculations one concludes that the quasilinearity of 𝛾tr(𝜔,𝑇) (and 𝛾tr) 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 𝛾𝑒𝑝 and 𝛾𝑒𝑝,tr (and 𝛾𝑒𝑝,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, low-temperature superconductors such as Al, Pb, and so forth—see more in [36] and references therein.

Let us discuss briefly the experimental results for 𝑅(𝜔) and 𝛾tr(𝜔,𝑇) and compare these with theoretical predictions obtained by using a single-band model with 𝛼2𝑒𝑝(𝜔)𝐹ph(𝜔) extracted from the tunnelling data with the EPI coupling constant 𝜆𝑒𝑝2 [4245]. In the case of YBCO the agreement between measured and calculated 𝑅(𝜔) is very good up to energies 𝜔<6000cm1, 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 [35]. In optimally doped Bi2Sr2CaCu2O6 (𝐵𝑖2212) [97, 98] the experimental results for 𝛾tr(𝜔,𝑇) are explained theoretically by assuming that the EPI spectral function 𝛼2𝑒𝑝(𝜔)𝐹(𝜔)𝐹ph(𝜔), where 𝐹ph(𝜔) is the phononic density of states in BSCO, with 𝜆𝑒𝑝=1.9 and 𝛾imp320cm1—see Figure 5(a). At the same time the fit of 𝛾tr(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 𝛾tr(𝜔,𝑇) (and 𝛾tr(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),𝛾tr(𝜔,𝑇), and 𝑚tr(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 𝛾tr(𝜔,𝑇) (and 𝛾tr(𝜔,𝑇)) for 𝜀=1 is linear up to much higher 𝜔 than in the case 𝜀>1.

However, in some experiments [100103] the extracted 𝛾tr(𝜔,𝑇) (and 𝛾tr(𝜔,𝑇)) is linear up to very high 𝜔1500cm1. This means that the ion background and interband transitions (contained in 𝜀) are not properly taken into account since too small 𝜀 (1) is assumed. The recent elipsometric measurements on YBCO [104] give the value 𝜀4-6, which gives much less spectacular linearity in the relaxation rates 𝛾tr(𝜔,𝑇) (and 𝛾tr(𝜔,𝑇)) than it was the case immediately after the discovery of HTSC cuprates, where much smaller 𝜀 was assumed.

Furthermore, we would like to comment two points related to 𝜎,𝛾tr, and 𝛾. First, the parametrization of 𝜎(𝜔) with the generalized Drude formula in (12) and its relation to the transport scattering rate 𝛾tr(𝜔,𝑇) and the transport mass 𝑚tr(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 𝛾etr(𝜔,𝑇), 𝑚etr(𝜔,𝑇) despite the fact that the inelastic electron-boson scattering is absent. To this end we stress that the single-band approach is justified for a number of HTSC cuprates such as LSCO, BSCO, and so forth. Second, at the beginning we said that 𝛾tr(𝜔,𝑇) and 𝛾(𝜔,𝑇) are physically different quantities and it holds that 𝛾tr(𝜔,𝑇)𝛾(𝜔,𝑇). In order to give the physical picture and qualitative explanation for this difference we assume that 𝛼2tr𝐹(𝜈)𝛼2𝐹(𝜈). In that case the renormalized quasiparticle frequency 𝜔(𝜔)=𝑍(𝜔)𝜔=𝜔Σ(𝜔) and the transport one 𝜔tr(𝜔)—defined in (12)—are related and at 𝑇=0 they are given by [32, 36] 𝜔tr1(𝜔)=𝜔𝜔0𝑑𝜔𝜔2𝜔.(19) (For the definition of 𝑍(𝜔) see Appendix A.) It gives the relation between 𝛾tr(𝜔) and 𝛾(𝜔) as well as𝑚tr(𝜔) and 𝑚(𝜔), respectively: 𝛾tr1(𝜔)=𝜔𝜔0𝑑𝜔𝛾𝜔,𝜔𝑚tr1(𝜔)=𝜔𝜔0𝑑𝜔2𝜔𝑚𝜔.(20) 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 𝜔𝜔, then they relax as quasiparticles with the renormalized frequency 𝜔. Since 𝜔 takes values 0<𝜔<𝜔, then the optical relaxation 𝜔tr(𝜔) is the energy-averaged 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] 𝜔tr1(𝜔)=𝜔0𝑑𝜔1𝑛𝐹𝜔𝑛𝐹𝜔𝜔𝜔2𝜔.(21)

(2) Inversion of the Optical Data and 𝛼2tr(𝜔)𝐹(𝜔)
In principle, the transport spectral function 𝛼2tr(𝜔)𝐹(𝜔) can be extracted from 𝜎(𝜔) (or 𝛾tr(𝜔)) only at 𝑇=0K, which follows from (18) as 𝛼2tr𝜔(𝜔)𝐹(𝜔)=2𝑝8𝜋2𝜕2𝜕𝜔21𝜔Re,𝜎(𝜔)(22) or equivalently as 𝛼2tr(𝜔)𝐹(𝜔)=(1/2𝜋)𝜕2(𝜔𝛾tr(𝜔))/𝜕𝜔2. However, real measurements are performed at finite 𝑇 (at 𝑇>𝑇𝑐 which is rather high in HTSC cuprates) and the inversion procedure is an ill-posed problem since 𝛼2tr(𝜔)𝐹(𝜔) is the deconvolution of the inhomogeneous Fredholm integral equation of the first kind with the temperature-dependent Kernel 𝐾2(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 𝛼2tr(𝜔)𝐹(𝜔)  get blurred (most peaks are washed out) in the extraction procedures and, second, the extracted 𝛼2tr(𝜔)𝐹(𝜔)  be temperature dependent even when the true 𝛼2tr(𝜔)𝐹(𝜔) is 𝑇 independent. This artificial 𝑇 dependence is especially pronounced in HTSC cuprates because 𝑇𝑐(~100 K) is very high. In the context of HTSC cuprates, this problem was first studied in [3336] where this picture is confirmed by the following results: (1) the extracted shape of 𝛼2tr(𝜔)𝐹(𝜔) in YBa2Cu3O7𝑥 as well as in other cuprates is not unique and it is temperature dependent, that is, at higher 𝑇>𝑇𝑐 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 𝛼2tr(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 𝛼2(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 𝛼2tr(𝜔)𝐹(𝜔) in both compounds coincide with the width of the phonon density of states 𝐹ph(𝜔) [3336, 99]. (2) The upper energy bound for 𝛼2tr(𝜔)𝐹(𝜔) is extracted in [3336] and it coincides approximately with the maximal phonon frequency in cuprates 𝜔maxph80meV as it is seen in Figures 7 and 8.

These results demonstrate the importance of EPI in cuprates [3336]. We point out that the width of 𝛼2tr(𝜔)𝐹(𝜔) which is extracted from the optical measurements [3336] coincides with the width of the quasiparticle spectral function 𝛼2(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<𝜔maxph(80meV). Since in cuprates this interval coincides with the width in the phononic density of states 𝐹(𝜔) and since the maxima of 𝛼2(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 [2427] it is assumed that 𝛼2tr(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 𝜔𝑐𝜔maxph, this should be seen in the width of the extracted spectral function 𝛼2tr(𝜔)𝐹(𝜔). In that respect in [2527] it is assumed that SFI dominates in the quasiparticle scattering and that 𝛼2tr(𝜔)𝐹(𝜔)𝑔2sfIm𝜒(𝜔) where 𝑑Im𝜒(𝜔)=2𝑘𝜒(𝐤,𝜔). This claim is based on reanalyzing of some IR measurements [2527] and the transport spectral function 𝛼2tr(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 𝛼2tr(𝜔)𝐹(𝜔), which is an artefact and due to the chosen method, in [25] it is assumed that 𝛼2tr(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 𝛼2tr(𝜔)𝐹(𝜔) does not reproduce the experimental curve Im𝜒(𝜔) [107] in some important aspects. First, the relative heights of the two peaks in the extracted spectral function 𝛼2tr(𝜔)𝐹(𝜔) at lower temperatures are opposite to the experimental curve Im𝜒(𝜔) [107]—see [25, Figure  1]. Second, the strong temperature dependence of the extracted 𝛼2tr(𝜔)𝐹(𝜔) found in [2527] 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 ill-posed problem of solving the Fredholm integral equation of the first kind with strong 𝑇-dependent kernel. Third, the extracted spectral weight 𝐼𝐵(𝜔)=𝛼2tr(𝜔)𝐹(𝜔) in [25] has much smaller values at larger frequencies (𝜔>100meV) than it is the case for the measured Im𝜒(𝜔), that is, (𝐼𝐵(𝜔>100meV)/𝐼𝐵(𝜔max))Im𝜒(𝜔>100meV)/Im𝜒(𝜔max)—see [25, Figure  1]. Fourth, the recent magnetic neutron scattering measurements on optimally doped large-volume crystals Bi2Sr2CaCu2O8+𝛿 [93] (where the absolute value of Im𝜒(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 [2527]. Namely, this experiment shows that the local spin susceptibility Im𝜒(𝜔)=𝑞Im𝜒(𝐪,𝜔) is temperature independent across the superconducting transition 𝑇𝑐=91K, that is, there is only a minimal change in Im𝜒(𝜔) between 10K and 100K. This 𝑇-independence of Im𝜒(𝜔) strongly opposes the (above discussed) results in [2427], where the fit of optic measurements gives strong 𝑇 dependence of Im𝜒(𝜔).

Fifth, the transport coupling constant 𝜆tr extracted in [25] is too large, that is, 𝜆tr>3 contrary to the previous findings that 𝜆tr1.5 [3336, 99]. Since in HTSC one has 𝜆>𝜆tr, this would probably give 𝜆6, which is not confirmed by other experiments. Sixth, the interpretation of 𝛼2tr(𝜔)𝐹(𝜔) in LSCO and BSCO solely in terms of Im𝜒(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 Im𝜒(𝐐,𝜔) is small in the normal state and its magnitude is even below the experimental noise. This means that if the assumption that 𝛼2tr(𝜔)𝐹(𝜔)𝑔2sfIm𝜒(𝜔) were correct then the contribution to Im𝜒(𝜔) from the momenta 0<𝑘𝑄 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 La0.5Sr0.5CoO3 and Ca0.5Sr0.5RuO3 which are nonsuperconducting [108] and in Ba1𝑥K𝑥BiO3 which is superconducting below 𝑇𝑐30K 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(Ω𝑐,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(Ω𝑐,𝑇)=𝑊(0)𝛽𝑇2 in the normal state can be explained in a natural way by the 𝑇 dependence of the EPI transport relaxation rate 𝛾𝑒𝑝tr(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 0𝑑𝜔𝜎𝑁1𝜔(𝜔)=2pl8=𝜋𝑛𝑒2,2𝑚(23) while in the superconducting state it is given by the Tinkham-Ferrell-Glover (TFG) sum rule 0𝑑𝜔𝜎𝑆1𝑐(𝜔)=28𝜆2𝐿++0𝑑𝜔𝜎𝑆1𝜔(𝜔)=2pl8.(24) Here,𝑛 is the total electron density, 𝑒 is the electron charge, 𝑚 is the bare electron mass, and 𝜆𝐿 is the London penetration depth. The first (singular) term 𝑐2/8𝜆2𝐿 in (24) is due to the superconducting condensate which contributes 𝜎𝑆1,cond(𝜔)=(𝑐2/4𝜆2𝐿)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 𝐻tot=𝑇𝑒+𝐻int where all electrons, electronic bands, and their interactions 𝐻int (Coulomb, EPI, with impurities, etc.) are accounted for. Here, 𝑇𝑒 is the kinetic energy of bare electrons: 𝑇𝑒=𝜎𝑑0𝑥0200𝑑3𝑥𝜓𝜎̂𝐩(𝑥)22𝑚𝜓𝜎(𝑥)=𝐩,𝜎𝐩22𝑚𝑒̂𝑐𝐩𝜎̂𝑐𝐩𝜎.(25)

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: 𝐻𝑣=𝐩,𝜎𝜉𝐩̂𝑐𝑣,𝐩𝜎̂𝑐𝑣,𝐩𝜎+𝑉𝑣,𝑐.(26)𝐻𝑣 contains the band energy with the dispersion 𝜖𝐩 (𝜉𝐩=𝜖𝐩𝜇) and the effective Coulomb interaction of the valence electrons 𝑉𝑣,𝑐. In this case the partial sum rule in the normal state reads [110] (for a general form of 𝜖𝐩) 0𝑑𝜔𝜎𝑁1,𝑣(𝜔)=𝜋𝑒22𝑉𝐩0𝑥0200𝑑̂𝑛𝑣,𝐩𝐻𝑣𝑚𝐩,(27) where the number operator ̂𝑛𝑣,𝐩=𝜎̂𝑐𝐩𝜎̂𝑐𝐩𝜎;1/𝑚𝐩=𝜕2𝜖𝐩/𝜕𝑝2𝑥 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 Ω𝑐 (of the order of the band plasma frequency). In this case the restricted sum rule has the form 𝑊Ω𝑐=,𝑇Ω𝑐0𝑑𝜔𝜎𝑁1,𝑣=𝜋(𝜔)2𝐾𝑑+Π(0)Ω𝑐0𝑑𝜔ImΠ(𝜔)𝜔,(28) where 𝐾𝑑 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(𝑖𝜔𝑛) (at the Matsubara frequency 𝜔𝑛=𝜋𝑇(2𝑛+1)) is given by [39, 40] Π(𝑖𝜔)=2𝐩𝜕𝜖𝐩𝜕𝐩2𝜔𝑚𝐺𝐩,𝑖𝜔+𝑛𝑚𝐺𝐩,𝑖𝜔𝑚,(29) where 𝜔+𝑛𝑚=𝜔𝑛+𝜔𝑚 and 𝐺(𝐩,𝑖𝜔𝑛)=(𝑖𝜔𝑛𝜉𝐩Σ(𝐩,𝑖𝜔𝑛))1 is the electron Green's function. In the case when the interband gap 𝐸𝑔 is the largest scale in the problem, that is, when 𝑊𝑏<Ω𝑐<𝐸𝑔, in this region one has approximately ImΠ(𝜔)0 and the limit Ω𝑐 in (28) is justified. In that case one has Π(0)Ω𝑐0(ImΠ(𝜔)/𝜔)𝑑𝜔 which gives the approximate formula for 𝑊(Ω𝑐,𝑇): 𝑊Ω𝑐=,𝑇Ω𝑐0𝑑𝜔𝜎𝑁1,𝑣𝜋(𝜔)2𝐾𝑑=𝑒2𝜋𝐩𝜕2𝜖𝐩𝜕𝐩2𝑛𝐩,(30) where 𝑛𝐩=̂𝑛𝑣,𝐩 is the quasiparticle distribution function in the interacting system. Note that 𝑊(Ω𝑐,𝑇) is cutoff dependent while 𝐾𝑑 in (30) does not depend on Ω𝑐. 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(Ω𝑐,𝑇) by using the exact result in (28) which apparently depends on Ω𝑐. However, (30) is useful for appropriately chosen Ω𝑐, 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 𝜖𝐩=2𝑡(cos𝑝𝑥𝑎+cos𝑝𝑦𝑎) which gives 1/𝑚𝐩=2𝑡𝑎2cos𝑝𝑥𝑎. It is straightforward to show that in this case (30) is reduced to a simpler one: 𝑊Ω𝑐=,𝑇Ω𝑐0𝑑𝜔𝜎𝑁1,𝑣(𝜔)𝜋𝑒2𝑎22𝑉𝑇𝑣,(31) where 𝑇𝑣𝐻𝑣=𝐩𝜖𝐩𝑛𝑣𝐻𝑣 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(Ω𝑐,𝑇) is a direct measure of the average band (kinetic) energy. In the superconducting state the partial band sum rule reads 𝑊𝑠Ω𝑐=𝑐,𝑇28𝜆2𝐿+Ω𝑐+0𝑑𝜔𝜎𝑆1,𝑣=(𝜔)𝜋𝑒2𝑎22𝑉𝑇𝑣𝑠.(32) In order to introduce the reader to (the complexity of) the problem of the 𝑇 dependence of 𝑊(Ω𝑐,𝑇), let us consider the electronic system in the normal state and in absence of the quasiparticle interaction. In that case one has 𝑛𝐩=𝑓𝐩 (𝑓𝐩 is the Fermi distribution function) and 𝑊𝑛(Ω𝑐,𝑇) increases with the decrease of the temperature, that is, 𝑊𝑛(Ω𝑐,𝑇)=𝑊𝑛(0)𝛽𝑏𝑇2 where 𝛽𝑏1/𝑊𝑏. To this end, let us mention in advance that the experimental value 𝛽exp is much larger than 𝛽𝑏, that is, 𝛽exp𝛽𝑏, thus telling us that the simple Sommerfeld-like smearing of 𝑓𝐩 by the temperature effects cannot explain quantitatively the 𝑇 dependence of 𝑊(Ω𝑐,𝑇). We stress that the smearing of 𝑓𝐩 by temperature lowers the spectral weight compared to that at 𝑇=0K, that is, 𝑊𝑛(Ω𝑐,𝑇)<𝑊𝑛(Ω𝑐,0). In that respect it is not surprising that there is a lowering of 𝑊𝑠(Ω𝑐,𝑇) in the BCS superconducting state, 𝑊BCS𝑠(Ω𝑐,𝑇𝑇𝑐)<𝑊𝑛(Ω𝑐,𝑇𝑇𝑐) since at low temperatures 𝑓𝐩 is smeared mainly due to the superconducting gap, that is, 𝑓𝐩=[1(𝜉𝐩/𝐸𝐩)th(𝐸𝐩/2𝑇)]/2, 𝐸𝐩=𝜉2𝐩+Δ2, 𝜉𝐩=𝜖𝐩𝜇. The maximal decrease of 𝑊𝑠(Ω𝑐,𝑇) is at 𝑇=0.

Let us enumerate and discuss the main experimental results for 𝑊(Ω𝑐,𝑇) in HTSC cuprates. (1) In the normal state (𝑇>𝑇𝑐) of most cuprates, one has 𝑊𝑛(Ω𝑐,𝑇)=𝑊𝑛(0)𝛽ex𝑇2 with 𝛽exp𝛽𝑏, that is, 𝑊𝑛(Ω𝑐,𝑇) is increasing by decreasing 𝑇, even at 𝑇 below 𝑇—the temperature for the opening of the pseudogap. The increase of 𝑊𝑛(Ω𝑐,𝑇) from the room temperature down to 𝑇𝑐 is no more than 5%. (2) In the superconducting state (𝑇<𝑇𝑐) of some underdoped and optimally doped Bi-2212 compounds [111, 113, 114] (and underdoped Bi-2212 films [115]) there is an effective increase of 𝑊𝑠(Ω𝑐,𝑇) with respect to that in the normal state, that is, 𝑊𝑠(Ω𝑐,𝑇)>𝑊𝑛(Ω𝑐,𝑇) for 𝑇<𝑇𝑐. This is a non-BCS behavior which is shown in Figure 9. Note that in the tight binding model the effective band (kinetic) energy 𝑇𝑣 is negative (𝑇𝑣<0) and in the standard BCS case (32) gives that 𝑊𝑠(𝑇<𝑇𝑐)  decreases due to the increase of 𝑇𝑣. Therefore the experimental increase of 𝑊𝑠(𝑇<𝑇𝑐) by decreasing 𝑇 is called the non-BCS behavior. The latter means a lowering of the kinetic energy 𝑇𝑣 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 𝑇𝑐. In [117] it is speculated that the latter case might be realized in underdoped cuprates.

However, in some optimally doped and in most overdoped cuprates, there is a decrease of 𝑊𝑠(Ω𝑐,𝑇) at 𝑇<𝑇𝑐 (𝑊𝑠(Ω𝑐,𝑇)<𝑊𝑛(Ω𝑐,𝑇)) which is the BCS-like behavior [112] as it is seen in Figure 10.

We stress that the non-BCS behavior of 𝑊𝑠(Ω𝑐,𝑇) for underdoped (and in some optimally doped) systems was obtained by assuming that Ω𝑐(1-1.2)eV. However, in [104] these results were questioned by claiming that the conventional BCS-like behavior was observed (𝑊𝑠(Ω𝑐,𝑇)<𝑊𝑛(Ω𝑐,𝑇)) in the optimally doped YBCO and slightly underdoped Bi-2212 by using larger cutoff energy Ω𝑐=1.5eV. This discussion demonstrates how risky is to make definite conclusions on some fundamental physics based on the parameter- (such as the cutoff energy Ω𝑐) 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 𝑊𝑠(Ω𝑐,𝑇)<𝑊𝑛(Ω𝑐,𝑇) was observed, while in LSCO it was found that 𝑊𝑠(Ω𝑐,𝑇)const, that is, 𝑊𝑠(Ω𝑐,𝑇<𝑇𝑐)𝑊𝑛(Ω𝑐,𝑇𝑐).

The first question is the following. How to explain the strong temperature dependence of 𝑊(Ω𝑐,𝑇) in the normal state? In [39, 40] 𝑊(𝑇) is explained solely in the framework of the EPI physics where the EPI relaxation 𝛾𝑒𝑝(𝑇) plays the main role in the 𝑇 dependence of 𝑊(Ω𝑐,𝑇). The main theoretical results of [39, 40] are the following: the calculations of 𝑊(𝑇) based on the exact (30) give that for Ω𝑐Ω𝐷 (the Debye energy) the difference in spectral weights of the normal and superconducting states is small, that is, 𝑊𝑛(Ω𝑐,𝑇)𝑊𝑠(Ω𝑐,𝑇) since 𝑊𝑛(Ω𝑐,𝑇)𝑊𝑠(Ω𝑐,𝑇)Δ2/Ω2𝑐. (2) In the case of large Ω𝑐 the calculations based on (30) give 𝑊Ω𝑐𝜔,𝑇2pl8𝛾1(𝑇)𝑊𝑏𝜋22𝑇2𝑊2𝑏.(33) In the case when EPI dominates one has 𝛾=𝛾𝑒𝑝(𝑇)+𝛾imp where 𝛾𝑒𝑝(𝑇)=0𝑑𝑧𝛼2(𝑧)𝐹(𝑧)coth(𝑧/2𝑇). It turns out that for 𝛼2(𝜔)𝐹(𝜔), shown in Figure 4, one obtains (i) 𝛾𝑒𝑝(𝑇)𝑇2 in the temperature interval 100K<𝑇<200K as it is seen in Figure 11 for the 𝑇 dependence of 𝑊(Ω𝑐,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 𝜆𝑒𝑝,tr=1.5 one obtains rather good agreement between the theory in [39, 40] and experiments in [104, 111, 113, 114]. At lower temperatures, 𝛾𝑒𝑝(𝑇) deviates from the 𝑇2 behavior and this deviation depends on the structure of the spectrum in 𝛼2(𝜔)𝐹(𝜔). It is seen in Figure 11 that, for a softer Einstein spectrum (with Ω𝐸=200K), 𝑊(Ω𝑐,𝑇) lies above the curve with the 𝑇2 asymptotic behavior, while the curve with a harder phononic spectrum (with Ω𝐸=400K) lies below it. This result means that different behavior of 𝑊(Ω𝑐,𝑇) in the superconducting state of cuprates for different doping might be simply related to different contributions of low- and high-frequency phonons. We stress that such a behavior of 𝑊(Ω𝑐,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(Ω𝑐,𝑇) above and below 𝑇𝑐 in systems at and near the optimal doping.

(4) 𝛼2(𝜔)𝐹(𝜔) and the In-Plane Resistivity 𝜌𝑎𝑏(𝑇)
The temperature dependence of the in-plane resistivity 𝜌𝑎𝑏(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 𝜌𝑎𝑏(𝑇) in the normal state. Our intention is not to discuss this problem, but only to demonstrate that the EPI spectral function 𝛼2(𝜔)𝐹(𝜔), which is obtained from tunnelling experiments in cuprates (see Section 1.3.4), is able to explain the temperature dependence of 𝜌𝑎𝑏(𝑇) in the optimally doped 𝑌𝐵𝐶𝑂. In the Boltzmann theory 𝜌𝑎𝑏(𝑇) is given by 𝜌𝑎𝑏(𝑇)=𝜌imp+4𝜋𝜔2𝑝𝛾tr(𝑇),(34) where 𝛾tr𝜋(𝑇)=𝑇0𝜔𝑑𝜔sinh2𝛼(𝜔/2𝑇)2tr(𝜔)𝐹(𝜔).(35) The residual resistivity 𝜌imp 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 𝜌𝑎𝑏(𝑇) is also dominated by EPI in some temperature region 𝑇>𝑇𝑐. This is indeed confirmed in the optimally doped 𝑌𝐵𝐶𝑂, where 𝜌imp is chosen appropriately and the spectral function 𝛼2tr(𝜔)𝐹(𝜔) is taken from the tunnelling experiments in [4245]. 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 𝛾tr(𝑇)𝑇 for 𝑇>𝛼Θ𝐷, 𝛼<1 depending on the shape of 𝛼2tr(𝜔)𝐹(𝜔) (for the simple Debye spectrum 𝛼0.2). In the linear regime one has 𝜌(𝑇)𝜌imp+8𝜋2𝜆𝑒𝑝,tr(𝑘𝐵𝑇/𝜔2𝑝)=𝜌imp+𝜌𝑇.

There is experimental constraint on 𝜆tr since 𝜆tr0.25𝜔2pl(eV)𝜌(𝜇Ωcm/K). For instance, for 𝜔pl(2-3)eV [108] and 𝜌0.6 in the oriented YBCO films and 𝜌0.3-0.4 in single crystals of BSCO, one obtains 𝜆tr0.6-1.4. In case of YBCO single crystals, there is a pronounced anisotropy in 𝜌𝑎𝑏(𝑇) [119] which gives 𝜌𝑥(𝑇)=0.6𝜇Ωcm/K and 𝜌𝑦(𝑇)=0.25𝜇Ωcm/K. The function 𝜆tr(𝜔pl) is shown in Figure 13 where the plasma frequency 𝜔pl can be calculated by LDA-DFT and also extracted from the width (𝜔pl) of the Drude peak at small frequencies, where 𝜔pl=𝜀𝜔pl. We stress that the rather good agreement of theoretical and experimental results for 𝜌𝑎𝑏(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 𝜌𝑎𝑏(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 [811], 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 𝛼2(𝜔)𝐹(𝜔) obtained from the tunnelling experiments (and optics) one obtains the correct 𝑇 dependence of 𝜌𝑎,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<0.2Θ𝐷 [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 𝜌𝑎𝑏(𝑇) is realized even at 𝑇<0.2Θ𝐷. The quasilinear behavior of the resistivity at 𝑇0.2Θ𝐷 has been observed in Bi2(Sr0.97Pr0.003)2CuO6 [125], in LSCO, and in 1-layer Bi-2201 [122, 123, 126, 127], where in all these systems the critical temperature is rather small, 𝑇𝑐10K. In that respect all existing theories based on the electron-boson scattering are plagued and having difficulties to explain this low-temperature behavior of 𝜌𝑎𝑏(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=10K but some others with approximately the same 𝑇𝑐 have the usual Bloch-Grüneisen-type behavior characteristic for the EPI scattering. In that respect it is very unlikely that the linear resistivity up to 𝑇=10K 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=10K 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 𝑞𝑐𝑘𝐹), which is due to strong correlations, may give rise to the linear behavior of 𝜌(𝑇) down to very low temperatures 𝑇Θ𝐷/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 [4254] in the framework of the Eliashberg theory one obtains the large EPI coupling constant 𝜆𝑒𝑝2-3.5 which implies that 𝜆tr(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 𝜆tr cannot be obtained within the LDA-DFT band-structure calculations, which means that 𝜆𝑒𝑝 and 𝜆tr contain renormalization which do not enter in the LDA-DFT theory. In Section 2 we will argue that the strong suppression of 𝜆tr may have its origin in strong electronic correlations [7880, 130] and in the long-range Madelung energy [36].

(5) Femtosecond Time-Resolved Optical Spectroscopy
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 (1fs=1015sec) laser pump excites in materials electron-hole pairs via interband transitions. These hot carriers release their energy via electron-electron (with the relaxation time 𝜏𝑒𝑒) and electron-phonon scattering reaching states near the Fermi energy within 10-100fs—see [131]. The typical energy density of the laser pump pulses with the wavelength 𝜆810 nm (𝜔=1.5eV) was around 𝐹1𝜇J/cm2 (the excitation fluence 𝐹) which produces approximately 3×1010 carriers per pulse (by assuming that each photon produces 𝜔/Δ carriers, Δ is the superconducting gap). By measuring photoinduced changes of the reflectivity in time, that is, Δ𝑅(𝑡)/𝑅0, 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 Δ𝑅(𝑡)𝑅0=𝑓(𝑡)𝐴𝑒𝑡/𝜏𝐴+𝐵𝑒𝑡/𝜏𝐵,+(36) where 𝑓(𝑡)=𝐻(𝑡)[1exp{𝑡/𝜏𝑒𝑒}] (𝐻(𝑡) 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 La2𝑥Sr𝑥CuO4, with 𝑥=0.1 and 0.15 and 𝑇𝑐=30K and 38K, respectively [41]. In that case one has 𝐴0 for 𝑇<𝑇𝑐 and 𝐴=0 for 𝑇>𝑇𝑐, while the signal 𝐵 was present also at 𝑇>𝑇𝑐. 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 𝜏𝐴>10ps at 𝑇=4.5K. At the so called threshold fluence (𝐹𝑇=4.2±1.7𝜇J/cm2 for 𝑥=0.1 and 𝐹𝑇=5.8±2.3𝜇J/cm2 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 𝜏𝑟0.8ps. The external fluence is distributed in the sample over the excitation volume which is proportional to the optical penetration depth 𝜆op(150 nm at 𝜆810nm) of the pump. The energy densities stored in the excitation volume at the vaporization threshold for 𝑥=0.1 and 𝑥=0.15 are 𝑈𝑝=𝐹𝑇/𝜆op=2.0±0.8K/Cu and 2.6±1.0K/Cu, respectively. The important fact is that 𝑈𝑝 is much larger than the superconducting condensation energy which is 𝑈cond0.12K/Cu for 𝑥=0.1 and 𝑈cond0.3K/Cu for 𝑥=0.15, that is, 𝑈𝑝𝑈cond. This means that the energy difference 𝑈𝑝𝑈cond must be stored elsewhere on the time scale 𝜏𝑟. 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 reservoir from 𝑇=4.5K to 𝑇𝑐 is 𝑈sf=𝑇𝑐𝑇𝐶sf(𝑇)𝑑𝑇. The measured specific heat 𝐶sf(𝑇) in La2CuO4 [41] gives very small value 𝑈sf0.01K. In the case of the phonon reservoir one obtains 𝑈ph=𝑇𝑐𝑇𝐶ph(𝑇)𝑑𝑇=9K/Cu for 𝑥=0.1 and 28K/Cu for 𝑥=0.15, where 𝐶ph is the phonon specific heat. Since 𝑈sf𝑈𝑝𝑈cond, the spin reservoir cannot absorb the rest energy 𝑈𝑝𝑈cond. The situation is opposite with phonons since 𝑈ph𝑈𝑝𝑈cond 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 𝜆tr and 𝜔pl? (ii) What is the reason that 𝜆tr𝜆 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)𝛾0+𝜆𝑐𝜔 for 𝜔>𝜔phmax with 𝜆𝑐1. 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.

1.3.3. 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 𝐸𝑖=5-1000eV 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 Δ𝐸1-2meV (for photon energies 20 eV) and angular resolution of Δ𝜃0.2. On the other side the ARPES method is surface-sensitive technique, since the average escape depth (𝑙esc) of the outgoing electrons is of the order of 𝑙esc10 Å, 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 Bi2Sr2CaCu2O8 (𝐵𝑖2212) and its single-layer counterpart Bi2Sr2CuO6 (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 𝐼tot(𝐤,𝜔)𝐼1𝐼2𝐼3 is the product of three independent terms: (1) 𝐼1 that describes optical excitation of the electron in the bulk, (2) 𝐼2 that describes the scattering probability of the travelling electrons, and (3) 𝐼3 that describes the transmission probability through the surface potential barrier. The central quantity in the three-step model is 𝐼1(𝐤,𝜔) and it turns out that for 𝐤=𝐤 it can be written in the form 𝐼1(𝐤,𝜔)𝐼0(𝐤,𝜐)𝑓(𝜔)𝐴(𝐤,𝜔) [22, 23] with 𝐼0(𝐤,𝜐)|𝜓𝑓|𝐩𝐀|𝜓𝑖|2 and the quasiparticle spectral function 𝐴(𝐤,𝜔)=Im𝐺(𝐤,𝜔)/𝜋: 1𝐴(𝐤,𝜔)=𝜋ImΣ(𝐤,𝜔)[]𝜔𝜉(𝐤)ReΣ(𝐤,𝜔)2+ImΣ2.(𝐤,𝜔)(37) Here, 𝜓𝑓|𝐩𝐀|𝜓𝑖 is the dipole matrix element which depends on 𝐤, polarization, and energy 𝐸𝑖 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=ReΣ+𝑖ImΣ are the quasiparticle Green's function and the self-energy, 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 𝜉𝐤𝑎,𝑏=𝜉𝐤±𝑡𝐤 with 𝑡𝐤=[𝑡(cos2𝑘𝑥cos2𝑘𝑦)+] 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 𝑇𝑐<𝑇<𝑇 there is a 𝑑-wave-like pseudogap Δ𝑝𝑔(𝐤)Δ𝑝𝑔,0(cos𝑘𝑥cos𝑘𝑦) in the quasiparticle spectrum where Δ𝑝𝑔,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>𝑇𝑐 there are kinks in the quasiparticle dispersion 𝜔(𝜉𝐤) 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 40meV [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 (Tl2Ba2CuO6), Na–CCOC (Ca2𝑥Na𝑥CuO2Cl2). These kinks exist also above 𝑇𝑐, which excludes the scenario with the magnetic resonance peak in Im𝜒𝑠(𝐐,𝜔). Moreover, since the tunnelling and magnetic neutron scattering measurements give small SFI coupling constant 𝑔sf<0.2eV, 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 𝐤𝑁 couple practically to a rather broad spectrum of phonons since at least three groups of phonons were extracted in the bosonic spectral function 𝛼2𝐹(𝐤𝑁,𝜔) from the ARPES effective self-energy in La2𝑥Sr𝑥CuO4 [135]—Figure 15.

The latter result is in a qualitative agreement with numerous tunnelling measurements [4254] which apparently demonstrate that the very broad spectrum of phonons couples with holes without preferring any particular phonons—see discussion below. (𝐍𝟕) Recent ARPES measurements in Bi2212 [92] show very different slope 𝑑𝜔/𝑑𝜉𝐤 of the quasiparticle energy 𝜔(𝜉𝐤) at small |𝜉𝐤|𝜔ph and at large energies |𝜉𝐤|𝜔ph—see Figure 16. The theoretical analysis [137] of these results gives the total coupling constant 𝜆𝑍=𝜆𝑍𝑒𝑝+𝜆𝑍𝑐3, and for the EPI coupling 𝜆𝑍𝑒𝑝2, while the Coulomb coupling (SFI is a part of it) is 𝜆𝑍𝑐1 [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(𝜉𝑘) at 𝜔0.3eV. The latter is apparently due to the Coulomb interaction. Although the position of the low-energy kink is not affected by this procedure (if 𝜔maxph𝜔𝑐), this subtraction procedure gives in fact an effective EPI self-energy Σ𝑒𝑝e(𝐤,𝜔) and the effective coupling constant 𝜆𝑍𝑒𝑝,e(𝐤) only. We demonstrate below that the 𝜆𝑍𝑒𝑝,e(𝐤) is smaller than the real EPI coupling constant 𝜆𝑍𝑒𝑝(𝐤). The total self-energy is Σ(𝐤,𝜔)=Σ𝑒𝑝(𝐤,𝜔)+Σ𝑐(𝐤,𝜔) where Σ𝑐 is the contribution due to the Coulomb interaction. At very low energies 𝜔𝜔𝑐 one has usually Σ𝑐(𝐤,𝜔)=𝜆𝑍𝑐(𝐤)𝜔, where 𝜔𝑐(~1 eV) is the characteristic Coulomb energies and 𝜆𝑍𝑐 is the Coulomb coupling constant. The quasiparticle spectrum 𝜔(𝐤) is determined from the condition 𝜔𝜉(𝐤)Re[Σ𝑒𝑝(𝐤,𝜔)+Σ𝑐(𝐤,𝜔)]=0 where 𝜉(𝐤) is the bare band-structure energy. At low energies 𝜔<𝜔maxph𝜔𝑐 it can be rewritten in the form 𝜔𝜉ren(𝐤)ReΣ𝑒𝑝e(𝐤,𝜔)=0,(38) with 𝜉ren(𝐤)=[1+𝜆𝑍𝑐(𝐤)]1𝜉(𝐤), ReΣ𝑒𝑝e(𝐤,𝜔)=ReΣ𝑒𝑝e(𝐤,𝜔)1+𝜆𝑍𝑐.(𝐤)(39) Since at very low energies 𝜔𝜔maxph one has ReΣ𝑒𝑝(𝐤,𝜔)=𝜆𝑍𝑒𝑝(𝐤)𝜔 and ReΣ𝑒𝑝e(𝐤,𝜔)=𝜆𝑍𝑒𝑝,e(𝐤)𝜔, then the real coupling constant is related to the effective one by 𝜆𝑍𝑒𝑝(𝐤)=1+𝜆𝑍𝑐𝜆(𝐤)𝑍𝑒𝑝,e(𝐤).(40) As a result one has 𝜆𝑍𝑒𝑝(𝐤)>𝜆𝑍𝑒𝑝,e(𝐤). At higher energies 𝜔maxph<𝜔<𝜔𝑐, where the EPI effects are suppressed and Σ𝑒𝑝(𝐤,𝜔) stops growing, one has ReΣ(𝐤,𝜔)ReΣ𝑒𝑝(𝐤,𝜔)𝜆𝑍𝑐(𝐤)𝜔. The measured ReΣexp(𝐤,𝜔) at 𝑇=10K near and slightly away from the nodal point in the optimally doped Bi-2212 with 𝑇𝑐=91K [136] is shown in Figure 16.

It is seen that ReΣexp(𝐤,𝜔) has two kinks—the first one at low energy 𝜔1𝜔highph50-70meV which is (as we already argued) most probably of the phononic origin [22, 23, 138, 139], while the second kink at higher energy 𝜔2𝜔𝑐350meV which is due to the Coulomb interaction. However, the important results in [136] are that the slopes of ReΣexp(𝐤,𝜔) at low (𝜔<𝜔highph) and high energies (𝜔highph<𝜔<𝜔𝑐) are very different. The low-energy and high-energy slope near the nodal point are shown in Figure 16 schematically (thin lines). From Figure 16 it is obvious that EPI prevails at low energies 𝜔<𝜔highph. More precisely digitalization of ReΣexp(𝐤,𝜔) in the interval 𝜔highph<𝜔<0.4eV gives the Coulomb coupling 𝜆𝑍𝑐1.1 while the same procedure at 20meV𝜔lowph<𝜔<𝜔highph50-70meV gives the total coupling constant (𝜆2)𝜆𝑍=𝜆𝑍𝑒𝑝+𝜆𝑍𝑐3.2 and the EPI coupling constant 𝜆𝑍𝑒𝑝(𝜆𝑍𝑒𝑝,high)2.1>2𝜆𝑍𝑒𝑝,e(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 𝜆𝑍𝑒𝑝(2.1)2𝜆𝑐𝑧>2𝜆𝑍sf, while at large energies (compared to 𝜔ph) the Coulomb interaction with 𝜆𝑍𝑐1.1 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 ReΣexp(𝐤,𝜔), which can be well described in the framework of the Migdal-Eliashberg theory for EPI [140]. At higher energies 𝜔>𝜔ph 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 Ca2CuO2Cl2 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 0.8eV 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 [144146], explains successfully this broad peak of the boson side band by the formation of small polarons due to the EPI coupling (𝜆𝑒𝑝1.2). Note that this value of 𝜆𝑒𝑝 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 [144146]. In [144146] it was stressed that even when the electron-magnon 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. (𝐍𝟏𝟎) Recent soft X-ray ARPES measurements on the electron-doped HTSC Nd1.85Ce0.15CuO4 [147], and Sm(2𝑥)Ce𝑥CuO4 (𝑥=0.1,0.15,0.18), Nd1.85Ce0.15CuO4, and Eu1.85Ce0.15CuO4 [148] show kink at energies 50-70meV 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 𝜆𝑒𝑝,e(<𝜆𝑒𝑝) is isotropic and 𝜆𝑒𝑝,e0.8-1. 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)Δ0(cos𝑘𝑥cos𝑘𝑦) with 2Δ0/𝑇𝑐5-6 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 (𝛼𝐤,±) with energies 𝐸𝛼±𝐤=𝜉2𝐤+|Δ𝐤|2 are a mixture of electron (̂𝑐𝐤,𝜎) and hole (̂𝑐𝐤,𝜎), that is, 𝛼𝐤,+=𝑢𝐤̂𝑐𝐤+𝑣𝐤̂𝑐𝐤, 𝛼𝐤,=𝑢𝐤̂𝑐𝐤+𝑣𝐤̂𝑐𝐤 where the coherence factors 𝑢𝐤, 𝑣𝐤 are given by |𝑢𝐤|2=1|𝑣𝐤|2=(1+𝜉𝐤/𝐸𝐤)/2. Note that |𝑢𝐤|2+|𝑣𝐤|2=1, which is exactly observed, together with 𝑑-wave pairing Δ(𝐤)=Δ0(cos𝑘𝑥cos𝑘𝑦), 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 (60-70)meV in the quasiparticle energy around the nodal point is not shifted (in energy) while the antinodal kink at 𝜔(40)ph40meV is shifted (in energy) in the superconducting state by Δ0(=(25-30)meV), that is, 𝜔(40)ph𝜔(40)ph+Δ0=(65-70)meV [22, 23]. To remind the reader, in the standard Eliashberg theory the kink in the normal state at 𝜔=𝜔ph should be shifted in the superconducting state to 𝜔ph+Δ0 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<𝑞𝑐𝑘𝐹) 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) 𝛼2𝐹(𝐤,𝐤,Ω)𝛼2𝐹(𝜑,𝜑,Ω) (𝜑 is the angle on the Fermi surface) has a pronounced forward scattering peak (at 𝛿𝜑=𝜑𝜑=0) due to strong correlations—see Section 2. Its width 𝛿𝜑𝑐 is narrow, that is, 𝛿𝜑𝑐2𝜋 and the angle integration goes over the region 𝛿𝜑𝑐 around the point 𝜑. In that case the kink is shifted (approximately) by the local gap Δ(𝜑)=Δmaxcos2𝜑—for more details see [151]. As a consequence, the antinodal kink is shifted by the maximal gap, that is, |Δ(𝜑AN𝜋/2)|=Δmax while the nodal gap is practically unshifted since |Δ(𝜑AN𝜋/4)|0. (𝐒𝟒) The recent ARPES spectra [152] in the undoped single crystalline 4-layered cuprate with the apical fluorine (F), Ba2Ca3Cu4O8F2 (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 𝑇𝑐=60K. Moreover, the ARPES data [152] reveal at least two metallic Fermi-surface sheets with corresponding volumes equally below and above half-filling—see Figure 20.

Second, one of the Fermi surfaces is due to the electron-like (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 self-doping. For instance, in HgBa2Ca𝑛Cu𝑛+1O2𝑛+2 (𝑛=2,3) and (Cu,C)Ba2Ca𝑛Cu𝑛+1O3𝑛+2 (𝑛=2,3,4), the inner CuO2 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 Ba2Ca3Cu4O8F2 the ratio is anomalous (Δ𝑁/Δ𝑃)2 and Δ𝑁 is an order of magnitude larger than in the electron-doped cuprate Nd2𝑥Ce𝑥CuO4. 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 (Δ𝑁/Δ𝑃)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<𝑇𝑐 (in the superconducting state) [154]. This effect can be explained by assuming that the small 𝑞-scattering (the forward scattering peak) dominates in the elastic impurity scattering as it is pointed in [7880, 130, 155, 156]. As a result, one finds that the impurity scattering rate in the superconducting state is almost zero, that is, 𝛾imp(𝐤,𝜔)=𝛾𝑛(𝐤,𝜔)+𝛾𝑎(𝐤,𝜔)0 for |𝜔|<Δ0 for any kind of pairing (𝑠-, 𝑝-, 𝑑-wave, etc.) since the normal (𝛾𝑛) and the anomalous (𝛾𝑎) 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 Anderson-like theorem for unconventional superconductors which is due to the dominance of the small 𝑞-scattering [7880, 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 𝑇𝑐 [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>𝜔ph. 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.

1.3.4. Tunnelling Spectroscopy and Spectral Function 𝛼2𝐹(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 𝐺NS(𝑉)=𝑑𝐼/𝑑𝑉 the so called tunnelling density of states in superconductors 𝑁𝑇(𝜔). Moreover, by measuring of 𝐺NS(𝑉) at voltages eV>Δ it is possible to determine the Eliashberg spectral function 𝛼2𝐹(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 𝜉0, then this kind of measurements in HTSC materials with small coherence length 𝜉0 (𝜉𝑎𝑏20 Å in the 𝑎𝑏 plane and 𝜉𝑐1-3 Å 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 (energy-dependent) 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 𝐺NS(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 𝐺NS(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 𝛼2𝐹(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 𝛼2𝐹(𝜔), which are extracted from the second derivative 𝑑2𝐼/𝑑𝑉2 at voltages above the superconducting gap, in most HTSC cuprates resemble the phonon density of states 𝐹ph(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 [4245] show that the peaks (and shoulders) in 𝑑2𝐼/𝑑𝑉2 (or dips-negative peaks in 𝑑2𝐼/𝑑𝑉2) coincide with the peaks (and shoulders) in the phonon density of states 𝐹ph(𝜔) measured by neutron scattering—see Figure 21.

The tunnelling spectra in Bi-2212 break junctions [4245], which are shown in Figure 21 indicates that the spectral function 𝛼2𝐹(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 𝛼2𝐹(𝜔) is partly due to the gapless spectrum of 𝑑-wave pairing in HTSC cuprates. Additionally, the tunnelling density of states 𝑁𝑇(𝜔) at very low 𝑇 and for 𝜔>Δ shows a pronounced gap structure. It was found that 2Δ/𝑇𝑐=6.2-6.5, where 𝑇𝑐=74-85Kand Δ is some average value of the gap. In order to obtain 𝛼2𝐹(𝜔) the inverse procedure was used by assuming 𝑠-wave superconductivity and the effective Coulomb parameter 𝜇0.1 [4245]. The obtained 𝛼2𝐹(𝜔) gives large EPI coupling constant 𝜆𝑒𝑝2.3. Although this analysis [4245] 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 𝑑2𝐼/𝑑𝑉2 at eV>Δ albeit introducing a broadening in it—see the physical meaning in Appendix A. We point out that the results obtained in [4245] 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 [4245] on break junctions. It is worth mentioning that several groups [4852] have obtained similar results for the shape of the spectral function 𝛼2𝐹(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 Bi2Sr2CaCu2O8 [46, 47, 53, 54] give impressive results. The Eliashberg spectral function 𝛼2𝐹(𝜔) of this compound was extracted from the measurements of 𝑑2𝐼/𝑑𝑉2 and by solving the inverse problem—see Appendix A. The extracted 𝛼2𝐹(𝜔) has several peaks in broad energy region up to 80meV as it is seen in Figures 22 and 23, which coincide rather well with the peaks in the phonon density of states 𝐹ph(𝜔)—more precisely the generalized phonon density of states 𝐺𝑃𝐷𝑆(𝜔) defined in Appendix A. In [53, 54] numerous peaks, from 𝑃1-𝑃13, in 𝛼2𝐹(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 (Δ𝑇𝑐) to 𝑇𝑐 as it is seen in Table 1. Note that due to the nonlinearity of the problem the sum of (Δ𝑇𝑐)𝑖,𝑖=1,2,,13, due to various modes is not equal to 𝑇𝑐.

The next remarkable result is that the extracted EPI coupling constant is very large, that is, 𝜆𝑒𝑝(=2𝑑𝜔𝛼2𝐹(𝜔)/𝜔)=𝑖𝜆𝑖3.5—see Table 1. It is obvious from Figure 24 and Table 1 that almost all phonon modes contribute to 𝜆𝑒𝑝 and 𝑇𝑐, which means that on the average each particular phonon mode is not too strongly coupled to electrons since 𝜆𝑖<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 𝑇𝑐. 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 𝜆𝜅1) which give appreciable contributions to 𝑇𝑐. It is also seen in Table 1 that the coupling constants 𝜆𝑖 of the high-energy phonons (𝑃9-𝑃13 with 𝜔70meV) have 𝜆𝑖1 and give moderate contribution to 𝑇𝑐—around 10%. These results give solid evidence for the importance of the low-energy modes related to the change of the Madelung energy in the ionic-metallic structure of HTSC cuprates—the idea advocated in [36] 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 𝑇𝑐. We stress that each peak 𝑃1-𝑃13 in 𝛼2𝐹(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 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 the structure of 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 CuO2 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 𝛼2𝐹(𝜔) of the slightly overdoped Bi2Sr2CaCu2O8 with the ratio (2Δ/𝑇𝑐)6.5 describe rather well numerous optical, transport, and thermodynamic properties [53, 54]. However, in underdoped systems with (2Δ/𝑇𝑐)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 high-frequency modes (mostly the oxygen modes) to 𝛼2𝐹(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)=𝑣𝐹𝛾1(𝜔). Since the relaxation time 𝛾1(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 𝛼2𝐹(𝜔) in HTSC cuprates comes out from tunnelling measurements on Andreev junctions (the BTK parameter 𝑍1—low barrier) and Giaver junctions (𝑍1—high barrier) in La2𝑥Sr𝑥CuO4 and YBCO compounds [160], where the extracted 𝛼2𝐹(𝜔) is in good accordance with the phonon density of states 𝐹ph(𝜔)—see Figure 28.

Note that the BTK parameter 𝑍 is related to the transmission and reflection coefficients for the normal metal (1+𝑍2)1 and 𝑍2(1+𝑍2)1, respectively.

Although most of the peaks in 𝛼2𝐹(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 𝛼2𝐹(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 𝐸𝑟 scales with 𝑇𝑐, that is, 𝐸𝑟=(5-6)𝑇𝑐 as shown in Figure 29.

This means that if one of the peaks in 𝛼2𝐹(𝜔) is due to the magnetic resonance at 𝜔=𝐸𝑟, 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 𝛼2𝐹(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 Pr0.88Ce0.12CuO4 with 𝑇𝑐=24K shows the existence of the bosonic mode at 𝜔𝐵=16meV which is significantly larger than the magnetic-resonance mode with 𝜔𝑟=(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 La1.85Sr0.15CuO4 films prepared by the molecular beam epitaxy on the [001]-symmetric SrTiO3 bicrystal substrates [56]. They give unique evidence for eleven peaks in the (negative) second derivative, that is, 𝑑2𝐼/𝑑𝑉2. Furthermore, these peaks coincide with the peaks in the intensities of the phonon Raman scattering data measured at 30K in single crystals of LSCO with 20% of Sr [57]. These results are shown in Figure 31. In spite of the 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 𝑑2𝐼/𝑑𝑉2 at eV>Δ was not seen below 𝑇𝑐, while the pseudogap structure is observed at temperatures near and above 𝑇𝑐. 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 [6163] give information on the possible nature of the bosonic mode which couples with electrons. In [6163] the local conductance 𝑑𝐼/𝑑𝑉(𝐫,𝐸) is measured where it is found that 𝑑2𝐼/𝑑𝑉2(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, Ω𝑀1/2O, with Ω16=52meV and Ω1848meV—as it is seen in Figure 32(b). This result is interpreted in [6163] as an evidence that the oxygen phonons are strongly involved in the quasiparticle scattering. A possible explanation is put forward in [6163] by assuming that this isotope effect is due to the 𝐵1𝑔 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 𝑑2𝐼/𝑑𝑉2(𝐫,𝐸)—as reported in [6163].

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 𝛼2𝐹(𝜔) as being the only one which dominates in pairing mechanism. This important result means that the high 𝑇𝑐 is not attributable to a particular phonon mode in the EPI mechanism but all phonon modes contribute to 𝜆𝑒𝑝. Having in mind that the phonon spectrum in HTSC cuprates is very broad (up to 80meV), then the large EPI constant (𝜆𝑒𝑝2) obtained in the tunnelling experiments is not surprising at all. Note that similar conclusion holds for some other oxide superconductors such as Ba9.6K0.4BiO3 with 𝑇𝑐=30K where the peaks in the bosonic spectral function/extracted from tunnelling measurements coincide with the peaks in the phononic density of states [165167].

1.3.5. 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 𝛼2𝐹(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 L2𝑥Sr𝑥CuO4—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, [69]—but 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 𝐷1(𝐪,𝜔)=𝐷01(𝐪,𝜔)Π(𝐪,𝜔)). In cases when the EPI coupling constant 𝑔𝑒𝑝(𝐤,𝐤) is a function on the transfer momentum 𝐪=𝐤𝐤 only, then Π(𝑞) (𝑞=(𝐪,𝑖𝜔𝑛)) depends on the quasiparticle charge susceptibility 𝜒𝑐(𝑞)=𝑃(𝑞)/𝜀𝑒(𝑞): ||𝑔Π(𝑞)=𝑒𝑝||(𝐪)2𝜒𝑐(𝑞),(41) and 𝑃(𝑞) is the irreducible electronic polarization given by 𝑃(𝑞)=𝑝𝐺(𝑝+𝑞)Γ𝑐(𝑝,𝑞)𝐺(𝑝).(42) The screening due to the long-range Coulomb interaction is contained in the electronic dielectric function 𝜀𝑒(𝑞) while the “screening” due to (strong) correlations is described by the charge vertex function Γ𝑐(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 𝜀𝑒=1 and 𝜒𝑐(𝑞)=𝑃(𝑞). However, in studying the phonon spectra in HTSC cuprates it is believed that this deficiency might be partly compensated by choosing the bare phonon frequency 𝜔0(𝐪) (contained in 𝐷01(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 𝑔𝑒𝑝(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 3d9 state) due to strong correlations, and since the suppressed value of 𝜒𝑐(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.

(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] and here we discuss briefly two important examples which demonstrate the inefficiency of the LDA-DFT-band structure calculations to treat quantitatively EPI in HTSC cuprates. For instance, the Cu–O bond-stretching phonon mode shows a substantial softening at 𝐪hb=(0.3,0,0) by doping of La1.85Sr0.15CuO4 and YBa2Cu3O7 [170, 172]—called the half-breathing phonon, and a large broadening by 5meV at 15% doping [173175] 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 𝜒𝑐(𝑞) and due to the absence of many-body effects which can increase the coupling constant 𝑔𝑒𝑝(𝐪)—see more in Section 2. The neutron scattering measurements in La1.85Sr0.15CuO4 give evidence for large (30%) softening of the O𝑍𝑍 with Λ1 symmetry with the energy 𝜔60meV, which is theoretically predicted in [177], and for the large line-width about 17meV 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 𝜆𝑒𝑝 in the framework of LDA-DFT give very small EPI coupling constant 𝜆𝑒𝑝0.3 [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 |𝑔𝐪| and charge susceptibility 𝜒𝑐, that is, 𝛿𝜔(𝐪)|𝑔𝑒𝑝(𝐪)|2Re𝜒𝑐, while the line-width is given by Γ𝜔(𝐪)|𝑔𝑒𝑝(𝐪)|2|Im𝜒𝑐|. 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)Γ(exp)𝜔(𝐪). However, it turns out that in strongly correlated systems doped by holes (with the concentration 𝛿1) the charge fluctuations are suppressed in which case the following sum rule holds [10, 11, 171]: 1𝜋𝑁𝐪00𝑥0200𝑑||𝑑𝜔Im𝜒𝑐||(𝐪,𝜔)=2𝛿(1𝛿)𝑁,(43) while in the LDA-DFT method one has 1𝜋𝑁𝐪0||𝑑𝜔Im𝜒𝑐||(𝐪,𝜔)(LDA)=(1𝛿)𝑁.(44) The inequality Γ(LDA)𝜔(𝐪)Γ(exp)𝜔(𝐪) (for some phonon modes) together with (43)-(44) means that for low doping 𝛿1 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 𝜒𝑐(𝐪,𝜔) (i.e., Γ𝑐(𝑝,𝑞)=Γ𝑐(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 Γ𝑐(𝐩,𝑞) is calculated [6, 7880, 130, 179, 180]. It turns out that in the adiabatic limit (𝜔=0) the vertex functions Γ𝑐(𝐩𝐹,𝐪) in these two methods have important differences. For instance, Γ𝑐(𝑋)(𝐩𝐹,𝑞) (in the X-method) is peaked at 𝐪=0—the so called forward scattering peak (FSP)—while Γ(SB)𝑐(𝐩𝐹,𝐪) has maximum at finite |𝐪|0 [181]—see Section 2.3.5. The enumerated properties of Γ𝑐(𝑋)(𝐩𝐹,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 𝑈𝑡. 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 𝜒𝑐(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 |𝑔𝑒𝑝(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 |𝑔𝑒𝑝(𝐪)|2|𝑔(LDA)𝑒𝑝(𝐪)|2. For instance, for the half-breathing mode, one has |𝑔𝑒𝑝(𝐪)|23|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, 7880, 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 𝑔𝑒𝑝(𝐪) in Σ(𝑘): Σ(𝑘)=𝑞||𝑔𝑒𝑝(𝐪)𝛾𝑐||(𝐤,𝑞)2𝐷(𝑞)𝑔(𝑘+𝑞),(45) where 𝑔(𝑘)(𝐺(𝑘)/𝑄) is the quasiparticle Green's function, 𝛾𝑐(𝐤,𝑞)=Γ𝑐(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|>𝑞𝜔=𝜔ph/𝑣𝐹 one has 𝛾𝑐(𝐤,𝑞)𝛾𝑐(𝐤,𝐪) and for 𝑞𝑞𝑐(𝛿𝜋/𝑎) the charge vertex is strongly suppressed (𝛾𝑐(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 (𝜆𝑑𝜆𝑠). Then in the presence of the residual Coulomb interaction EPI gives rise to 𝑑-wave pairing. On the other side the charge vertex Γ𝑐(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 Γ𝑐(𝐤,𝐪) at 𝐪=0 may be (partly) responsible that the maximal experimental softening and broadening of the stretching (half-breathing) mode in La1.85Sr0.15CuO4 and YBa2Cu3O7 is at 𝐪(exp)hb=(0.3,0,0) [170] and not at 𝐪hb=(0.5,0) for which 𝑔𝑒𝑝(𝐪hb) 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.

(2) The Phonon Raman Scattering
The phonon Raman scattering gives an indirect evidence for importance of EPI in cuprates [184188]. We enumerate some of them—see more in [6] and references therein. (i) There is a pronounced asymmetric line-shape (of the Fano resonance) in the metallic state. For instance, in YBa2Cu3O7 two Raman modes at 115cm1 (Ba dominated mode) and at 340cm1 (O dominated mode in the CuO2 planes) show pronounced asymmetry which is absent in YBa2Cu3O6. This asymmetry means that there is an appreciable interaction of Raman active phonons with continuum states (quasiparticles). (ii) The phonon frequencies for some 𝐴1𝑔 and 𝐵1𝑔 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 remarkable correlation between the electronic Raman cross-section 𝑆exp(𝜔) and the optical conductivity in the 𝑎𝑏 plane 𝜎𝑎𝑏(𝜔), that is, 𝑆exp(𝜔)𝜎𝑎𝑏(𝜔) [6]. In previous subsections it is argued that EPI with the very broad spectral function 𝛼2𝐹(𝜔) (0<𝜔80meV) explains in a natural way the 𝜔 and 𝑇 dependence of 𝜎𝑎𝑏(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 𝛼2𝐹(𝜔) extracted from the tunnelling measurements in YBa2Cu3O6+𝑥 and Bi2Sr2CaCu2O8+𝑥 [6, 4254]. 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 La0.5Sr0.5CoO3 and Ca0.5Sr0.5RuO3 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.

1.3.6. Isotope Effect in 𝑇𝑐 and Σ(𝑘,𝜔)

The isotope effect 𝛼𝑇𝐶 in the critical temperature 𝑇𝑐 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 mono-atomic systems and in few polyatomic systems: the hydrogen isotope effect in PdH, the Mo and Se isotope shift of 𝑇𝑐 in Mo6Se8, and the isotope effect in Nb3Sn and MgB2. We point out that very small (𝛼𝑇𝐶0 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 𝛼𝑇𝐶 cannot be considered as the smoking gun effect since it is sensitive to numerous influences. For instance, in MgB2 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, 𝛼𝑇𝐶0.3 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, 𝛼Σ, which will be discussed below.

(1) Isotope Effect 𝛼𝑇𝐶 in 𝑇𝑐
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 𝛼𝑇𝐶=𝑖,𝑝𝛼𝑖(𝑝)=𝑖,𝑝𝑑ln𝑇𝑐/𝑑ln𝑀𝑖(𝑝), where 𝑀𝑖(𝑝) 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, 𝜇𝑐=0, the theory predicts 𝛼𝑇𝐶=1/2. From this formula one can deduce that the relative change of 𝑇𝑐,𝛿𝑇𝑐/𝑇𝑐, for heavier elements should be rather small—for instance, it is 0.02 for 135Ba138Ba, 0.03 for 63Cu65Cu, and 0.07 for 138La139La. This means that the measurements of 𝛼𝑖 for heavier elements are confronted with the ability of the present experimental techniques. Therefore most isotope effect measurements were done by substituting light atoms 16O by 18O only. It turns out that in most optimally doped HTSC cuprates 𝛼O is rather small. For instance, 𝛼O0.02-0.05 in YBa2Cu3O7 with 𝑇𝑐,max91K, but it is appreciable in La1.85Sr0.15CuO4 with 𝑇𝑐,max35K where 𝛼O0.1-0.2. In Bi2Sr2CaCu2O8 with 𝑇𝑐,max76K one has 𝛼O0.03-0.05 while 𝛼O0.03 and even negative (0.013) in Bi2Sr2Ca2Cu2O10 with 𝑇𝑐,max110K. The experiments on Tl2Ca𝑛1BaCu𝑛O2𝑛+4 (𝑛=2,3) with 𝑇𝑐,max121K are still unreliable and 𝛼O is unknown. In the electron-doped (Nd1𝑥Ce𝑥)2CuO4 with 𝑇𝑐,max24K one has 𝛼O<0.05 while in the underdoped materials 𝛼O increases. The largest 𝛼O is obtained even in the optimally doped compounds like in systems with substitution, such as La1.85Sr0.15Cu1𝑥M𝑥O4, 𝑀=Fe,Co, where 𝛼O1.3 for 𝑥0.4%. In La2𝑥M𝑥CuO4 there is a Cu-isotope effect which is of the order of the oxygen one, that is, 𝛼Cu𝛼O giving 𝛼Cu+𝛼O0.25-0.35 for optimally doped systems (𝑥=0.15). In case when 𝑥=0.125 with 𝑇𝑐𝑇𝑐,max one has𝛼Cu0.81 with 𝛼Cu+𝛼O1.8 [190, 191]. The appreciable copper isotope effect in La2𝑥M𝑥CuO4 tells us that vibrations other than oxygen ions are important in giving high 𝑇𝑐. 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 𝛼2𝐹(𝐤,𝜔) and according to these results the oxygen modes give moderate contribution to 𝑇𝑐 [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 𝛼Σ 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 𝛿𝜔𝑘,70(10-15)meV of the nodal kink at 𝜔𝑘,7070meV for the 16O18O substitution. In [139] it is reported that the isotope shift of the self-energy 𝛿Σ=Σ16Σ1810meV is very pronounced at large energies 𝜔=100-300meV. Concerning the latter result, there is a dispute since it is not confirmed in other experiments [141, 142]. However, the isotope effect in ReΣ(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 latter allowed very good precision in measuring the isotope effect in the nodal point of Bi-2212 with 𝑇𝑐16=92.1K and 𝑇𝑐18=91.1K [192]. They observed a shift in the maximum of ReΣ(𝐤𝑁,𝜔)—at 𝜔𝑘,7070meV (it corresponds to the half-breathing or to the breathing phonon)—by 𝛿𝜔𝑘,703.4±0.5meV as shown in Figure 34.

By analyzing the shift in ImΣ(𝐤𝑁,𝜔)—shown in Figure 34—one finds similar result for 𝛿𝜔𝑘,703.2±0.6meV. The similar shift was obtained in STM measurements [6163] 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 𝑇𝑐100K the oxygen isotope effect in 𝑇𝑐 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 𝑇𝑐—see the discussion in Subsection 1.3.4 on the tunnelling spectroscopy. (ii) In ARPES measurements of [192] the effective EPI coupling constant 𝜆𝑒𝑝,e0.6 is extracted, while the theory in Subsection 1.3.3 gives that the real coupling constant is larger, that is, 𝜆𝑒𝑝>1.2. This value is significantly larger than the LDA-DFT theory predicts 𝜆𝑒𝑝,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<𝜆𝑒𝑝<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 𝛼2𝐹(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 𝐹ph(𝜔). So obtained spectral function when inserted in the Eliashberg equations provides sufficient strength for obtaining high critical temperature 𝑇𝑐100K. 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 (𝜔𝜔𝑐) 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 𝜆𝑒𝑝, that is, 𝜆tr𝜆𝑒𝑝/3, which points to some peculiar scattering mechanism not met in low-temperature superconductors. The different renormalization of the quasiparticle and transport self-energies 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 𝑇𝑐 is practically unchanged. This implies that the real SFI coupling constant 𝜆sf(𝑔2sf) is small since the experimental value 𝑔(exp)sf<0.2eV 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.

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 𝑇𝑐 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 high-energy processes are incorporated in the (so called) ideal band-structure (nonlocal) potential 𝑉IBS(𝐱,𝐲) and the vertex function Γ. This program of calculations of 𝑉IBS(𝐱,𝐲),Γ, and the EPI coupling (matrix elements) 𝑔𝑒𝑝(𝐱,𝐲) is not realized in HTSC superconductors due to its complexity. However, one pragmatical way out is to calculate 𝑔𝑒𝑝 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 𝑈𝑊𝑏) 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 (𝑉𝑒𝑝,e(𝐪,𝜔)) at small transfer momenta 𝑞(𝑞𝑐𝑘𝐹) [6, 7880, 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-two-dimensionality 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 cuprates—which 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.

2.1. Microscopic Theory of Superconductivity
2.1.1. 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: 𝐻𝐻=𝑒+𝐻𝑖+𝐻𝑒-𝑖. The electronic Hamiltonian 𝐻𝑒, which describes the kinetic energy and the Coulomb interactions of electrons, is given by 𝐻𝑒=𝑑3𝑟𝜓(𝐫)̂𝜏3𝜖0(+1̂𝑝)𝜓(𝐫)2𝑑3𝑟𝑑3𝑟𝜓(𝐫)̂𝜏3𝜓(𝐫)𝑉𝑐𝐫𝐫𝜓𝐫̂𝜏3𝐫𝜓,(46) where 𝜖0(̂𝑝)=̂𝑝2/2𝑚 is the kinetic energy of electron and 𝑉𝑐(𝐫𝐫)=𝑒2/|𝐫𝐫| is the electron-electron Coulomb interaction. Note that in the electron-hole space the pseudospin (Nambu) matrices ̂𝜏𝑖, 𝑖=0,1,2,3 are Pauli matrices. Since we will discuss only the electronic properties, the explicit form of the lattice Hamiltonian 𝐻𝑖 [6, 194] is omitted here. The electron-ion Hamiltonian describes the interaction of electrons with the equilibrium lattice and with its vibrations, respectively: 𝐻𝑒-𝑖=𝑛𝑑0𝑥0200𝑑3𝑟𝑉𝑒-𝑖𝐫𝐑0𝑛𝜓(𝐫)̂𝜏3+𝑑𝜓(𝐫)3𝑟Φ(𝐫)𝜓(𝐫)̂𝜏3𝜓(𝐫).(47) Here, 𝑉𝑒-𝑖(𝐫𝐑0𝑛) 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 𝑉𝑒-𝑖(𝐫𝐑0𝑛)=𝑍𝑒2/|𝐫𝐑0𝑛| where 𝑍𝑒 is the ionic charge. The second term which is proportional to the lattice distortion operator Φ(𝐫)=𝑛,𝛼̂𝑢𝛼𝑛𝛼𝑉𝑒-𝑖(𝐫𝐑0𝑛Φ)+anh(𝐫) (because of convenience it includes also the EPI coupling 𝛼𝑉𝑒-𝑖) describes the interaction of electrons with harmonic (̂𝑢𝛼𝑛) (or anharmonic Φanh(𝐫)) lattice vibrations.

Dyson's equations for the electron and phonon Green's functions 𝐺(1,2)=𝑇𝜓(1)𝜓(2), 𝐷(12)=𝑇Φ(1)Φ(2) are 𝐺1𝐺(1,2)=01(1,2)Σ(1,2) and 𝐷1𝐷(1,2)=01(1,2)Π(1,2), where the 𝐺01(1,2)=[(𝜕/𝜕𝜏1𝜖0(𝐩1)+𝜇)̂𝜏0𝑢e(1)̂𝜏3]𝛿(12) is the bare inverse electronic Green's function. Here, 1=(𝐫1,𝜏1), where 𝜏1 is the imaginary time in the Matsubara technique, and the effective one-body potential 𝑢e(1)=𝑉𝑒-𝑖(1)+𝑉𝐻++Φ(1), where 𝑉𝐻 is the Hartree potential. The electron and phonon self-energies Σ(1,2) and Π(1,2) take into account many-body dynamics of the interacting system. The electronic self-energy ΣΣ(1,2)=𝑐Σ(1,2)+𝑒𝑝(1,2) is obtained in the form Σ(1,2)=𝑉e1,1̂𝜏3𝐺1,2Γe2,2;1,(48) where integration (summation) over the bar indices is understood. The effective retarded potential 𝑉e(1,1) in (48) contains the screened (by the electron dielectric function 𝜀𝑒(1,2)) Coulomb and EPI interactions: 𝑉e(1,2)=𝑉𝑐11𝜀𝑒11,2+𝜀𝑒11,1𝐷1,2𝜀𝑒12,2.(49) The inverse electronic dielectric permeability 𝜀𝑒1(1,2) = 𝛿(12)+𝑉𝑐(11)𝑃(1,2)𝜀𝑒1(2,2) is defined via the irreducible electronic polarization operator 𝑃(1,2) = 𝑆𝑝{̂𝜏3𝐺(1,Γ2)e(2,3;2)𝐺(3,1+)}. The vertex function Γe(1,2;3) = 𝛿𝐺(1,2)/𝛿𝑢e(3) in (48) is the solution of the complicated (and practically unsolvable) integro-differential functional equation Γe(1,2;3)=̂𝜏3+𝛿𝛿(12)𝛿(13)Σ(1,2)𝛿𝐺1,2𝐺1,3𝐺4,2Γe3,.4;3(50) Note that the effective vertex function Γe(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.

2.1.2. Low-Energy Migdal-Eliashberg Theory

If the vertex function Γe 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 low-energy phenomena (with energies |𝜔𝑛|,𝜉𝜔𝑐 and for momenta 𝑘=𝑘𝐹+𝛿𝑘 in the shell 𝛿𝑘𝛿𝑘𝑐 near the Fermi momentum 𝑘𝐹;𝜔𝑐 and 𝛿𝑘𝑐 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,𝜔𝑛)=[𝑖𝜔𝑛(𝐤2/2𝑚𝜇)̂𝜏3Σ(𝐤,𝜔𝑛)]1 can be formally written in the form 𝐺𝐤,𝜔𝑛=𝐺low𝐤,𝜔𝑛+𝐺high𝐤,𝜔𝑛,(51) where 𝐺low(𝐤,𝜔𝑛)=𝐺(𝐤,𝜔𝑛)Θ(𝜔𝑐|𝜔𝑛|)Θ(𝛿𝑘𝑐𝛿𝑘) is the low-energy Green's function and 𝐺high(𝐤,𝜔𝑛)=𝐺(𝐤,𝜔𝑛)Θ(|𝜔𝑛|𝜔𝑐)Θ(𝛿𝑘𝛿𝑘𝑐) is the high-energy one and analogously 𝐷=𝐷low(𝐤,𝜔𝑛)+𝐷high(𝐤,𝜔𝑛). By introducing the small parameter of the theory 𝑠(𝜔/𝜔𝑐)(𝛿𝑘/𝛿𝑘𝑐)1 one has in leading order 𝐺low(𝐤,𝜔𝑛)𝑠1, 𝐺high(𝐤,𝜔𝑛)1 and 𝐷low(𝐤,𝜔𝑛)𝑠0, 𝐷high(𝐤,𝜔𝑛)𝑠2. Note that the coupling constants (𝑉𝑒𝑖,𝑉𝑒𝑖,𝑉𝑖𝑖, etc.) are of the order 𝑠0=1.

The procedure of separating low-energy and high-energy processes lies also behind the adiabatic approximation since in most materials the characteristic phonon (Debye) energy 𝜔𝐷 of lattice vibrations is much smaller than the characteristic electronic Fermi energy 𝐸𝐹 (𝜔𝐷𝐸𝐹). In the 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 ΓeΓ(1,2;3)𝑐Γ(1,2;3)+𝛿𝑒𝑝(1,2;3) [1, 2], where the Coulomb charge vertex Γ𝑐(1,2;3)=̂𝜏3Σ𝛿(12)𝛿(13)+𝛿𝑐(1,2)/𝛿𝑢e(3) contains correlations due to the Coulomb interaction only but does not contain EPI and phonon propagator 𝐷 explicitly. The part 𝛿Γ𝑒𝑝Σ(1,2;3)=𝛿𝑒𝑝(1,2)/𝛿𝑢e(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 𝛿Γ𝑒𝑝(1,2;3) which is linear in 𝐷 one can express the self-energy in terms of the charge (Coulomb) vertex Γ𝑐(1,2;3) only. As a result of this approximation, the part of the self-energy due to Coulomb interaction is given by Σ𝑐(1,2)=𝑉sc𝑐1,1̂𝜏3𝐺1,2Γ𝑐2,2;1,(52) where 𝑉sc𝑐(1,2)=𝑉𝑐(1,2)𝜀𝑒1(2,2) is the screened Coulomb interaction. The part which is due to EPI has the following form: Σ𝑒𝑝(1,2)=𝑉𝑒𝑝1,2Γ𝑐1,3;1𝐺3,4Γ𝑐4,2;2,(53) where 𝑉𝑒𝑝(1,2)=𝜀𝑒1(1,1)𝐷(1,2)𝜀𝑒1(2,2) is the screened EPI potential. Note that Σ𝑒𝑝(1,2) depends now quadratically on the charge vertex Γ𝑐, which is due to the adiabatic theorem.

It is well known that the Coulomb self-energy Σ𝑐(1,2) is the most complicating part of the electronic dynamics, but since we are interested in low-energy physics when 𝑠1, then the term Σ𝑐(1,2) can be further simplified by separating it in two parts: Σ𝑐Σ(1,2)=𝑐()Σ(1,2)+𝑐(𝑙)(1,2).(54) The term Σ𝑐()(1,2) is due to high-energy processes contained in the product 𝐺high(1,Γ2)high𝑐(2,2;1) (e.g., due to the large Hubbard 𝑈 in strongly correlated systems) and Σ𝑐(𝑙)(1,2) is due to low-energy processes. The leading part of Σ𝑐()(1,2) is 1, that is, Σ𝑐()(1,2)𝑠0, while Σ𝑐(𝑙)(1,2) is small of order 1, that is, Σ𝑐(𝑙)(1,2)𝑠1. For further purposes we define the quantity 𝑉0 as 𝑉0𝑉(1,2)=𝑒-𝑖(1)+𝑉𝐻𝜏(1)3𝛿Σ(12)+𝑐()(1,2),(55) where 𝑉𝑒-𝑖, 𝑉𝐻 are also of order 𝑠0. After the Fourier transform with respect to time (and for small |𝜔𝑛|𝜔𝑐) Σ𝑐() is given by Σ𝑐()𝐱1,𝐱2,𝜔𝑛Σ()𝑐0𝐱1,𝐱2,0̂𝜏3+Σ()𝑐0𝐱1,𝐱2,0𝑖𝜔𝑛.(56) As we said, Σ()𝑐0𝑠0 while (Σ()𝑐0)𝜔𝑛𝑠1 because 𝜔𝑛𝑠1. From (52) it is seen that the part Σ𝑐(𝑙)(1,2) contains the low-energy Green's function 𝐺low(1,2) and this skeleton diagram is of order 𝑠1. The similar analysis based on (53) for Σ𝑒𝑝(1,2) gives that the leading order is 𝑠1 which describes the low-energy part of EPI. After the separations of terms (of 𝑠0 and 𝑠1 orders) the Dyson equation in the low-energy region has the form 𝑖𝜔𝑛𝑍𝑐𝐱,𝐱𝐻0𝐱,𝐱Σ𝑐(𝑙)𝐱,𝐱,𝜔𝑛Σ𝑒𝑝𝐱,𝐱,𝜔𝑛×𝐺low𝐱,𝐲,𝜔𝑛=𝛿(𝐱𝐲)̂𝜏0,(57) where 𝐱 means integration 𝑑3𝑥 over the crystal volume. The Coulomb renormalization function 𝑍𝑐(𝐱,𝐲)=𝛿(𝐱𝐲)(Σ()0𝑐)(𝐱,𝐲,0) and the single-particle Hamiltonian 𝐻0(𝐱,𝐲) collect formally all high-energy processes which are unaffected by superconductivity (which is low-energy process) where 𝐻01(𝐱,𝐲)=2𝑚2𝐱𝜇𝛿(𝐱𝐲)+𝑉0()(𝐱,𝐲,0)̂𝜏3(58) with 𝑉0()𝑉(𝐱,𝐲,0)=𝑒-𝑖(𝐱)+𝑉𝐻(𝐱)𝛿(𝐱𝐲)+Σ()𝑐0(𝐱,𝐲,0).(59) One can further absorb 𝑍𝑐(𝐱,𝐲) into the renormalized Green's function 𝐺𝑟𝑥,𝐲,𝜔𝑛=𝑍𝑐1/2𝐱,𝐱𝐺low𝑥,𝐲,𝜔𝑛𝑍𝑐1/2,𝐲,𝐲(60) the renormalized vertex function Γren(1,2;3)=𝑍𝑐1/2Γ𝑐𝑍𝑐1/2, and the renormalized self-energies Σ(𝑙)𝑟;𝑐,𝑒𝑝𝐱,𝐲,𝜔𝑛=𝑍𝑐1/2𝐱,𝐱Σ(𝑙)𝑐,𝑒𝑝𝑥,𝐲,𝜔𝑛𝑍𝑐1/2𝐲,𝐲(61) and introduce the ideal band-structure Hamiltonian 0(𝑥,𝐲)=𝑍𝑐1/2(𝐱,𝐻𝐱)0(𝑥,𝐲)𝑍𝑐1/2(𝐲,𝐲) given by 01(𝐱,𝐲)=2𝑚2𝐱𝜇𝛿(𝐱𝐲)+𝑉IBS(𝐱,𝐲)̂𝜏3.(62) Here, 𝑉IBS(𝐱,𝐲)=𝑍𝑐1/2𝐱,𝐱𝑉0()𝑥,𝐲𝑍𝑐1/2𝐲,𝐲(63) is the ideal band-structure potential (sometimes called the excitation potential) and apparently nonlocal quantity, which is contrary to the standard local potential 𝑉𝑔(𝐱) in the LDA-DFT theories—see Section 2.2. The static potential 𝑉IBS(𝐱,𝐲) is of order 𝑠0 and includes high-energy processes.

Finally, we obtain the matrix Dyson equation for the renormalized Green's function 𝐺𝑟(𝑥,𝐲,𝜔𝑛) which is the basis for the (strong-coupling) Migdal-Eliashberg theory in the low-energy region 𝑖𝜔𝑛𝛿𝐱𝐱0𝐱,𝐱Σ(𝑙)𝑐,𝑟𝐱,𝐱,𝜔𝑛Σ𝑒𝑝,𝑟𝐱,𝐱,𝜔𝑛×𝐺𝑟𝐱,𝐲,𝜔𝑛=𝛿(𝐱𝐲)̂𝜏0,(64) where Σ(𝑙)𝑐,𝑟 and Σ𝑒𝑝,𝑟 have the same form as (52)-(53) but with the renormalized Green's and vertex functions 𝐺𝑟,Γ𝑟 instead of Γ𝐺,. 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 𝛼2𝐤𝐩𝐹(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 0(𝐱,𝐲)—given by (62) which contains many-body (excitation) ideal band-structure nonlocal periodic crystal potential 𝑉IBS(𝐱,𝐲). The Hamiltonian 0(𝐱,𝐲) determines the ideal energy spectrum 𝜖(𝐤) of the conduction electrons and the wave function 𝜓𝑖,𝐩(𝐱) through 0𝐱,𝐲𝜓𝑖,𝐤𝐲=𝜖𝑖𝜓(𝐤)𝜇𝑖,𝐤(𝐱),(65) where 𝜇 is the chemical potential. We stress that the Hamiltonian 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 𝑔𝑒𝑝,𝑟=𝑔𝑒𝑝,0Γren𝜀𝑒1) in the basis of 𝜓𝑖,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 𝑉IBS(𝐱,𝐲) (enters the ideal band-structure Hamiltonian 0(𝑥,𝐲)) and the renormalized vertex function (entering the EPI coupling constant 𝑔𝑒𝑝,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 many-body excitation potential 𝑉IBS(𝐱,𝐲) is replaced by some (usually local) potential 𝑉LDA(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.

2.2. 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 𝜆𝑍𝑒𝑝=𝜆𝑠+𝜆𝑑+ is rather large, that is, 1<𝜆𝑍𝑒𝑝<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, 𝜆𝑑𝜆𝑠. 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 𝜆𝑑𝜆𝑠. 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 𝜆𝑍𝑒𝑝. It turns out that at present it is an illusory task to calculate 𝜆𝑍𝑒𝑝 and 𝜆𝑑 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 (𝜔𝑐𝜔𝑎𝑏).

The main task of the LDA-DFT theory in obtaining the EPI matrix elements is to calculate the change of the ground-state (self-consistent) potential 𝛿𝑉𝑔(𝐫)/𝛿𝑅𝛼 and the EPI coupling constant (matrix element) 𝑔LDA𝛼(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 𝑔LDA𝛼(𝐤,𝐤) is inserted into the many-body Eliashberg equations. By knowing 𝑔LDA𝛼(𝐤,𝐤) one can define the total (𝜆) and partial (𝜆𝐪,𝜈) EPI coupling constants for the 𝜈th mode, respectively [195], as 1𝜆=𝑁𝑝𝐪,𝜈𝜆𝐪,𝜈,𝜆𝐪,𝜈=𝑝𝛾𝐪,𝜈𝜋𝑁(0)𝜔𝐪,𝜈,(66) 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 𝛾𝐪,𝜈 is defined in the Migdal-Eliashberg theory by 𝛾𝐪,𝜈=2𝜋𝜔𝐪,𝜈1𝑁𝑙𝑙𝐤12𝑀𝜔𝐤𝐪,𝜈||𝑒𝛼𝜈(𝐪)𝑔𝛼,𝑙𝑙||(𝐤,𝐤𝐪)2×𝑛𝐹𝜉𝑙,𝐤𝑛𝐹𝜉𝑙,𝐤+𝜔𝐪,𝜈𝜔𝐪,𝜈𝜉×𝛿𝑙,𝐤𝐪𝜉𝑙,𝐤𝜔𝐪,𝜈.(67) Here, 𝑒𝛼𝜈(𝐪) is the phonon polarization vectors; 𝑛𝐹 is the Fermi function. Since the ideal energy spectrum from (65) 𝜉𝑙,𝐤=𝐸𝑙,𝐤𝜇 and the corresponding eigenfunctions 𝜓𝐤𝑙 are unknown, then instead of these one sets in (67) the LDA-DFT eigenvalues for the 𝑙th band 𝜉(LDA)𝑙,𝐤 and 𝜓(LDA)𝐤𝑙. In the LDA-DFT method the EPI matrix element 𝑔(LDA)𝛼,𝑙𝑙 is defined by the change of the ground-state potential 𝛿𝑉𝑔(𝐫)/𝛿𝑅𝛼: 𝑔(LDA)𝛼,𝑙𝑙𝐤,𝐤=𝜓(LDA)𝐤𝑙|||||𝑛𝛿𝑉𝑔(𝐫)𝛿𝑅𝑛𝛼|||||𝜓(LDA)𝐤𝑙.(68) 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 𝛿𝑉𝑔(𝐫)/𝛿𝑅𝛼 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 𝑉𝑔(𝐫) can be considered as a sum of ionic potentials 𝑉𝑔(𝐫)=𝑛𝑉𝑔(𝐫𝐑𝑛) where the ion potential 𝑉𝑔(𝐫𝐑𝑛) and the electron density 𝜌𝑒(𝐫) are carried rigidly with the ion at 𝐑𝑛 during the ion displacement (𝐑𝑛=𝐑0𝑛+̂𝑢𝛼𝑛). In the RI approximation the change of 𝑉𝑔(𝐫) is given by 𝛿𝑉𝑔(𝐫)=𝑛0𝑥0200𝑑𝛼𝑉𝑔𝐫𝐑0𝑛𝑢𝛼𝑛,𝛿𝑉𝑔(𝐫)𝛿𝑅𝑛𝛼=𝛼𝑉𝑔𝐫𝐑0𝑛,(69) 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 rigid-ion model is even further simplified by using the rigid muffin-tin approximation (RMTA) (or similar version with the rigid-atomic sphere)—see discussions in [195198]. The RMTA assumes that the ground-state potential and the electron density follow ion displacements rigidly inside the Wigner-Seitz cell while outside it 𝑉𝑔(𝐫) is not changed because of the assumed very good metallic screening (e.g., in simple metals): 𝛼𝑉𝑔𝐫𝐑𝑛=𝛼𝑉𝑔𝐫𝐑𝑛,𝐫incell𝑛,0,outside.(70) 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 𝑔LDA𝛼,𝑛(𝐤,𝐤) is calculated by the wave function centered at the given ion 𝐑0𝑛 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, 𝐫𝜓(RMTA)𝐤=𝑙𝑚𝑛𝐶𝑙𝑚𝑘,𝐑0𝑛𝜌𝑙||𝐫𝐑0𝑛||𝑌𝑙𝑚(𝜙,𝜃)(71) (the angles 𝜙,𝜃 are related to the vector ̂𝑟=(𝐫𝐑0𝑛)/|𝐫𝐑0𝑛|). The radial function 𝜌𝑙(|𝐫𝐑0𝑛|) is zero outside the muffin-tin sphere. In that case the EPI matrix element is given by 𝑔RMTA𝛼,𝑛(𝐤,𝐤)𝑌𝑙𝑚|̂𝑟|𝑌𝑙𝑚 and because ̂𝑟 is vector the selection rule implies that only terms with Δ𝑙𝑙𝑙=±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 Madelung-like interaction, are important in HTSC cuprates, then additional terms contribute also to the coupling constant 𝑔𝛼,𝑛, that is, 𝑔𝛼,𝑛(𝐤,𝐤)=𝑔RMTA𝛼,𝑛(𝐤,𝐤)+𝑔nonloc𝛼,𝑛(𝐤,𝐤), where a part (𝛿𝑔nonloc𝛼,𝑛) of the nonlocal contribution to 𝑔nonloc𝛼,𝑛 is represented schematically:𝛿𝑔nonloc𝛼,𝑛𝐤,𝐤𝑌𝑙𝑚||𝐑0𝑛𝐑0𝑚𝛼||𝑌𝑙𝑚.(72) From (72) comes out the selection rule Δ𝑙=𝑙𝑙=0 for the nonlocal part of the 𝐸𝑃 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 long-range 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 𝜆RMTA0.1 in 𝑌𝐵𝐶𝑂, which is in apparent contradiction with the experimental finding that 𝜆𝑒𝑝 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 𝜆𝑒𝑝 in La2𝑥M𝑥CuO4. 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 𝜆𝜈 (for the considered phonon 𝜈th mode) is found. More precisely speaking, in this approach the matrix elements of 𝛿𝑉𝑔(𝐫)/𝛿𝑅𝜅0,𝛼 are determined from the finite difference of the ground-state potential Δ𝑉𝑔,𝐪,𝜈(𝐫)=𝑉𝑔(𝐑𝜅0,𝐿+Δ𝝉𝜅𝐪,𝜈(𝐿))𝑉𝑔(𝐑𝜅0,𝐿)=𝐿,𝜅Δ𝝉𝜅𝐪,𝜈(𝐿)𝜕𝑉𝑔(𝐑𝜅0,𝐿)/𝜕𝐑𝜅0,𝐿, where 𝐿,𝜅 enumerate elementary lattice cells and atoms in the unit cell, respectively. The frozen-in atomic displacements of the phonon Δ𝝉𝜅𝐪,𝜈(𝐿) of the 𝜈th mode are given by Δ𝝉𝜅𝐪,𝜈(𝐿)=Δ𝑢𝐪,𝜈(/2𝑀𝜅𝜔𝐪,𝜈)1/2Re[𝐞𝜅,𝜈(𝐪)𝑒𝑖𝐪𝐑] where Δ𝑢𝐪,𝜈 is the dimensionless phonon amplitude and the phonon polarization (eigen)vector 𝐞𝜅,𝜈(𝐪) fulfills the condition 𝜅𝐞𝜅,𝜈(𝐪)𝐞𝜅,𝜈(𝐪)=𝛿𝜈,𝜈. Based on this approach various symmetric 𝐴𝑔 (and some 𝐵3𝑔) modes of La2𝑥M𝑥CuO4 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 𝜆𝑒𝑝1.37 is the consequence of the following three main 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 (𝑛1021cm3) 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 𝜆𝑒𝑝. Note that the calculated density of states on the (heavy) Cu and (light) O in-plane atoms, 𝑁Cu(0)0.54states/eV and 𝑁O𝑥𝑦(0)0.35states/eV, is of same order of magnitude as in some LTS materials. For instance, in NbC where 𝑇𝑐11K one has on (the heavy) Nb atom 𝑁Nb(0)0.58states/eV and on (the light) C atom 𝑁C(0)0.25states/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 CuO2 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 O𝑥𝑦 atoms which comes out from the band-structure calculations [200], that is, Δ=|𝜖Cu𝜖O𝑥𝑦|3eV. The latter value gives rise to strong covalent mixing (the hybridization parameter 𝑡𝑝𝑑) of the Cu𝑑𝑥22𝑦 and O𝑝𝑥,𝑦 states, that is, 𝑡𝑝𝑑=1.85eV. 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 Cu𝑑𝑥22𝑦 states across the transition-metal series. So, the natural closeness of the atomic energy levels of the Cu𝑑𝑥22𝑦 and O𝑝𝑥,𝑝𝑦 states is this distinctive feature of HTSC cuprates which basically allows achievement of high 𝑇𝑐. (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 (long-range) Madelung-like interaction to EPI. It turns out that because of the ionicity of the structure the La and O𝑧  axial modes are strongly coupled with charge carriers in the CuO2 planes despite the fact that the local density of states on these atoms is very small [195], that is, 𝑁La(0)=0.13states/eV and 𝑁O𝑧(0)=0.016states/eV. (For comparison, on planar atoms Cu and O𝑥𝑦 one has 𝑁Cu(0)=0.54states/eV and 𝑁O𝑥𝑦(0)=0.35states/eV.) These calculations show that the lanthanum mode (with 𝜔𝐪,𝜈=202cm1) at the 𝑞=(0,0.2𝜋/𝑐) zone boundary (fully symmetric 𝑍-point) has ten times larger coupling constant 𝜆La𝐪,𝜈(𝐹𝐼𝑃)=4.8 than it is predicted in the RMT approximation 𝜆La𝐪,𝜈(RMT)=0.48. The similar increase holds for the average coupling constant, where 𝜆La𝜈,average(FIP)=1.0 but 𝜆La𝜈,average(RMT)=0.1. Note that for the 𝐪0 La-mode one obtains 𝜆La𝜈(FIP)=4.54 compared to 𝜆La𝜈(RMT)=0.12. Similar results hold for the axial apex-oxygen 𝑞=(0,0.2𝜋/𝑐) mode (O𝑧) with 𝜔𝐪,𝜈=396cm1 where the large (compared to the RMT method) coupling constant is obtained: 𝜆O𝑧𝐪,𝜈=14 and 𝜆O𝑧𝜈,average=4.7, while for 𝑞0 axial apex-oxygen modes with 𝜔𝐪,𝜈=415cm1 one has 𝜆O𝑧𝜈,average=11.7. After averaging over all calculated modes it was estimated that 𝜆=1.37 and 𝜔log400K. By assuming that 𝜇=0.1 one obtains 𝑇𝑐=49K by using Allen-Dynes formula for 𝑇𝑐0.83𝜔logexp{1.04(1+𝜆)/[𝜆𝜇(1+0.62𝜆)]} with 𝜔log=2𝑑𝜔𝑑𝜔𝛼2(𝜔)𝐹(𝜔)ln𝜔/𝜆𝜔. For 𝜇=0.15 and 0.2 one obtains 𝑇𝑐=41 and 32K, respectively. We stress that the rather large 𝜆𝑒𝑝 (and 𝑇𝑐) 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 𝜆𝑒𝑝 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 𝜆𝑒𝑝 is an extrapolated value. The all-𝐪 calculations of 𝜆𝑒𝑝,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 (Ca1𝑥Sr𝑥)1𝑦CuO2 for 𝑥0.7 and 𝑦0.1 with 𝑇𝑐=110K [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 CaCuO2 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 𝜆𝐿(𝐪)=𝑀𝐤,𝜈𝑌𝐿||𝑔(𝐤+𝐪)𝐤,𝐪,𝜈||2𝑌𝐿𝜉(𝐤)𝛿𝐤+𝐪𝛿𝜉𝐤.(73) Here, 𝜉𝐤 is the quasiparticle energy, 𝑔𝐤,𝐪,𝜈 is the EPI coupling constant (matrix element) with the 𝜈th branch, 𝑌𝐿(𝐤) is the 𝐿-channel wave function, and the normalization factor 𝑀𝑁𝐿1(0) with the partial density of states is 𝑁𝐿(0)𝐤𝑌2𝐿(𝐤)𝛿(𝜉𝐤). The total coupling constant in the 𝐿-channel is an average of 𝜆𝐿(𝐪) over the whole 2D Brillouin zone (over 𝐪), that is, 𝜆𝐿(𝑞)=𝜆𝐿(𝐪)BZ. We stress three important results of [205]. First, the 𝑠- and 𝑑-coupling constants, 𝜆𝑠(𝐪), 𝜆𝑑(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 |𝑔𝐿,𝐪|2=𝜆𝐿(𝐪)/𝜒𝐿(𝐪) (with 𝜒𝐿(𝐪)𝐤𝑌𝐿(𝐤+𝐪)𝑌𝐿(𝐤)𝛿(𝜉𝐤+𝐪)𝛿(𝜉𝐤)) for 𝐿=𝑠,𝑑 is similar to that of 𝜆𝐿(𝐪), that is, these are peaked at small transfer momenta 𝑞2𝑘𝐹. Both of these results mean that the structural properties of HTSC cuprates imply the dominance of small-q EPI scattering. Third, the calculations give similar values for 𝜆𝑠(𝑞=0) and 𝜆𝑑(𝑞=0), that is, 𝜆𝑠=0.47 for 𝑠-wave and 𝜆𝑑=0.36 for 𝑑-wave pairing [205]. The result that 𝜆𝑑𝜆𝑠 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 [7880, 130, 179, 180] mean that strong correlations and the peculiar structural properties of HTSC cuprates make EPI conform with 𝑑-wave pairing, either as its main cause or as its supporter. We stress that the obtained coupling constant 𝜆𝑑=0.36 is rather small to give 𝑑-wave pairing with large 𝑇𝑐 and on the first glance this result is against the EPI mechanism of pairing in cuprates. However, it is argued throughout this paper that the LDA methods applied to strongly correlated systems overestimate the screening effects and underestimate the coupling constant and therefore their quantitative predictions are not reliable.

2.3. EPI and Strong Correlations in HTSC Uprates
2.3.1. Minimal Model Hamiltonian

The minimal microscopic model for HTSC cuprates must include at least three orbitals: one 𝑑𝑥2𝑦2-orbital of the Cu-ion and two 𝑝-orbitals (𝑝𝑥,𝑦) of the O-ion since they participate in transport properties of these materials—see more in [6] and references therein. The electronic part of the Hamiltonian (of the minimal model) is 𝐻𝐻=0+𝐻int—usually called the Emery model (or the 𝑝-𝑑 model) [206], where the one-particle tight-binding Hamiltonian 𝐻0 describes the lowering of the kinetic energy in the 𝑝-𝑑 model (with three bands or orbitals): 𝐻0=𝑖,𝜎𝜖0𝑑𝑑𝜇𝑖𝜎𝑑𝑖𝜎+𝑗,𝛼,𝜎𝜖0𝑝𝛼𝑝𝜇𝑗𝛼𝜎𝑝𝑗𝛼𝜎+𝑖,𝑗,𝛼,𝜎𝑡𝑝𝑑𝑖𝑗𝛼𝑑𝑖𝜎𝑝𝑗𝛼𝜎+𝑗,𝑗,𝛼,𝛽,𝜎𝑡𝑝𝑝𝑗𝑗,𝛼𝛽𝑝𝑗𝛼𝜎𝑝𝑗𝛽𝜎.(74) Here 𝑡𝑝𝑑𝑖𝑗𝛼 (𝑖,𝑗 enumerate the Cu- and O-sites, resp.) is the hopping integral between the 𝑝𝛼(𝛼=𝑥,𝑦)—and 𝑑-states and 𝑡𝑝𝑝𝑗𝑗𝛼𝛽 between the 𝑝𝛼- and 𝑝𝛽-states—while 𝜖0𝑑 and 𝜖0𝑝𝛼 are the bare 𝑑- and 𝑝-local energy levels and 𝜇 is the chemical potential. This tight-binding Hamiltonian is written in the electronic notation where the charge-transfer energy Δ𝑑𝑝,0𝜖0𝑑𝜖0𝑝>0 by assuming that there is one 3𝑑𝑥2𝑦2 electron on the copper (Cu2+) while electrons in the 𝑝-levels of the O2 ions occupy filled bands. 𝐻0 contains the main ingredients coming from the comparison with the LDA-DFT band-structure calculations. The LDA-DFT results are reproduced by assuming that 𝑡𝑝𝑝𝑡𝑝𝑑 (and 𝜖0𝑝𝛼=𝜖0𝑝 ) where the good fit to the LDA-DFT band structure is found for Δ𝑑𝑝,0𝜖0𝑑𝜖0𝑝3.2eV and 𝑡𝑝𝑑(𝑡𝑝𝑑)=(3/2)(𝑝𝑑𝜎),(𝑝𝑑𝜎)=1.8eV. The total LDA bandwidth 𝑊𝑏=(42)|𝑡𝑝𝑑|9eV [207].

The electron interaction is described by 𝐻int: 𝐻int=𝑈𝑑𝑖𝑛𝑑𝑖𝑛𝑑𝑖+𝑈𝑝𝑗,𝛼𝑛𝑝𝑗𝛼𝑛𝑝𝑗𝛼+𝑉𝑐+𝑉𝑒𝑝,(75) where 𝑈𝑑 and 𝑈𝑝 are the on-site Coulomb repulsion energies at Cu and O sites, respectively, while 𝑉𝑐 and 𝑉𝑒𝑝 describe the long-range part of the Coulomb interaction of electrons (holes) and EPI, respectively. Note that the Hubbard repulsion 𝑈𝑑 on the Cu-ion is different from its bare atomic value 𝑈𝑑0(16 eV for Cu) due to various kinds of screening effects in solids [208210]. It turns out that in most transition metal oxides one has 𝑈𝑑𝑈𝑑0. This problem is thoroughly studied in [208210] and applied to HTSC cuprates. The estimation from the numerical cluster calculations [211] gives 𝑈𝑑=9-11eV and 𝑈𝑝=4-6eV but because 𝑛𝑑𝑖𝑈𝑑𝑛𝑝𝑗𝑈𝑝 the on-site repulsion on the oxygen ion is usually neglected at the first stage of the analysis.

Note that in the case of large 𝑈𝑑(≫𝑡𝑝𝑑,Δ𝑑𝑝,0) the hole notation is usually used where in the parent compound (and for |𝑡𝑝𝑑|Δ𝑑𝑝,0) one has 𝑛𝑑𝑖=1, that is, one hole in the 3D-shell (in the 3𝑑𝑥2𝑦2 state) in the ground state. In the limit of large 𝑈𝑑 the doubly occupancy on the Cu atoms is forbidden and only two copper states are possible: Cu2+—described by the quantum state 𝑑𝑖𝜎|0 with one hole in the 3D shell and Cu1+—described by |0 with zero holes in the filled 3D shell. In this (hole) notation the oxygen 𝑝-level is fully occupied by electrons, that is, there are no holes (𝑛𝑝𝑗=0) in the occupied oxygen 2𝑝-shell of O2. In this notation the vacuum state |0𝑣 (not the ground state) of the Hamiltonian 𝐻 for large 𝑈𝑑 corresponds to the closed-shell configuration Cu1+O2. In the hole notation the hole 𝑝-level 𝜖0ph lies higher than the hole 𝑑-level 𝜖0𝑑, that is, Δ𝑝𝑑,0𝜖0ph𝜖0𝑑>0 (note that in the electron notation it is opposite) and 𝑈𝑑 means repulsion of two holes (in the 3𝑑𝑥2𝑦2 orbital) with opposite spins—3𝑑8 configuration of the Cu3+ ion. Note that 𝜖0ph=𝜖0𝑝,𝜖0𝑑=𝜖0𝑑, and 𝑡𝑝𝑑,=𝑡𝑝𝑑. In the following the index in 𝑡𝑝𝑑, is omitted. The reason for 𝜖0ph>𝜖0𝑑 is partly in different energies for the hole sitting on the oxygen and copper, respectively [207]. From this model one can derive in the limit 𝑈 the 𝑡-𝐽 model for the 2D lattice in the CuO2 plane [212, 213], where now each lattice site corresponds to a Cu-atom. In the presence of one hole in the 3D-shell then in the undoped (no oxygen holes) HTSC cuprate each lattice site is occupied by one hole. By doping the system with holes the additional holes go onto O-sites. Furthermore, due to the strong Cu–O covalent binding the energetics of the system implies that an O-hole forms a Zhang-Rice singlet with a Cu-hole [212]. In the 𝑡-𝐽 model the Zhang-Rice singlet is described by an empty site. Since in the 𝑡-𝐽 model the doubly occupancy is forbidden, one introduces annihilation (Hubbard) operator of the composite fermion 𝑋𝑖𝜎0=𝑐𝑖𝜎(1𝑛𝑖,𝜎) which describes creation of a hole (in the 3D-shell of the Cu-atoms) on the 𝑖th site if this site is previously empty (thus excluding doubly occupancy), that is, the constraint 𝑛𝑖,𝜎+𝑛𝑖,𝜎1 must be fulfilled on each lattice site. In this picture the doped-hole concentration 𝛿 means at the same time the concentration of the oxygen holes, that is, of the Zhang-Rice singlets.

In order not to confuse the reader we stress the difference in the meaning of the hole in the (𝑝-𝑑) three-band Emery model and in the single-band (effective) 𝑡-𝐽 model. In the Emery model the hole means the absence of the electron in the filled shell—the 3D shell for Cu atoms(ions) and 2𝑝 shell for O atoms(ions). On the other side the hole on the 𝑖th lattice site in the 𝑡-𝐽 model means the presence of the Zhang-Rice singlet on this site.

The bosonic-like operators 𝑋𝜎1𝜎2𝑖=𝑋𝜎10𝑖𝑋0𝜎2𝑖 for 𝜎1𝜎2 create a spin fluctuation at the 𝑖th site and the spin operator is given by 𝑋𝐒=𝜎10𝑖(𝜎)𝜎1𝜎2𝑋0𝜎2𝑖 where summation over the bar indices is understood. The operator 𝜎𝑋𝑖𝜎𝜎 has the meaning of the hole number on the 𝑖th site. It is useful to introduce the operator 𝑋𝑖00=𝑋𝑖0𝜎𝑋𝑖𝜎0 at the 𝑖th lattice site which is the number of Zhang-Rice singlets on the 𝑖th site. For 𝑋𝑖00|0=1|0 the 𝑖th site is occupied by the Zhang-Rice singlet, while for 𝑋𝑖00|1=0|1 there is no Zhang-Rice singlet on the 𝑖th site (i.e., this site is occupied only by one 3𝑑9 hole on the Cu site). This property of 𝑋𝑖00 is due to the local constraint 𝑋𝑖00+𝜎=𝑋𝑖𝜎𝜎=1,(76) which forbids doubly occupancy of the 𝑖th site by holes. By projecting out doubly occupied (high-energy) states the 𝑡-𝐽 model reads 𝐻𝑡-𝑗=𝑖,𝜎𝜖0𝑖𝑋𝑖𝜎𝜎𝑖,𝑗,𝜎𝑡𝑖𝑗𝑋𝑖𝜎0𝑋𝑗0𝜎+𝑖,𝑗𝐽𝑖𝑗𝐒𝑖𝐒𝑗14̂𝑛𝑖̂𝑛𝑗+𝐻3.(77) The first term (𝜖0𝑖) describes an effective local energy of the hole (or the Zhang-Rice singlet), the second one (𝑡𝑖𝑗) describes hopping of the holes, and the third one (𝐽𝑖𝑗) is the Heisenberg-like exchange energy between two holes. The theory [212] predicts that |𝜖0𝑖||𝑡𝑖𝑗|. This property is very important in the study of EPI. 𝐻3 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: 𝑋𝑖𝛼𝛽,𝑋𝑗𝛾𝜆±=𝛿𝑖𝑗𝛿𝛾𝛽𝑋𝑖𝛼𝜆±𝛿𝛼𝜆𝑋𝑖𝛾𝛽,(78) where 𝛿𝑖𝑗 is the Kronecker symbol. Note that the Hubbard operators possess the projection properties with 𝑋𝑖𝛼𝛽𝑋𝑖𝛾𝜆=𝛿𝛽𝛾𝑋𝑖𝛼𝜆. 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 𝑋𝑖0𝜎=𝑓𝑖𝜎𝑏𝑖,𝑋𝜎1𝜎2𝑖=𝑓𝑖𝜎1𝑓𝑖𝜎2 are represented in terms of the fermion (spinon) operator 𝑓𝑖𝜎 which annihilates the spin on the 𝑖th and the boson (holon) operator 𝑏𝑖 which creates the Zhang-Rice singlet.

In the minimal theoretical model the electron-phonon interaction (EPI) contains in principle two leading terms: 𝐻𝑒𝑝=𝐻ion𝑒𝑝+𝐻cov𝑒𝑝,(79) which are the “ionic” one (𝐻ion𝑒𝑝) and the “covalent” one (𝐻cov𝑒𝑝). 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, 7880, 130] 𝐻ion𝑒𝑝=𝑖,𝜎Φ𝑖𝑋𝑖𝜎𝜎,(80) where the “displacement” operator Φ𝑖=𝐿𝜅𝜖𝐑0𝑖𝐑0𝐿𝜅+̂𝐮𝑖̂𝐮𝐿𝜅𝐑𝜖0𝑖𝐑0𝐿𝜅(81) (which as in Section 2.1 includes the bare coupling constant) describes the change of the hole (or Zhang-Rice singlet) energy 𝜖0𝑎,𝑖 by displacing atoms in the lattice by the vector ̂𝐮𝐿𝜅. In the harmonic approximation the EPI potential is given by Φ𝑖=𝑔𝑖(𝐪,𝜆)exp{𝑖𝐪𝐑𝑖}[𝑏𝐪,𝜆+𝑏𝐪,𝜆] where 𝑏𝐪,𝜆 and 𝑏𝐪,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 O-hole and Cu-hole bare energies 𝜖0ph,𝜖0𝑑 in the three-band model due to lattice vibrations, (2) the change of the long-range 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 𝑡𝑝𝑑 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 Φ𝑖 is calculated in the harmonic approximation (Φ̂𝑢) 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 Φ𝑖 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 𝑔HFhb=±4𝑡𝑝𝑑𝜕𝑡𝑝𝑑𝜕𝑅Cu-O1𝜖𝑑𝜖𝑝𝑢0,(82) while the coupling constant in the 𝑡-𝐽 model 𝑔𝑡-𝐽hb(=𝜕𝜖0/𝜕𝑅Cu-O) is given by 𝑔𝑡-𝐽hb=±4𝑡𝑝𝑑𝜕𝑡𝑝𝑑𝜕𝑅Cu-O2𝑝21𝜖𝑑𝜖𝑝+2𝑝2𝑈𝑑||𝜖𝑑𝜖𝑝||𝑢0,(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 𝜆hb is proportional to |𝑔hb|2, one obtains that the bare 𝑡-𝐽 coupling constant is almost three times larger than the LDA-DFT one: 𝜆𝑡-𝐽hb3𝜆HFhb.(84) 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: 𝐻cov𝑒𝑝=𝑖,𝑗,𝜎𝜕𝑡𝑖𝑗𝜕𝐑0𝑖𝐑0𝑗̂𝐮𝑖̂𝐮𝑗𝑋𝑖𝜎0𝑋𝑗0𝜎+𝑖,𝑗,𝜕𝐽𝑖𝑗𝜕𝐑0𝑖𝐑0𝑗̂𝐮𝑖̂𝐮𝑗𝐒𝑖𝐒𝑗.(85) Here, we will not go into details but only stress that since |𝜖0𝑖||𝑡𝑖𝑗| 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 𝐻ion𝑒𝑝 will be considered [6, 7880, 130].

2.3.2. 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 X-method—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 [7880, 130]. In the paramagnetic and homogeneous state (with finite doping) the Green's function 𝐺𝜎1𝜎2(12) is diagonal, that is, 𝐺𝜎1𝜎2(12)=𝛿𝜎1𝜎2𝐺(12) where 𝑇𝑋𝐺(12)=0𝜎𝑋(1)𝜎0(2)=𝑔(12)𝑄,(86) with the Hubbard spectral weight 𝑋𝑄=00𝑋+𝜎𝜎. The function 𝑔(12) plays the role of the quasiparticle Green's function—see more in [6, 7880, 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 𝑋𝑖00+𝑁𝜎=1𝑋𝑖𝜎𝜎=𝑁2.(87) In order to reach the convergence of physical quantities in the limit 𝑁 the hopping and exchange energy are also rescaled, that is, 𝑡𝑖𝑗=𝑡0,𝑖𝑗/𝑁 and 𝐽𝑖𝑗=𝐽0,𝑖𝑗/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 [7880, 130, 179, 180] it is developed a systematic 1/𝑁 expansion for the quasiparticle Green's function 𝑔(12)(=𝑔0+𝑔1/𝑁+), 𝑄(=𝑁𝑞0+𝑞1+) (also for 𝐺(12)) and the self-energy. For large 𝑁() the leading term is 𝐺0(12)=𝑔0(12)𝑄0=𝑂(𝑁) with 𝑔0=O(1) and 𝑄0𝑋=𝑖00=𝑁𝛿/2. 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 equation 1𝛿=2𝐩𝑛𝐹(𝐩) with 𝑛𝐹(𝐩)=(𝑒𝜖0(𝐤)𝜇+1)1. The quasiparticle Green's function 𝑔0(𝐤,𝜔) and the quasiparticle spectrum 𝜖0(𝐤) in the leading order are given by 𝑔0(𝐺𝐤,𝜔)0(𝐤,𝜔)𝑄0=1𝜖𝜔0,𝜖(𝐤)𝜇(88)0(𝐤)=𝜖𝑐𝛿𝑡(𝐤)𝐩𝐽0(𝐤+𝐩)𝑛𝐹(𝐩).(89) The level shift is 𝜖𝑐=𝜖0+2𝐩𝑡(𝐩)𝑛𝐹(𝐩) and 𝑡(𝐩) is the Fourier transform of the hopping integral 𝑡𝑖𝑗—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, 7880, 130, 179, 180]. (i) The Green's function 𝑔0(𝐤,𝜔) describes the coherent motion of quasiparticles whose contribution to the total spectral weight of the Green's function 𝐺0(𝐤,𝜔) is 𝑄0=𝑁𝛿/2. The coherent motion of quasiparticles is described in leading order by 𝐺0(𝐤,𝜔)=𝑄0𝑔0(𝐤,𝜔) and the quasiparticle residuum 𝑄0 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 𝜖0(𝐤) plays the same role as the eigenvalues of the ideal band-structure Hamiltonian 0(𝐱,𝐲) (it contains the excitation potential 𝑉IBS(𝐱,𝐲) which is due to high-energy processes of the Coulomb interaction). So, if we would consider 𝜖𝑡𝑏(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 𝑉𝑥𝑐(𝑥)𝛿(𝑥𝑦)), then one can define a nonlocal excitation potential 𝑉𝑡𝐽IBS𝑉(𝐱,𝐲)=𝑡𝐽IBS(𝐱,𝐲)+𝑉𝑥𝑐(𝐱)𝛿(𝐱𝐲) which mimics strong correlations in the 𝑡-𝐽 model 𝑉𝑡𝐽IBS(𝐱,𝐲)𝑉0𝛿(𝐱𝐲)+(1𝛿)𝑡(𝐱𝐲)𝐽(𝐱𝐲).(90) Here, 𝑉0=2𝐩𝑡(𝐩)𝑛𝐹(𝐩) and 𝑡(𝐱𝐲) is the Fourier transform of 𝑡(𝐤) while 𝐽(𝐱𝐲) is the Fourier transform of the third term in (89). The relative excitation potential 𝑉𝑡𝐽IBS(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 𝜖0(𝐤) is dominated by the exchange parameter if 𝐽0>𝛿𝑡0. However, in the case when 𝐽0𝛿𝑡0 there is a band narrowing by lowering the hole-doping 𝛿, where the band width is proportional to the hole-concentration 𝛿, that is, 𝑊𝑏=𝑧𝛿𝑡0. (iv) The O(1)-order quasiparticle Green's function 𝑔0(𝐤,𝜔) and the quasiparticle spectrum 𝜖0(𝐤) in the X-method have similar form as the spinon Green's function 𝑔0,𝑓(𝐤,𝜔)=𝑇𝑓𝜎𝑓𝜎𝐤,𝜔 and the spinon energy 𝜖𝑠(𝐤) in the SB method. However, in the SB method there is a broken gauge symmetry in the metallic state (with 𝛿0) which is characterized by ̂𝑏𝑖0. 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 X-method where Green's function 𝐺0(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 (̂𝑏𝑖0). The assumption of the broken symmetry ̂𝑏𝑖0 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 method—see 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 𝑋- and SB-methods differ substantially. As a consequence, the SB method [217] predicts false superconductivity in the 𝑡-𝐽 model (for 𝐽=0) with large 𝑇𝑐 (due to the kinematical interaction), while the 𝑋-method gives extremely small 𝑇𝑐(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.

2.3.3. 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 Σ𝑒𝑝 is given by the expression [6, 7880, 130] Σ𝑒𝑝(1,2)=𝑉𝑒𝑝12𝛾𝑐1,3;1𝑔034𝛾𝑐4,2;2,(91) where the screened (by the dielectric constant) EPI potential 𝑉𝑒𝑝(12)=𝜀𝑒111𝑉0𝑒𝑝12𝜀𝑒122(92) and 𝑉0𝑒𝑝(12)=𝑇Φ(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 𝛾𝑐(1,2;3)=𝛿𝑔01(1,2)/𝛿𝑢e(3) corresponds to the the renormalized vertex Γ𝑐,𝑟 in (53) and it describes the screening by strong correlations. It depends on the relative excitation potential 𝑉𝑡𝐽IBS(𝐱,𝐲). The electronic dielectric function 𝜀𝑒(12) 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 𝑔0𝑒𝑝 and lattice displacement ̂𝑢, that is, Φ𝑔0𝑒𝑝̂𝑢—see more in [6]. (Note that in the above equations summation and integration over bar indices are understood.) The self-energy Σ𝑒𝑝(𝐤,𝜔) due to EPI reads Σ𝑒𝑝(𝐤,𝜔)=0𝛼𝑑𝜈2𝐹𝐤,𝐤,𝜈𝐤𝑅(𝜔,𝜈),(93) with 𝑅(𝜔,𝜈)=2𝜋𝑖(𝑛B(𝜈)+1/2)+𝜓(1/2+𝑖)𝜓(1/2𝑖(𝜈+𝜔)/2𝜋𝑇) where 𝑛B(𝜈) is the Bose distribution function and 𝜓 is di-gamma function. The Eliashberg spectral function is given by 𝛼2𝐹𝐤,𝐤,𝜔=𝑁(0)𝜈||𝑔𝜈𝐤,𝐤𝐤||2×𝛿𝜔𝜔𝜈𝐤𝐤𝛾2𝑐𝐤,𝐤𝐤,(94) where 𝑔𝜈(𝐤,𝐩) is the EPI coupling constant for the 𝜈th mode, where the renormalization by long-range Coulomb interaction is included, that is, 𝑔𝜈(𝐤,𝐩)=𝑔0𝑒𝑝,𝜈(𝐤,𝐩)/𝜀𝑒(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 𝛾𝑐(𝐤,𝐤𝐤) which, as we will see in Section 2.3.4, changes the properties of 𝑉𝑒𝑝(𝐪,𝜈) drastically compared to weakly correlated systems. In fact the charge vertex depends on frequency 𝜔 but in the adiabatic limit (𝜔ph𝑊) and for 𝑞𝑣𝐹>𝜔ph it is practically frequency independent, that is, 𝛾𝑐(𝑎𝑑)(𝐤,𝐪,𝜔)𝛾𝑐(𝐤,𝐪,𝜔=0) where the latter is real quantity. For 𝐽=0 in the 𝑡-𝑡 model the 1/𝑁 expansion gives 𝑁(0)=𝑁0(0)/𝑞0 where 𝑞0=𝛿/2. For 𝐽0 the density of states 𝑁(0) does not diverge for 𝛿0 where 𝑁(0)(1/𝐽0)>𝑁0(0). The bare density of states 𝑁0(0) is calculated in absence of strong correlations, for instance, by the LDA-DFT method.

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 𝛼2𝐹(𝐤,𝐤,𝜔) enter the Eliashberg equations. Assuming that the superconducting order parameter transforms according to the representation Γ𝑖 of the point group 𝐶4𝑣 of the square lattice (in the CuO2 planes) the appropriate symmetry-projected spectral function is given by 𝛼2𝐹𝑖̃̃𝐤𝐤,=𝑁,𝜔(0)8𝜈,𝑗|||𝑔𝜈̃̃𝐤,𝐤𝑇𝑗̃𝐤|||2×𝛿𝜔𝜔𝜈̃𝐤𝑇𝑗̃𝐤×𝛾2𝑐̃̃𝐤,𝐤𝑇𝑗̃𝐤𝐷𝑖(𝑗)(95) where ̃𝐤 and ̃𝐤 are momenta on the Fermi line in the irreducible Brillouin zone (1/8 of the total Brillouin zone). 𝑇𝑗,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 𝑠- and 𝑑-wave symmetry of the full rotation group, respectively. 𝐷𝑖(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 𝛼2𝐹𝑖(𝜔): 𝛼2𝐹𝑖𝛼(𝜔)=2𝐹𝑖̃̃𝐤𝐤,,𝜔̃𝐤̃𝐤,(96) which governs the transition temperature for the order parameter with the symmetry Γ𝑖. For instance, 𝛼2𝐹3(𝜔) is the pairing spectral function in the 𝑑-channel and it gives the coupling for 𝑑-wave superconductivity (the irreducible representation Γ3—sometimes labelled as 𝐵1𝑔). Performing similar calculations for the phonon-limited resistivity, one finds that the resistivity is related to the transport spectral function 𝛼2𝐹tr(𝜔): 𝛼2tr𝛼𝐹(𝜔)=2𝐹𝐤,𝐤𝐤,𝜔𝐯(𝐤)𝐯2𝐤𝐤2𝐯2(𝐤)𝐤𝐤.(97) The effect of strong correlations on EPI was discussed in [130] within the model where 𝑔𝜈(𝐤,𝐩) and the phonon frequencies 𝜔𝜈(̃̃𝐤𝐤) are weakly momentum dependent. In order to elucidate the main effect of strong correlations on EPI and 𝛼2𝐹𝑖(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 Λ𝑖: Λ𝑖=18𝑁(0)𝑁0(0)8𝑗=1|||𝛾𝑐̃̃𝐤,𝐤𝑇𝑗̃𝐤|||2̃𝐤̃𝐤𝐷𝑖(𝑗).(98) Similarly, the resistivity 𝜌(𝑇)(𝜆trΛtr) is renormalized by the correlation effects where the transport coupling constant Λtr is given by Λtr=𝑁(0)𝑁0|||𝛾(0)𝑐̃̃𝐤,𝐤𝑇𝑗̃𝐤|||2𝐤𝐯(𝐤)𝐯2𝐤𝐤2𝐯2(𝐤)𝐤𝐤.(99)

As we see, all projected spectral functions 𝛼2𝑖𝐹(𝜔) depend on the charge vertex function 𝛾𝑐(𝐤,𝐪) which describes the screening (renormalization) of EPI due to strong correlations (suppression of doubly occupancy) [7880, 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].

2.3.4. Charge Vertex and the EPI Coupling

The charge vertex function 𝛾𝑐(𝐤,𝐪) (in the adiabatic approximation) has been calculated in [7880, 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 𝛾𝑐(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 𝛾𝑐(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 [7880, 130, 179, 180] 𝛾𝑐(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: 𝛾𝑐(𝐤,𝑞)=16𝛼=10𝑥0200𝑑6𝛽=10𝑥0200𝑑𝐹𝛼̂(𝐤)1+𝜒(𝑞)1𝛼𝛽𝜒𝛽2(𝑞),(100) where 𝜒𝛼𝛽(𝑞)=𝑝𝐺𝛼(𝑝,𝑞)𝐹𝛽(𝐩),𝐹𝛼(𝐤)=[𝑡(𝐤),1, 2𝐽0cos𝑘𝑥,2𝐽0sin𝑘𝑥,2𝐽0cos𝑘𝑦,2𝐽0sin𝑘𝑦], and 𝐺𝛼(𝑝,𝑞) = [1,𝑡(𝐩+𝐪),cos𝑝𝑥,sin𝑝𝑥,cos𝑝𝑦,sin𝑝𝑦]Π(𝑝,𝑞). Here, Π(𝑘,𝑞) = 𝑔(𝑘)𝑔(𝑘+𝑞) and 𝑞=(𝐪,𝑖𝑞𝑛),𝑞𝑛=2𝜋𝑛𝑇,𝑝=(𝐩,𝑖𝑝𝑚),𝑝𝑚=𝜋𝑇(2𝑚+1). The physical meaning of the vertex function 𝛾𝑐(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 𝑉𝑡𝐽IBS(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 𝛾𝑐(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 𝛾𝑐(𝐤𝐹,𝐪,𝜔=0) is shown. The peak at 𝑞=0 is very narrow at very small doping since its width 𝑞𝑐 is proportional to the doping 𝛿, that is, 𝑞𝑐𝛿(𝜋/𝑎) where 𝑎 is the lattice constant. It is interesting that 𝛾𝑐(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 [7880, 130, 179, 180].

The existence of the forward scattering peak in 𝛾𝑐(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 𝛾𝑐(𝐤,𝑞) at large 𝑞(𝑘𝐹) 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 𝜉ch(𝜋/𝑞𝑐)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 𝛾𝑐(𝐤,𝑞) one can calculate the relative coupling constants Λ1Λ𝑠, Λ3Λ𝑑, Λtr, and so forth. In the absence of correlations and for an isotropic band one has Λ1=Λtr=1, Λ𝑖=0 for 𝑖>1. The averages in Λ𝑠, Λ𝑑, and Λtr 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 Λ𝑠 approaches 1 in the empty-band limit 𝛿1, when strong correlations are absent. Note that the superconducting critical temperature 𝑇𝑐 in the weak coupling limit and in the 𝑖th channel scales like 𝑇𝑐(𝑖)exp(1/(𝜆0Λ𝑖𝜇𝑖) where 𝜆0 is some effective coupling constant which depends on microscopic details. The parameter 𝜇𝑖 is the effective residual Coulomb repulsion in the 𝑖th superconducting channel. We 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 𝑔0𝜆(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 𝐻ion𝑒𝑝 (see (80)) and corresponding EPI depends only on the momentum transfer 𝐪, that is, 𝑔0𝜆(𝐤,𝐪)𝑔0𝜆(𝐪) while for the much smaller covalent coupling 𝐻cov𝑒𝑝 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=|𝐤𝐤| 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 |𝑔0(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 𝜔br which is supposed to be important in HTSC cuprates. The bare coupling constant (squared) for this mode is approximately given by |𝑔0br(𝐤,𝐪)|2=|𝑔0br|2[sin2(𝑞𝑥𝑎/2)+sin2(𝑞𝑦𝑎/2)] which reaches maximum for large 𝐪=(𝜋/𝑎,𝜋/𝑎). By extracting the component in the 𝑑-channel one has ||𝑔0br𝐤𝐤||2=||𝑔0br||2114𝜓𝑑(𝐤)𝜓𝑑𝐤+(101) with 𝜓𝑑(𝐤)=cos𝑘𝑥𝑎cos𝑘𝑦𝑎.(102) This gives the repulsive coupling constant 𝜆0𝑑 in the 𝑑-channel, that is, 𝜆0𝑑=2𝜔br𝜓𝑑||𝑔(𝐤)0br𝐤𝐤||2𝜓𝑑𝐤<0.(103) However, in the presence of strong correlations one expects that the effective coupling constant is given approximately by |𝑔ebr(𝐤,𝐤𝐤)||𝑔0br(𝐤𝐤)|2𝛾2𝑐(𝐤𝐹,𝐤𝐤) which is at small doping 𝛿 suppressed substantially at large 𝑞 since 𝛾2𝑐 starts to fall off drastically at 𝑞𝑞𝑐𝛿(𝜋/𝑎). The latter property makes the effective coupling constant (in the 𝑑-channel) 𝜆e𝑑 for these modes less negative or even positive (depending on the ratio 𝜉ch/𝑎1/𝛿), that is, one has 𝜆e𝑑>𝜆0𝑑. We stress again that this analysis is only qualitative (and semiquantitative) since it is based on the 𝑡-𝐽 model while the better quantitative results are expected in the strongly correlated three-band Emery model with 𝑈𝑑𝑡,Δ𝑝𝑑—see [6, Appendix  D]. Unfortunately, these calculations are not finalized until now.

(2) In weakly correlated systems (or, e.g., in the empty-band limit 𝛿1) the relative 𝑑-wave coupling constant Λ𝑑 is much smaller than the 𝑠-wave coupling constant Λ𝑠, that is, Λ𝑑Λ𝑠 as it is seen in Figure 36. Furthermore, Λ𝑠 decreases with decreasing doping.

(3) It is indicative that independently on the value of 𝑡0 or 𝑡=0 the coupling constant Λ𝑠 and Λ𝑑  meet each other (note that Λ𝑠>Λ𝑑 for all 𝛿) at some small doping 𝛿0.1-0.2 where Λ𝑠Λ𝑑. We would like to stress that such a unique situation (with Λ𝑠Λ𝑑) was practically never realized in low-temperature and weakly correlated superconductors and in that respect the strong momentum-dependent EPI in HTSC cuprates is an exclusive but very important phenomenon.

(4) By taking into account the residual Coulomb repulsion of quasiparticles then the 𝑠-wave superconductivity (which is governed by Λ𝑠) is suppressed, while the 𝑑-wave superconductivity (which is governed by Λ𝑑) stays almost unaffected, since 𝜇𝑠𝜇𝑑. In that case the 𝑑-wave superconductivity which is mainly governed by EPI becomes more stable than the 𝑠-wave one at sufficiently low doping 𝛿. This transition between 𝑠- and 𝑑-wave superconductivity is triggered by electronic correlations because in the model calculations [7880, 130] the bare EPI coupling is assumed to be momentum independent, that is, the bare coupling constant contains the 𝑠-wave symmetry only.

(5) The calculations of the charge vertex 𝛾𝑐 are performed in the adiabatic limit, that is, for 𝜔<𝐪𝐯𝐹(𝐪) the frequency 𝜔 in 𝛾𝑐 can be neglected. In the nonadiabatic regime, that is, for 𝜔>𝐪𝐯𝐹(𝐪), the function 𝛾2𝑐(𝐤𝐹,𝐪,𝜔) may be substantially larger compared to the adiabatic case because 𝛾𝑐(𝐤𝐹,𝐪,𝜔) tends to the bare value 1 for 𝑞=0. This means that EPI for different phonons (with different energies 𝜔) is differently affected by strong correlations. For a given 𝜔 the EPI coupling to those phonons with momenta 𝑞<𝑞𝜔=𝜔/𝑣𝐹 will be (relatively) enhanced since 𝛾𝑐(𝐤𝐹,𝐪,𝜔)1, while the coupling to those with 𝑞>𝑞𝜔=𝜔/𝑣𝐹 will be substantially reduced due to the suppression of the backward scattering by strong correlations [37, 38]. These results are a consequence of the Ward identities and generally hold in the Landau-Fermi liquid theory [219].

(6) The transport EPI coupling constant Λtr is significantly reduced in the presence of strong correlations especially for low doping (𝛿1) where Λtr<Λ/3. This result is physically plausible since the resistivity is dominated by the backward scattering processes (large 𝑞𝑘𝐹) which are suppressed by strong correlations—the suppression of 𝛾𝑐(𝐤𝐹,𝐪,𝜔) at large 𝑞.

The theory based on the forward scattering peak in EPI is a good candidate to explain the linear temperature behavior of the resistivity down to very small temperatures 𝑇(Θ𝐷/30)10K in some cuprates with low 𝑇𝑐(10 K) [6, 128, 129]. One physically rather plausible model, which is based on the forward scattering peak in EPI, is elaborated in [128]. It takes into account (i) the quasiparticle scattering on acoustic (a) and on optic (o) phonons, (ii) the extended van Hove singularity in the quasiparticle density of states 𝑁(𝜉) which in some cuprates is very near the Fermi surface, and (iii) the umklapp and “undulation” (due to the flat regions at the Fermi surface) processes with 𝐯𝐤𝐯𝐤—this condition can partly increase the EPI coupling. The transport Eliashberg function 𝛼2tr𝐹(𝜔) is calculated similarly to (97) by using the definition of 𝛼2𝐹(𝐤,𝐤,𝜔) in (95) with the renormalized coupling constant 𝑔𝜈(𝑟)(𝐤𝐤)=𝑔𝜈(𝐤𝐤)𝛾𝑐(𝐤𝐤) of the 𝜈=𝑎,𝑜 mode, respectively. In [128] it is assumed a phenomenological form for the forward scattering peak in 𝛾𝑐(𝐤𝐤) with the cutoff 𝑞𝑐𝑘𝐹 (and which mimics the exact results from [7880, 130, 179, 180]). Since the scattering of the quasiparticles on phonons (with the sound velocity 𝑣𝑠) is limited to small-𝑞 transfer processes (with 𝑞<𝑞𝑐), then the maximal energy of the acoustic branch is not the Debye energy Θ𝐷(𝑘𝐹𝑣𝑠) but the effective Debye energy Θ𝐴(𝑞𝑐𝑣𝑠)Θ𝐷. In the case of Bi2201 in [128] it is taken (from the numerical results in [7880, 130, 179, 180]) that 𝑞𝑐𝑘𝐹/10 which gives Θ𝐴(30-50)𝐾. As a result the calculated 𝛼2tr𝐹(𝜔) gives that 𝜌𝑎𝑏(𝑇)𝑇 down to very low 𝑇(0.2Θ𝐴)10K. The slope (𝑑𝜌𝑎𝑏/𝑑𝑇) is governed by the effective EPI coupling constant for acoustic phonons. In systems with the extended van Hove singularity (in 𝑁(𝜉)) near the Fermi surface, which is the case in Bi-2201, the effective coupling constant for acoustic phonons can be sufficiently large to give experimental values for the slope (𝑑𝜌𝑎𝑏/𝑑𝑇)(0.5-1)𝜇Ωcm/K—for details see [128]. This physical picture is applicable also to cuprates near and at the optimal doping but since in these systems 𝑇𝑐 is large the linearity of 𝜌𝑎𝑏(𝑇) down to very low 𝑇 is “screened” by the appearance of superconductivity.

(7) The width of the forward scattering peak in 𝛾𝑐(𝐤𝐹,𝐪) is very narrow in underdoped cuprates—with the width 𝑞𝑐𝛿(𝜋/𝑎)—which may have further interesting consequences. For instance, HTSC cuprates are characterized not only by strong correlations but also by the relatively small Fermi energy 𝐸𝐹, which is in underdoped systems not much larger than the characteristic (maximal) phonon frequency 𝜔maxph, that is, 𝐸𝐹0.1-0.3eV, 𝜔maxph80meV. Due to the appreciable magnitude of 𝜔𝐷/𝐸𝐹 it is necessary to correct the Migdal-Eliashberg theory by the non-Migdal vertex corrections due to the EPI. It is well known that these vertex corrections lower 𝑇𝑐 in systems with the isotropic EPI. However, the non-Migdal vertex corrections in systems with the forward scattering peak in the EPI coupling with the cutoff 𝑞𝑐𝑘𝐹 may increase 𝑇𝑐 which can be appreciable. The corresponding calculations [220, 221] give two interesting results: (i) there is an appreciable increase of 𝑇𝑐 by lowering 𝑄𝑐=𝑞𝑐/2𝑘𝐹, for instance, 𝑇𝑐(𝑄𝑐=0.1)4𝑇𝑐(𝑄𝑐=1); (ii) even small values of 𝜆𝑒𝑝<1 can give large 𝑇𝑐. The latter results open a new possibility in reaching high 𝑇𝑐 in systems with appreciable ratio 𝜔𝐷/𝐸𝐹 and with the forward scattering peak in EPI. The difference between the Migdal-Eliashberg and the non-Migdal theory can be explained qualitatively in the framework of an approximative McMillan formula for 𝑇𝑐 (for not too large 𝜆) which reads 𝑇𝑐𝜔𝑒1/(̃𝜆𝜇). The Migdal-Eliashberg theory predicts ̃𝜆(ME)𝜆,1+𝜆(104) while the non-Migdal theory [220, 221] gives ̃𝜆(n-ME)𝜆(1+𝜆).(105) For instance, 𝑇𝑐100K in HTSC oxides can be explained by the Migdal-Eliashberg theory for 𝜆(ME)2, while in the non-Migdal theory much smaller coupling constant is needed, that is, 𝜆(n-ME)0.5.

(8) The existence of the forward scattering peak in EPI can in a plausible way explain the ARPES puzzle that the antinodal kink is shifted by the maximal superconducting gap Δmax while the nodal kink is unshifted. The reason is (as explained in Section 1.3.3) that due to strong correlations the EPI spectral function 𝛼2𝐹(𝐤,𝐤,Ω)𝛼2𝐹(𝜑𝜑,Ω) is strongly peaked at 𝜑𝜑=0 [151].

(9) The scattering potential on nonmagnetic impurities is renormalized by strong correlations giving also the forward scattering peak in the impurity scattering potential (amplitude) [155]. The latter effect gives large 𝑑-wave channel in the renormalized impurity potential, which is the reason that 𝑑-wave pairing in HTSC cuprates is robust in the presence of nonmagnetic impurities (and defects) [6, 155].

2.3.5. EPI and Strong Correlations—Other Methods

The calculations of the static (adiabatic) charge-vertex 𝛾𝑐(𝐤𝐹,𝐪) in the X-method are done for the case 𝑈= [7880, 130, 179, 180] where it is found that it is peaked at 𝑞=0—the forward scattering peak (FSP). This result is confirmed by the numerical Monte Carlo calculations for the finite-𝑈 Hubbard model [182], where it is found that FSP exists for all 𝑈, but it is especially pronounced in the limit 𝑡𝑈. These results are additionally confirmed in the calculations [183] within the four slave-boson method of Kotliar-Rückenstein where 𝛾𝑐(𝐤𝐹,𝐪) is again peaked at 𝐪=0 and the peak is also pronounced at 𝑡𝑈.

There are several calculations of the charge vertex in the one slave-boson method [219, 222224] which is invented to study the limit 𝑈. It is interesting to compare the results for the charge vertex in the X-method [7880, 130, 179, 180] and in the one slave-boson theory [222] which are calculated in 𝑂((1/𝑁)0) order. For instance, for 𝐽=0 one has 𝛾𝑐(𝑋)(𝐤,𝐪)=1+𝑏(𝐪)𝑎(𝐪)𝑡(𝐤)1+𝑏(𝐪)2,𝛾𝑎(𝐪)𝑐(𝐪)(SB)𝑐(𝐤,𝐪)=1+𝑏(𝐪)𝑎(𝐪)𝑡(𝐤)+𝑡(𝐤+𝐪)/21+𝑏(𝐪)2.𝑎(𝐪)𝑐(𝐪)(106) The explicit expressions for the “bare” susceptibilities 𝑎(𝐪),𝑏(𝐪), and 𝑐(𝐪) can be found in [7880, 130]. It is obvious from (106) that 𝛾𝑐(𝑋)(𝐤,𝐪=0)=𝛾(SB)𝑐(𝐤,𝐪=0) but the calculations give that max{𝛾𝑐(𝑋)(𝐤,𝐪)} is for 𝐪=0, while max{𝛾(SB)𝑐(𝐤,𝐪)} is for |𝐪|0 [181]. So, the SB vertex is peaked at finite 𝑞 which is in contradiction with the numerical Monte Carlo results for the Hubbard model [182] and with the four slave-boson theory [183]. The reason for the discrepancy of the one slave-boson (SB) in studying EPI with the numerical results and the X-method is not quite clear and might be due to the symmetry breaking of the local gauge invariance in leading order of the SB theory.

2.4. Summary of Section 2

The experimental results in HTSC cuprates which are exposed in Section 1 imply that the EPI coupling constant is large and in order to be conform with 𝑑-wave pairing this interaction must be very nonlocal (long range), that is, weakly screened and peaked at small transfer momenta. In absence of quantitative calculations in the framework of the ab initio microscopic many-body theory the effects of strong correlations on EPI are studied within the minimal 𝑡-𝐽 model where this pronounced nonlocality is due to two main reasons: (1) strong electronic correlations and (2) the combined metallic-ionic layered structure in these materials. In case (1) the pronounced nonlocality of EPI, which is found in the 𝑡-𝐽 model system, is due to the suppression of doubly occupancy at the Cu lattice sites in the CuO2 planes, which drastically weakens the screening effect in these systems. The pronounced nonlocality and suppression of the screening are mathematically expressed by the charge vertex function 𝛾𝑐(𝐤𝐹,𝐪,𝜔) which multiplies the bare EPI matrix element. The vertex function is peaked at 𝑞=0 and strongly suppressed at large 𝑞, especially for low (oxygen) hole-doping 𝛿1 near the Mott-Hubbard transition. Such a structure of 𝛾𝑐 gives that the 𝑑-wave and 𝑠-wave coupling constants are of the same order of magnitude around and below some optimal doping 𝛿op0.1, that is, 𝜆𝑑𝜆𝑠. This is very peculiar situation never met before. Since the residual effective (low-energy) Coulomb interaction is much smaller in the 𝑑-channel than in the 𝑠-channel, that is, 𝜇𝑠𝜇𝑑 (with the possibility that 𝜇𝑑<0), then the critical temperature for 𝑑-wave pairing is much larger than for the 𝑠-wave one, that is, 𝑇𝑐(𝑑)𝑇𝑐(𝑠). Since all charge fluctuation processes are modified by strong correlations, then the quasiparticle scattering on nonmagnetic impurities is also drastically changed; the pair-breaking effect on 𝑑-wave pairing is drastically reduced. This nonlocal effect, which is not discussed here—see more in [6] and references therein—is one of the main reasons for the robustness of 𝑑-wave pairing in HTSC oxides in the presence of nonmagnetic impurities and numerous local defects. The development of the forward scattering peak in 𝛾𝑐(𝐤𝐹,𝐪) and suppression at large 𝑞(𝑞𝑐=𝛿(𝜋/𝑎)) give rise to the suppression of the transport coupling constant 𝜆tr making it much smaller than the self-energy coupling constant 𝜆, that is, one has 𝜆tr𝜆/3 near the optimal doping 𝛿=0.1-0.2. Thus the behavior of the vertex function and the dominance of EPI in the quasiparticle scattering resolve the experimental puzzle that the transport and the self-energy coupling constant take very different values, 𝜆tr,𝑒𝑝𝜆𝑒𝑝. Note that this is not the case with the SFI mechanism which is dominant at large 𝐪𝐐=(𝜋,𝜋) thus giving 𝜆tr,sf𝜆sf. This result means that if in the SFI mechanism one fits the temperature-dependent resistivity (governed by 𝜆tr,sf) then one obtains very low 𝑇𝑐.

We stress that the strength of the EPI coupling constants 𝜆𝑒𝑝,𝜆𝑒𝑝,𝑑 is at present impossible to calculate since it is difficult to incorporate strong correlations and numerous structural effects in a tractable microscopic theory.

2.5. Discussions and Conclusions

Numerous experimental results related to tunnelling, optics, ARPES, inelastic neutron, and X-ray scattering measurements in HTSC cuprates at and near the optimal doping give evidence for strong electron-phonon interaction (EPI) with the coupling constant 1<𝜆𝑒𝑝<3.5. The tunnelling measurements furnish evidence for strong EPI which give that the peaks in the bosonic spectral function 𝛼2𝐹(𝜔)  coincide well with the peaks in the phonon density of states 𝐹ph(𝜔). The tunnelling spectra show that almost all phonons contribute to 𝑇𝑐 and that no particular phonon mode can be singled out in the spectral function 𝛼2𝐹(𝜔) as being the only one which dominates in pairing mechanism. In light of these results the small oxygen isotope effect in optimally doped systems can be partly due to this effect, thus not disqualifying the important role of EPI in pairing mechanism. The compatibility of the strong EPI with 𝑑-wave pairing implies an important constraint on the EPI pairing potential—it must be nonlocal, that is, peaked at small transfer momenta. The latter is due to (a) strong electronic correlations and (b) the combined metallic-ionic structure of these materials. If the EPI scattering is the main player in pairing in HTSC cuprates, then this nonlocality implies that at and below some optimal doping (𝛿op0.1) the magnitude of the EPI coupling constants in 𝑑-wave and 𝑠-wave channel must be of the same order, that is, 𝜆𝑒𝑝,𝑑𝜆𝑒𝑝,𝑠. This result in conjunction with the fact that the residual effective Coulomb coupling in 𝑑-wave channel is much smaller than in the 𝑠-wave one, that is, 𝜇𝑠𝜇𝑑 (with the possibility that 𝜇𝑑<0) gives that the critical temperature for 𝑑-wave pairing is much larger than for 𝑠-wave pairing.

The numerous tunnelling, ARPES, optics, and magnetic neutron scattering measurements give sufficient evidence that the spin-fluctuation interaction (SFI) plays a secondary role in pairing in HTSC cuprates. Especially important evidence for the smallness of SFI (in pairing) comes from the magnetic neutron scattering measurements which show that by varying doping slightly around the optimal one there is a huge reconstruction of the SFI spectral function Im𝜒(𝐪,𝜔) (imaginary part of the spin susceptibility) for 𝐪𝐐, while there is very small change in the critical temperature 𝑇𝑐. These experimental results imply important constraints on the pairing scenario for systems at and near optimal doping: (1) the strength of the 𝑑-wave pairing potential is provided by EPI (i.e., one has 𝜆𝑒𝑝,𝑑𝜆𝑒𝑝,𝑠) while the role of the residual Coulomb interaction and SFI, together, is to trigger 𝑑-wave pairing; (2) the Migdal-Eliashberg theory, but with the pronounced momentum dependent of EPI, is a rather good starting theory.

The ab initio microscopic theory of pairing in HTSC cuprates fails at present to calculate 𝑇𝑐 and to predict the magnitude of the 𝑑-wave order parameter. From that point of view it is hard to expect a significant improvement of this situation at least in the near future. However, the studies of some minimal (toy) models, such as the single-band 𝑡-𝐽 model, allow us to understand part of the physics in cuprates on a qualitative and in some cases even on a semiquantitative level. In that respect the encouraging results come from the theoretical studies of the EPI scattering in the 𝑡-𝐽 model by using controllable mathematical methods in the X-method formulated in terms of the Hubbard operators [7880, 130, 179, 180]. This theory predicts dressing of quasiparticles by strong correlations which dig up a large-scale correlation hole of the size 𝜉ch𝑎/𝛿 for 𝛿1. These quasiparticles respond to lattice vibrations in such a way to produce an effective long-range electron interaction (due to EPI), that is, the effective pairing potential 𝑉e(𝐪,𝜔) is peaked at small transfer momenta 𝑞—the forward scattering peak. This theory (of the toy model) is conform with the experimental scenario by predicting the following results: (i) the EPI coupling constants in 𝑑-wave and 𝑠-wave channels are of the same order, that is, 𝜆𝑒𝑝,𝑑𝜆𝑒𝑝,𝑠, at some optimal doping 𝛿op0.1; (ii) the transport coupling is much smaller than the pairing one, that is, 𝜆tr𝜆/3; (iii) due to strong correlations there is forward scattering peak in the potential for scattering on nonmagnetic impurities, thus making 𝑑-wave pairing robust in materials with a lot of defects and impurities. Applied to HTSC superconductors at and near the optimal doping, this theory is a realization of the Migdal-Eliashberg theory but with strongly momentum dependent EPI coupling, which is conform with the proposed experimental pairing scenario. This scenario which is also realized in the 𝑡-𝐽 toy model may be useful in making a (phenomenological) theory of pairing in cuprates. However, all present theories are confronted with the unsolved and challenging task—the calculation of 𝑇𝑐. From that point of view we do not have at present a proper microscopic theory of pairing in HTSC cuprates.

Appendix

A. Spectral Functions

A.1. Spectral Functions 𝛼2𝐹(𝑘,𝑘,𝜔) and 𝛼2𝐹(𝜔)

The quasiparticle bosonic (Eliashberg) spectral function 𝛼2𝐹(𝐤,𝐤,𝜔) and its Fermi surface average 𝛼2𝐹(𝜔)=𝛼2𝐹(𝐤,𝐤,𝜔)𝐤,𝐤 enter the quasiparticle self-energy Σ(𝐤,𝜔), while the transport spectral function 𝛼2𝐹tr(𝜔) enters the transport self-energy Σtr(𝐤,𝜔) and dynamical conductivity 𝜎(𝜔). Since the Migdal-Eliashberg theory for EPI is well defined, we define the spectral functions for this case and the generalization to other electron-boson interaction is straightforward. In the superconducting state Matsubara Green's functions 𝐺(𝐤,𝜔𝑛) and Σ(𝐤,𝜔𝑛) are 2×2 matrices with the diagonal elements 𝐺11𝐺(𝐤,𝜔𝑛),Σ11Σ(𝐤,𝜔𝑛) and the off-diagonal elements 𝐺12𝐹(𝐤,𝜔𝑛),Σ12Φ(𝐤,𝜔𝑛) which describe superconducting pairing. By defining 𝑖𝜔𝑛[1𝑍(𝐤,𝜔𝑛)]=[Σ(𝐤,𝜔𝑛)Σ(𝐤,𝜔𝑛)]/2 and 𝜒(𝐤,𝜔𝑛)=[Σ(𝐤,𝜔𝑛)+Σ(𝐤,𝜔𝑛)]/2, the Eliashberg functions for EPI in the presence of the Coulomb interaction (in the singlet pairing channel) read [70, 225227] 𝑍𝐤,𝜔𝑛𝑇=1+𝑁𝐩,𝑚𝜆𝑍𝐤𝐩𝜔𝑛𝑚𝜔𝑚𝑁(𝜇)𝜔𝑛𝑍𝐩,𝜔𝑚𝐷𝐩,𝜔𝑚,𝜒𝐤,𝜔𝑛𝑇=𝑁𝐩,𝑚𝜆𝑍𝐤𝐩𝜔𝑛𝑚𝑁(𝜇)𝜖(𝐩)𝜇+𝜒𝐩,𝜔𝑚𝐷𝐩,𝜔𝑚,Φ𝐤,𝜔𝑛=𝑇𝑁𝐩,𝑚𝜆Δ𝐤𝐩𝜔𝑛𝑚𝑁(𝜇)𝑉𝐤𝐩Φ𝐩,𝜔𝑚𝐷𝐩,𝜔𝑚,(A.1) where 𝜔𝑛𝑚𝜔𝑛𝜔𝑚, 𝜔𝑛=𝜋𝑇(2𝑛+1), Φ(𝐤,𝜔𝑛)𝑍(𝐤,𝜔𝑛)Δ(𝐤,𝜔𝑛), 𝐷=𝜔2𝑚𝑍2+(𝜖𝜇+𝜒)2+Φ2, and 𝑁(𝜇) is the density of states at the Fermi surface. (In studying some problems, such as optics, it is useful to define the renormalized frequency 𝑖𝜔𝑛(𝑖𝜔𝑛)(𝑖𝜔𝑛𝑍(𝜔𝑛))=𝜔𝑛Σ(𝜔𝑛) or its analytical continuation 𝜔(𝜔)=𝑍(𝜔)𝜔=𝜔Σ(𝜔)). These equations are supplemented with the electron number equation 𝑛(𝜇) (𝜇 is the chemical potential): 𝑛(𝜇)=2𝑇𝑁𝐩,𝑚𝐺𝐩,𝜔𝑚𝑒𝑖𝜔𝑚0+=12𝑇𝑁𝐩,𝑚𝜖(𝐩)𝜇+𝜒𝐩,𝜔𝑚𝐷𝐩,𝜔𝑚.(A.2) Note that in the case of EPI one has 𝜆Δ𝐤𝐩(𝜈𝑛)=𝜆𝑍𝐤𝐩(𝜈𝑛)(𝜆𝐤𝐩(𝜈𝑛)) (with 𝜈𝑛=𝜋𝑇𝑛) where 𝜆𝐤𝐩(𝜈𝑛) is defined by 𝜆𝐤𝐩𝜈𝑛=20𝜈𝑑𝜈𝜈2+𝜈2𝑛𝛼2𝐤𝐩𝛼𝐹(𝜈),(A.3)2𝐤𝐩𝐹(𝜈)=𝑁(𝜇)𝜅||𝑔ren𝜅,𝐤𝐩||2𝐵𝜅(𝐤𝐩,𝜈),(A.4) where 𝐵𝜅(𝐤𝐩;𝜈) is the phonon spectral function of the 𝜅th phonon mode related to the phonon propagator 𝐷𝜅𝐪,𝑖𝜈𝑛=0𝜈𝑑𝜈𝜈2+𝜈2𝑛𝐵𝜅(𝐪,𝜈).(A.5) However, very often it is measured the generalized phonon density of states 𝐺𝑃𝐷𝑆(𝜔)(𝐺(𝜔)) (see Section 1.3.4) defined by 𝐺(𝜔)=𝑖(𝜎𝑖/𝑀𝑖)𝐹𝑖(𝜔)/𝑖(𝜎𝑖/𝑀𝑖). Here, 𝜎𝑖 and 𝑀𝑖 are the cross-section and the mass of the 𝑖th nucleus and 𝐹𝑖(𝜔=(1/𝑁)𝑞|𝜀𝑖𝑞|2𝛿(𝜔𝜔𝑞) is the amplitude-weighted density of states.

The renormalized coupling constant 𝑔ren𝜅,𝐤𝐩(𝑔0𝜅,𝐤𝐩𝛾𝜀𝑒1) in (A.4) comprises the screening effect due to long-range Coulomb interaction (𝜀𝑒1—the inverse electronic dielectric function) and short-range strong correlations (𝛾—the vertex function)—see more in Section 2. Usually in the case of low-temperature superconductors (LTS) with 𝑠-wave pairing the anisotropy is rather small (or in the presence of impurities it is averaged out) which allows an averaging of the Eliashberg equations [70, 225227]:𝑍𝜔𝑛=1+𝜋𝑇𝜔𝑛𝑚𝜆𝜔𝑛𝑚𝜔𝑚𝜔2𝑚+Δ2𝜔𝑚,𝑍𝜔𝑛Δ𝜔𝑛=𝜋𝑇𝑚𝜆𝜔𝑛𝑚𝜔𝜇𝑐𝜃𝜔𝑐||𝜔𝑚||×Δ𝜔𝑚𝜔2𝑚+Δ2𝜔𝑚,𝜆𝜔𝑛𝑚=20𝜈𝑑𝜈𝜈2+(𝜔𝑛𝑚)2𝛼2𝐹(𝜈),(A.6) where 𝜔𝑛𝑚=𝜔𝑛𝜔𝑚,𝛼2𝐹(𝜔)=𝛼2𝐹(𝐤,𝐤,𝜔)𝐤,𝐤, and 𝐤,𝐤 is the average over the Fermi surface. The above equations can be written on the real axis by the analytical continuation 𝑖𝜔𝑚𝜔+𝑖𝛿 where the gap function is complex, that is, Δ(𝜔)=Δ𝑅(𝜔)+𝑖Δ𝐼(𝜔). The solution for Δ(𝜔) allows the calculation of the current-voltage characteristic 𝐼(𝑉) and tunnelling conductance 𝐺NS(𝑉)=𝑑𝐼NS/𝑑𝑉 in the superconducting state of the NIS tunnelling junction where 𝐼NS(𝑉) is given by 𝐼NS(𝑉)=2𝑒𝐤,𝐩||𝑇𝐤,𝐩||2𝑑𝜔,𝐴2𝜋𝑁(𝐤,𝜔)𝐴𝑆([].𝐩,𝜔+eV)𝑓(𝜔)𝑓(𝜔+eV)(A.7) Here, 𝐴𝑁,𝑆(𝐤,𝜔)=2Im𝐺𝑁,𝑆(𝐤,𝜔) are the spectral functions of the normal metal and superconductor, respectively, and 𝑓(𝜔) is the Fermi distribution function. Since the angular and energy dependence of the tunnelling matrix elements |𝑇𝐤,𝐩|2 is practically unimportant for 𝑠-wave superconductors, then the relative conductance 𝜎NS(𝑉)𝐺NS(𝑉)/𝐺NN(𝑉) is proportional to the tunnelling density of states 𝑁𝑇𝐴(𝜔)=𝑆(𝐤,𝜔)𝑑3𝑘/(2𝜋)3, that is, 𝜎NS(𝜔)𝑁𝑇(𝜔) where 𝑁𝑇(𝜔)=Re𝜔+𝑖̃𝛾(𝜔)(𝜔+𝑖̃𝛾(𝜔))2𝑍2(𝜔)Δ(𝜔)2.(A.8) Here, 𝑍(𝜔)=𝑍(𝜔)/Re𝑍(𝜔), ̃𝛾(𝜔)=𝛾(𝜔)/Re𝑍(𝜔),𝑍(𝜔)=Re𝑍(𝜔)+𝑖𝛾(𝜔)/𝜔, and the quasiparticle scattering rate in the superconducting state 𝛾𝑠(𝜔,𝑇)=2ImΣ(𝜔,𝑇) is given by 𝛾𝑠(𝜔,𝑇)=2𝜋0𝑑𝜈𝛼2𝐹(𝜈)𝑁𝑠×(𝜈+𝜔)2𝑛𝐵(𝜈)+𝑛𝐹(𝜈+𝜔)+𝑛𝐹(𝜈𝜔)+𝛾imp,(A.9) where 𝑁𝑠(𝜔)=Re{𝜔/(𝜔2Δ2(𝜔))1/2 is the quasiparticle density of states in the superconducting state; 𝑛𝐵,𝐹(𝜈) are Bose and Fermi distribution function, respectively. Since the structure of the phonon spectrum is contained in 𝛼2𝐹(𝜔), it is reflected on Δ(𝜔) for 𝜔>Δ0 (the real gap obtained from Δ0=ReΔ(𝜔=Δ0)) which gives the structure in 𝐺𝑆(𝑉) at 𝑉=Δ0+𝜔ph. On the contrary one can extract the spectral function 𝛼2𝐹(𝜔) from 𝐺NS(𝑉) by the inversion procedure proposed by Kulić [6] and McMillan and Rowell [228]. It turns out that in low-temperature superconductors the peaks of 𝑑2𝐼/𝑑𝑉2 at 𝑒𝑉𝑖=Δ+𝜔ph,𝑖 correspond to the peak positions of 𝛼2𝐹(𝜔) and 𝐹(𝜔). However, we would like to point out that in HTSC cuprates the gap function is unconventional and very anisotropic, that is, Δ(𝐤,𝑖𝜔𝑛)cos𝑘𝑥𝑎cos𝑘𝑦𝑎. Since in this case the extraction of 𝛼2𝐹(𝐤,𝐤,𝜔) is difficult and at present rather unrealistic task, then an “average” 𝛼2𝐹(𝜔) is extracted from the experimental curve 𝐺𝑆(𝑉). There is belief that it gives relevant information on the real spectral function such as the energy width of the bosonic spectrum (0<𝜔<𝜔max) and positions and distributions of peaks due to bosons. It turns out that even such an approximate procedure gives valuable information in HTSC cuprates—see discussion in Section 1.3.4.

Note that in the case when both EPI and spin-fluctuation interaction (SFI) are present one should make difference between 𝜆𝑍𝐤𝐩(𝑖𝜈𝑛) and 𝜆Δ𝐤𝐩(𝑖𝜈𝑛) defined by 𝜆𝑍𝐤𝐩𝑖𝜈𝑛=𝜆sf,𝐤𝐩𝑖𝜈𝑛+𝜆𝑒𝑝,𝐤𝐩𝑖𝜈𝑛,𝜆Δ𝐤𝐩=𝜆𝑒𝑝,𝐤𝐩𝑖𝜈𝑛𝜆sf,𝐤𝐩𝑖𝜈𝑛.(A.10) In absence of EPI, 𝜆𝑍𝐤𝐩(𝑖𝜈𝑛) and 𝜆Δ𝐤𝐩(𝑖𝜈𝑛) differ by sign, that is, 𝜆𝑍𝐤𝐩(𝑖𝜈𝑛)=𝜆Δ𝐤𝐩(𝑖𝜈𝑛)>0 since the SFI potential is repulsive in the singlet pairing channel.

A.1.1. Inversion of Tunnelling Data

Phonon features in the conductance 𝜎NS(𝑉) at eV=Δ0+𝜔ph make the tunnelling spectroscopy a powerful method in obtaining the Eliashberg spectral function 𝛼2𝐹(𝜔). Two methods were used in the past for extracting 𝛼2𝐹(𝜔).

The first method is based on solving the inverse problem of the nonlinear Eliashberg equations. Namely, by measuring 𝜎NS(𝑉), one obtains the tunnelling density of states 𝑁𝑇(𝜔)(𝜎NS(𝜔)) and by the inversion procedure one obtains 𝛼2𝐹(𝜔) [228]. In reality the method is based on the iteration procedure—the McMillan-Rowell (𝑀𝑅) inversion, where in the first step an initial 𝛼2𝐹ini(𝜔),𝜇ini, and Δini(𝜔) are inserted into Eliashberg equations (e.g., Δini(𝜔)=Δ0 for 𝜔<𝜔0 and Δini(𝜔)=0 for 𝜔>𝜔0) and then 𝜎ini(𝑉) is calculated. In the second step the functional derivative 𝛿𝜎(𝜔)/𝛿𝛼2𝐹(𝜔) (𝜔eV) is found in the presence of a small change of 𝛼2𝐹ini(𝜔) and then the iterated solution 𝛼2𝐹(𝟏)(𝜔)=𝛼2𝐹ini(𝜔)+𝛿𝛼2𝐹(𝜔) is obtained, where the correction 𝛿𝛼2𝐹(𝜔) is given by 𝛿𝛼2𝐹(𝜔)=𝑑𝜈𝛿𝜎ini(𝑉)𝛿𝛼2𝐹(𝜈)1𝜎ex𝑝(𝜈)𝜎ini.(𝜈)(A.11) The procedure is iterated until 𝛼2𝐹(𝑛)(𝜔) and 𝜇(𝑛) converge to 𝛼2𝐹(𝜔) and 𝜇 which reproduce the experimentally obtained conductance 𝜎expNS(𝑉). In such a way the obtained 𝛼2𝐹(𝜔) for 𝑃𝑏 resembles the phonon density of states 𝐹𝑃𝑏(𝜔), which is obtained from neutron scattering measurements. Note that the method depends explicitly on 𝜇 but on the contrary it requires only data on 𝜎NS(𝑉) up to the voltage 𝑉max=𝜔maxph+Δ0 where 𝜔maxph is the maximum phonon energy (𝛼2𝐹(𝜔)=0 for 𝜔>𝜔maxph) and Δ0 is the zero-temperature superconducting gap. One pragmatical feature for the interpretation of tunnelling spectra (and for obtaining the spectral pairing function 𝛼2𝐹(𝜔)) in LTS and HTSC cuprates is that the negative peaks of 𝑑2𝐼/𝑑𝑉2 (or peaks in 𝑑2𝐼/𝑑𝑉2) are at the peak positions of 𝛼2𝐹(𝜔) and 𝐹(𝜔). This feature will be discussed later on in relation with experimental situation in cuprates.

The second method has been invented in [229, 230] and it is based on the combination of the Eliashberg equations and dispersion relations for Green’s functions—we call it GDS method. First, the tunnelling density of states is extracted from the tunnelling conductance in a more rigorous way [231]: 𝑁𝑇𝜎(𝑉)=NS(𝑉)𝜎NN1(𝑉)𝜎(𝑉)𝑉0×𝑑𝑢𝑑𝜎(𝑢)𝑁𝑑𝑢𝑇(𝑉𝑢)𝑁𝑇,(𝑉)(A.12) where 𝜎(𝑉)=exp{𝛽𝑉}𝜎NN(𝑉) and the constant 𝛽 are obtained from 𝜎NN(𝑉) at large biases—see [229, 230]. 𝑁𝑇(𝑉) under the integral can be replaced by the BCS density of states. Since the second method is used in extracting 𝛼2𝐹(𝜔) in a number of LTSC as well as in HTSC cuprates—see below—we describe it briefly for the case of isotropic EPI at 𝑇=0K. In that case the Eliashberg equations are given by [70, 225227, 229, 230]: 𝑍(𝜔)Δ(𝜔)=Δ0𝑑𝜔Δ𝜔Re𝜔2Δ2(𝜔)1/2×𝐾+𝜔,𝜔𝜇𝜃𝜔𝑐,1𝜔1𝑍(𝜔)=𝜔Δ0𝑑𝜔𝜔Re𝜔2Δ2(𝜔)1/2𝐾𝜔,𝜔,(A.13) where 𝐾±𝜔,𝜔=𝜔maxphΔ0𝑑𝜈𝛼2×1𝐹(𝜈)𝜔+𝜔+𝜈+𝑖0+±1𝜔𝜔+𝜈𝑖0+.(A.14) Here 𝜇 is the Coulomb pseudopotential, the cutoff 𝜔𝑐 is approximately (5-10)𝜔maxph, and Δ0=Δ(Δ0) is the energy gap. Now by using the dispersion relation for the matrix Green’s functions 𝐺(𝐤,𝜔𝑛) one obtains [229, 230] Im𝑆(𝜔)=2𝜔𝜋Δ0𝑑𝜔𝑁𝑇𝜔𝑁BCS𝜔𝜔2𝜔2,(A.15) where 𝑆(𝜔)=𝜔/[𝜔2Δ2(𝜔)]1/2. From (A.13) one obtains 𝜔Δ00𝑑𝜈𝛼2𝜈𝐹(𝜔𝜈)ReΔ(𝜈)2Δ2(𝜈)1/2=ReΔ(𝜔)𝜔𝜔Δ00𝑑𝜈𝛼2𝐹(𝜈)𝑁𝑇(𝜔𝜈)+ImΔ(𝜔)𝜋+ImΔ(𝜔)𝜋0𝑑𝜔𝑁𝑇𝜔𝜔maxph0𝑑𝜈2𝛼2𝐹(𝜈)(𝜔)+𝜈2𝜔2.(A.16)

Based on (A.12)-(A.16) one obtains the scheme for extracting 𝛼2𝐹(𝜔): 𝜎NS(𝑉),𝜎NN(𝑉)𝑁𝑇(𝑉),Im𝑆(𝜔)Δ(𝜔)𝛼2𝐹(𝜔).(A.17) The advantage in this method is that the explicit knowledge of 𝜇 is not required [229, 230]. However, the integral equation for 𝛼2𝐹(𝜔) is linear Fredholm equation of the first kind which is ill defined—see the discussion in Section 1.3.2 item (2)

A.1.2. Phonon Effects in 𝑁𝑇(𝜔)

We briefly discuss the physical origin for the phonon effects in 𝑁𝑇(𝜔) by considering a model with only one peak, at 𝜔0, in the phonon density of states 𝐹(𝜔) by assuming for simplicity 𝜇=0 and neglecting the weak structure in 𝑁𝑇(𝜔) at 𝑛𝜔0+Δ0, 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 Δ𝑅(𝜔) reaches a maximum at 𝜔0+Δ0 then decreases and becomes negative and zero, while Δ𝐼(𝜔) is peaked slightly beyond 𝜔0+Δ0 that is the consequence of the effective electron-electron interaction via phonons.

It follows that for 𝜔<𝜔0+Δ0 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 𝜔𝜔0+Δ0 the charge fluctuations are in resonance with ion vibrations giving rise to the peak in Δ𝑅(𝜔). For 𝜔0+Δ0<𝜔 the ions move out of phase with respect to the charge fluctuations giving rise to repulsion and negative Δ𝑅(𝜔). This is shown in Figure 37(b). The structure in Δ(𝜔) is reflected on 𝑁𝑇(𝜔) as shown in Figure 37(c) which can be reconstructed from the approximate formula for 𝑁𝑇(𝜔) expanded in powers of Δ/𝜔: 𝑁𝑇(𝜔)1𝑁(0)1+2Δ𝑅(𝜔)𝜔2Δ𝐼(𝜔)𝜔2.(A.18) As Δ𝑅(𝜔) increases above Δ0, this gives 𝑁𝑇(𝜔)>𝑁BCS(𝜔), while for 𝜔𝜔0+Δ0 the real value Δ𝑅(𝜔) decreases while Δ𝐼(𝜔) rises and 𝑁𝑇(𝜔) decreases giving rise for 𝑁𝑇(𝜔)<𝑁BCS(𝜔).

A.2. Transport Spectral Function 𝛼2𝑡𝑟𝐹(𝜔)

The spectral function 𝛼2tr𝐹(𝜔) enters the dynamical conductivity 𝜎𝑖𝑗(𝜔) (𝑖,𝑗=𝑎,𝑏,𝑐 axis in HTS systems) which generally speaking is a tensor quantity given by the formula 𝜎𝑖𝑗𝑒(𝜔)=2𝜔𝑑4𝑞(2𝜋)4𝛾𝑖(𝑞,𝑘+𝑞)×𝐺(𝑘+𝑞)Γ𝑗(𝑞,𝑘+𝑞)𝐺(𝑞),(A.19) where 𝑞=(𝐪,𝜈) and 𝑘=(𝐤=0,𝜔) and the bare current vertex 𝛾𝑖(𝑞,𝑘+𝑞;𝐤=0) is related to the Fermi velocity 𝑣𝐹,𝑖, that is, 𝛾𝑖(𝑞,𝑘+𝑞;𝐤=0)=𝑣𝐹,𝑖. The vertex function Γ𝑗(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, Γ𝑗(𝑞,𝑘+𝑞)Γ𝑗(𝐪,𝑖𝜔𝑛,𝑖𝜔𝑛+𝑖𝜔𝑚) is the solution of an approximative integral equation written in the symbolic form [118] Γ𝑗=𝑣𝑗+𝑉e𝐺𝐺Γ𝑗. The effective potential 𝑉e (due to EPI) is given by 𝑉e=𝜅|𝑔ren𝜅|2𝐷𝜅, where 𝐷𝜅 is the phonon Green's function. In such a case the Kubo theory predicts 𝜎intra𝑖𝑖(𝜔) (𝑖=𝑥,𝑦,𝑧): 𝜎𝑖𝑖𝜔(𝜔)=2𝑝,𝑖𝑖4𝑖𝜋𝜔0𝜔𝑑𝜈th𝜔+𝜈𝑆2𝑇1+(𝜔,𝜈)0𝑑𝜈th𝜔+𝜈𝜈2𝑇th𝑆2𝑇1,(𝜔,𝜈)(A.20) where 𝑆(𝜔,𝜈)=𝜔+Σtr(𝜔+𝜈)Σtr(𝜈)+𝑖𝛾imptr, and 𝛾imptr is the impurity contribution. In the following we omit the tensor index 𝑖𝑖 in 𝜎𝑖𝑖(𝜔). In the presence of several bosonic scattering processes the transport self-energy Σtr(𝜔)=ReΣtr(𝜔)+𝑖ImΣtr(𝜔) is given by Σtr(𝜔)=𝑙0𝑑𝜈𝛼2tr,𝑙𝐹𝑙𝐾(𝜈)1(𝜔,𝜈)+𝑖𝐾2,𝐾(𝜔,𝜈)1Ψ1(𝜔,𝜈)=Re2+𝑖𝜔+𝜈12𝜋𝑇Ψ2+𝑖𝜔𝜈,𝐾2𝜋𝑇2𝜋(𝜔,𝜈)=2𝜈2𝑐th2𝑇th𝜔+𝜈2𝑇+th𝜔𝜈.2𝑇(A.21) Here 𝛼2tr,𝑙𝐹𝑙(𝜈) is the transport spectral function which measures the strength of the 𝑙th (bosonic) scattering process and Ψ is the di-gamma function. The index 𝑙 enumerates EPI, charge, and spin-fluctuation scattering processes. Like in the case of EPI, the transport bosonic spectral function 𝛼2tr,𝑙𝐹(Ω) defined in (97) is given explicitly by 𝛼2tr,𝑙𝐹1(𝜔)=𝑁2(𝜇)𝑑𝑆𝐤𝑣𝐹,𝐤𝑑𝑆𝐤𝑣𝐹,𝐤×𝑣1𝑖𝐹,𝐤𝑣𝑖𝐹,𝐤𝑣𝑖𝐹,𝐤2𝛼2𝐤𝐤,𝑙𝐹(𝜔).(A.22) We stress that in the phenomenological SFI theory [1217] one assumes that 𝛼2𝐤𝐤𝐹(𝜔)𝑁(𝜇)𝑔2sfIm𝜒(𝐤𝐩,𝜔), which, as we have repeated several times, can be justified only for small 𝑔sf, that is, 𝑔sf𝑊𝑏 (the bandwidth).

In case of weak coupling (𝜆<1), 𝜎(𝜔) can be written in the generalized (extended) Drude form as discussed in Section 1.3.2.

Acknowledgments

The authors devote this paper to their great teacher and friend Vitalii Lazarevich Ginzburg who passed away recently. His permanent interest in their work and support in many respects over many years are unforgettable. M. L. Kulić is thankful to Karl Bennemann for inspiring discussions on many subjects related to physics of HTSC cuprates. They also thank Godfrey Akpojotor for careful reading of the manuscript. M. L. Kulić is thankful to the Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, Berlin, for the hospitality and financial support during his stay where part of this work has been done.