Manifestations of the Electron-Phonon Coupling in the Spectroscopy of High-Temperature Superconductors

Abstract

A review of experimental and theoretical results on the spectroscopy of high-temperature superconductors is presented. The models where hole doped into antiferromagnet interacts both with magnetic subsystem and with phonons are considered. Theoretical results of these models for phonon spectra, angle resolved photoemission spectra, and optical conductivity are presented. Comparison with experimental data gives evidence for the strong electron-phonon coupling in the undoped and weakly doped high-temperature superconductors. The strength of electron-phonon coupling decreases with doping though at the optimal doping the compounds are still in the intermediate coupling regime.

1. Introduction

The role of the electron-phonon interaction (EPI) in the high-temperature superconductors is debated for many years. Firm opinions range from the complete negation of its role [1] up to the statement that the polaron binding energy is larger than the characteristic magnetic energy by an order of magnitude and, hence, the magnetic subsystem is not important [2, 3]. Of course there is an opinion that both magnetic and lattice subsystems are important [4].

Up to the very recent times it seemed that the spectral properties of cuprates can be explained considering only magnetic subsystem. The dispersion of the peak in the Angle Resolved Photoemission (ARPES) spectra was well reproduced by tt′t′′-J model [5] which is the model with a hole moving on the antiferromagnetic background [6–8]. Also, t-J model predicted a peak in the Optical Conductivity (OC) [9–22], situated roughly at the same energy as the Mid Infrared (MIR) band observed in experiments [23–28]. However, improvement of the experimental technique and advances in theoretical methods make them capable of seeing fine details of spectra which occurred to be quite different in experiment and theory. In experiment, the resolution of ARPES technique reached few meV [5] and the ellipsometry technique [29, 30] gave possibility to measure OC without Kramers-Kronig relation which introduce uncertainty into the experimental data. In theory, recently developed Diagrammatic Monte Carlo (DMC) method [31–56] became capable of avoiding serious approximations in a significant set of models describing strongly correlated systems.

One of the main and most evident contradiction between theory and experiment was the linewidth of the ARPES peak in undoped cuprates. Although the dispersion of the peak is well reproduced by tt′t′′-J model, its width is very broad in experiment [57] and very narrow in theory [37]. Naively, contribution of EPI cannot explain the large width since the coupling to phonons, in addition to broadening, must also change the dispersion of the particle which, in turn, is already well described by the pure tt′t′′-J model. However, as was shown in [42, 58], in the strong coupling regime of EPI the situation is exactly the same as in experiment. The polaron quasiparticle has very small weight and cannot be seen in experiment while shake-off Franck-Condon peak completely reproduces the dispersion of the pure magnetic model without EPI. Naturally, in such case the chemical potential must be pinned not to the observed broad shake-off peak but to the real invisible quasiparticle. Such decoupling of the chemical potential from the broad peak was observed in experiment [59] a few months after prediction had been made in [42].

Further theoretical and experimental studies brought more evidences of the importance of EPI in cuprates. One of the evidences is the two-peak structure of the MIR part of OC in the underdoped compounds which is easily reproduced by taking the EPI into account [51]. Another confirmation is the anomalous temperature dependence of the width of the ARPES peak which can be explained only by the interplay of magnetic and lattice system [50, 60]. Various estimates for the EPI strength [50, 61–63] give the value 𝜆≈1 for undoped compounds. The strength of EPI decreases with increase of the concentration of holes reaching the intermediate coupling regime at optimal doping [51, 64].

In Section 2 we introduce the models describing the physics of the interplay between magnetic and lattice degrees of freedom in cuprates. Then, we discuss different aspects of manifestation of the EPI in the phonon spectra, ARPES, and OC in Section 3. The influence of strong electronic correlations on the EPI is discussed in Section 4 and the final conclusions are presented in Section 5.

2. Models

The prototypical model for high-temperature superconductors is the three band model [65]. This model contains one 3𝑑𝑥²−𝑦² orbital of Cu and two oxygen 𝑝 orbitals in the CuO₂ plain. The Hamiltonian 𝐻_3B=𝜀_{O𝐢𝜹𝜎}𝑎^†_𝐢𝜹𝜎𝑎_𝐢𝜹𝜎+𝜀_𝑑𝐢𝜎𝑐^†_𝐢𝜎𝑐_𝐢𝜎+𝑈_𝐢𝑛_𝐢↑𝑛_𝐢↓+𝑡_{𝑝𝑑𝐢𝜹𝜎}𝑃_𝜹(𝑐^†_𝐢𝜎𝑎_𝐢𝜹𝜎+h.c.) (1) includes term with Coulomb repulsion 𝑈 on Cu ion and term describing hopping between Cu and O with amplitude 𝑡_𝑝𝑑. The vector 𝜹 defines oxygen position in the unit cell and runs over (𝑎/2,0)and(0,𝑎/2) in the second term and over (±𝑎/2,0)and(0,±𝑎/2) in the last term. 𝑃_𝜹=−𝑃_−𝜹 is even operator. Operator 𝑎^†_𝐢𝜹𝜎 (𝑐^†_𝐢𝜎) is the creation operator of electron with spin 𝜎 on the oxygen (cooper) ion. Energy 𝜀_𝑑 (𝜀_𝑂) corresponds to Cu (O) orbital.

The t-J model is derived from the three-band model. The site 𝑖 in the t-J model corresponds to the position of cooper ion. In the undoped system all Cu ions have configuration 𝑑⁹ and each site is occupied by one hole. Doped holes go mostly to the O sites and form with hole the Zhang-Rice singlet [66]. This singlet is an empty state in the t-J model described by the following Hamiltonian: 𝐻_t-J=−𝑡_{⟨𝐢𝐣⟩𝜎}(𝑐̃^†_𝐢𝜎𝑐̃_𝐣𝜎+h.c.)+𝐽_{⟨𝐢𝐣⟩}(𝐒_𝐢𝐒_𝐣+𝑛_𝐢𝑛_𝐣4). (2) Here 𝑐̃_𝐣𝜎 is projected (to avoid double occupancy) fermionic annihilation operator, 𝑛_𝐢<2 is the number operator, 𝐒_𝐢 is spin 1/2 operator, 𝐽 is the exchange integral, and ⟨𝐢𝐣⟩ is restricted to near neighbors in two-dimensional lattice.

Expressing spin operators in terms of spin waves and making Fourier and Bogoliubov transformations one can derive the t-J Hamiltonian in the spin wave approximation [22, 67–71]. Adding hoppings to the second 𝑡^′ and third 𝑡^′′ near neighbors 𝑡^′ and 𝑡^′′ [6–8, 72–76] one arrives to the Hamiltonian of the t𝑡^′𝑡^′′-J model (3) in the spin wave approximation. A hole (ℎ_𝐤 is its annihilation operator) with dispersion 𝜀(𝐤)=4𝑡^′cox(𝑘_𝑥)cos(𝑘_𝑦)+2𝑡^′′[cos(2𝑘_𝑥)+cos(2𝑘_𝑦)] moves in the field of magnons (𝛼_𝐤 is annihilation operator of the magnon): 𝐻⁰_t-J=_𝐤𝜀(𝐤)ℎ^†_𝐤ℎ_𝐤+_𝐤𝜔_𝐤𝛼^†_𝐤𝛼_𝐤. (3) The dispersion of magnons is 𝜔_𝐤=2𝐽1−𝛾²_𝐤, where 𝛾_𝐤=(cos𝑘_𝑥+cos𝑘_𝑦)/2. The hole is scattered by magnons 𝐻^h-m_t-J=𝑁^−1/2_𝐤,𝐪𝑀_𝐤,𝐪[ℎ^†_𝐤ℎ_𝐤−𝐪𝛼_𝐤+h.c.] (4) with the scattering vertex 𝑀_𝐤,𝐪. Amplitudes 𝑡, 𝑡^′, and 𝑡^′′ describe hoppings to the nearest, next nearest, and next next nearest neighbors. The simplest t-J model corresponds to the case when 𝑡^′ and 𝑡^′′ are set to zero. For hole (electron) doping the signs of the hopping amplitudes are 𝑡>0, 𝑡^′<0, and 𝑡^′′>0 (𝑡<0, 𝑡^′>0, 𝑡^′′<0) [76–79]. Note that t-J model can be also derived from the Hubbard model [80, 81] 𝐻_H=−𝑡_{⟨𝐢𝐣⟩𝜎}(𝑐̃^†_𝐢𝜎𝑐̃_𝐣𝜎+h.c.)+𝑈_𝐢𝑛_𝐢↑𝑛_𝐢↓ (5) with large 𝑈→∞.

In the generalized tt′t′′-J-Holstein (tt′t′′-J-H) model the short range EPI with dispersionless phonons with frequency Ω₀, 𝐻^ph=Ω_0𝐤𝑏^†_𝐤𝑏_𝐤, (6) is described by Holstein Hamiltonian: 𝐻^e-ph=𝑁^−1/2_𝐤,𝐪𝜎2𝑀Ω₀[ℎ^†_𝐤ℎ_𝐤−𝐪𝑏_𝐪+h.c.]. (7) Here 𝜎 is the coupling constant which is determined by the strength of coupling of the hole to the lattice displacement and which does not depend on the mass of isotope. The expression in front of square bracket is the standard Holstein coupling constant 𝛾=𝜎/(2𝑀Ω₀). Another frequently used coupling constant is 𝑔=𝛾/Ω₀. The most frequently used measure for the strength of the EPI is the dimensionless coupling constant 𝜆=𝛾²/4𝑡Ω₀.

The reason for the strong EPI in the system described by the t-J model is the large energy involved into formation of the Zhang-Rice singlet. This energy of several eVs in the rigid lattice is trivial constant. However, when the lattice is deformed by phonons, change of the amplitudes 𝑡_𝑝𝑑 leads to the strong EPI [61, 82–93]. Estimate for EPI obtained from the three-band model is 𝜆≈1 [62].

Another source of EPI is the Fröhlich interaction with the phonons polarized along 𝑐 axis which broaden and soften with increase of doping [94, 95]. These phonons recently attracted interest [96] driven by new ARPES data on Bi2Sr2CuO6 [97]. However, importance of such phonons was noted long ago [98].

Large EPI in cuprates arises due to strong electronic correlations [61, 89] because calculations by Local Density Approximation (LDA) method give the coupling constants which are smaller by an order of magnitude [99]. However, these results are doubtful since the phonon linewidth, obtained by LDA [100], is considerably smaller that the linewidth seen in experiment [101]. The authors of [102] claim that the kink cannot be driven by EPI because effect of EPI on ARPES spectra is negligible. However, as it was shown in [103], calculations by the method used in [102] do not reproduce the linewidth of phonons seen in experiment [104–108]. Hence, conclusions of [102] about kinks are at least doubtful.

3. Spectroscopy

There is no adopted agreement whether kinks in ARPES dispersion are driven by interaction with phonons or magnetic resonance mode. However, recent experiments on the isotope effect of the ARPES spectra give confidence that the kinks are due to interaction with phonons. Another evidence for the phonon origin of the kinks follows from the ARPES spectra measurements in the electron doped compounds where kinks and magnetic resonance mode are located on absolutely different energies. The results concerning kinks are presented in Section 3.1.

The most obvious and undebatable evidence for EPI can be found in phonon spectra where EPI is manifested in the softening and broadening of particular phonons. Experimental and theoretical results on the phonon anomalies are reviewed in Section 3.2.

Theoretically, ARPES signal measured in the undoped compounds corresponds to the Lehmann Function (LF) of a hole in the tt′t′′-J model. The LF of this model [6, 37, 72] has a narrow 𝛿-functional peak at low-energies and high-energy incoherent continuum. Dispersion of the low-energy peak is in perfect agreement with the experimental momentum dependence. However, even the narrowest width of the experimental peak in the nodal point (𝜋/2,𝜋/2) is larger than the dispersion bandwidth [57, 59]. This is the main contradiction of the experiment with the tt′t′′-J model because the theoretical width of the peak in the nodal point is zero [37]. The interpretation of the linewidth in terms of EPI [109] was suggested long ago but confronted with the fact that EPI must not only broaden the line but considerably change the dispersion. Solution to the problem of the large linewidth of ARPES line is presented in Section 3.3. For 𝜆>0.4 in the t-J-H model the agreement with experiment is perfect. At strong EPI the quasiparticle loses its weight, becomes dispersionless, and cannot be seen in ARPES spectra. The whole weight of the quasiparticle is transformed into the broad Franck-Condon shake-off peak whose dispersion inherits the dispersion of the pure t-J model.

The estimate of 𝜆 was done by several methods. The most detailed and convincing estimates were done in [62] where 𝜆 was determined from the calculation of the change of Zhang-Rice singlet energy with lattice deformation, from the linewidth of the peak, and the distance of the Franck-Condon peak from the chemical potential. All these methods give 𝜆∼1 which is enough to bring the system into the strong coupling regime. These and other methods to determine 𝜆 are described in the Section 3.4.

One of the most important experiments to reveal the role of EPI is the isotope effect on the ARPES spectra. Experimentally, the isotope effect on the ARPES of doped compounds was measured many times [110–113] but the most reliable result is obtained in [113]. Keeping in mind how many contradictions were overcome to measure the isotope effect on the doped compounds it is clear why the isotope effect on the undoped compounds, where preparation of specimens is more difficult, was not measured up to now. However, there is a theoretical prediction for the undoped case [46] which is presented in Section 3.5.

It was noted long ago [60] that the net influence of the polaronic effect is unable to explain anomalous temperature dependence of the ARPES spectra. Also, the pure t-J model is unable to explain the experimental data too [114]. However, the joint influence of the magnetic and lattice degrees of freedom, as it is shown in Section 3.6, perfectly explains the anomalous temperature dependence.

One more proof of the interplay of magnetic and lattice effects can be got from the study of the OC of the weakly doped compounds. Two-peak structure of the MIR OC was tacitly resolved in many measurements [23, 27, 115–120]. Recent measurement by ellipsometry resolved the two-peak structure clearly [51]. Section 3.7 reviews results of the calculation of the OC in the framework of the t-J-H model [51] where theoretical OC shows two-peak structure due to interplay of magnetic and lattice degrees of freedom. Dependence of the OC on doping gives possibility to estimate dependence of the coupling constant 𝜆 on doping. It is shown that with 𝜆∼1 in the weakly doped systems the effective EPI decreases with doping coming to the moderate coupling regime 𝜆∼0.5 at the optimal doping.

There are many evidences that the local EPI of the Holstein model cannot explain many features of the high-temperature superconductors. First, it is well known that the very nature of coupling to the Zhang-Rice singlet and Fróhlich coupling to the c-axis phonons leads to nonlocal coupling vertexes. Second, the local coupling to phonons leads to huge masses of the holes which are not observed in experiment. One can continue with the fact that the local EPI leads to the contradicting to experiment stability of the antiferromagnetism in doped systems though above motivation is enough to look at the models where EPI is not local. Results concerning nonlocal EPI are presented in Section 3.8.

3.1. Kinks

The nature of the kings is debated since its' discovery. Reasonable explanation of kinks is a battlefield between groups explaining its nature by the interaction with phonons and others prescribing the anomaly to the interaction with the magnetic resonance mode. Typical kinks in ARPES spectra are shown in Figure 1. Early studies [64, 121–127] indicated that the anomaly is located on the energy 70 meV. Later, refined analysis showed [97, 128] that there are structures with smaller energy. Kinks are observed not only along the nodal direction (0,0)−(𝜋,𝜋) but also in other directions [125–127, 129, 130].

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1 

Quasiparticle dispersion in the direction (0,0)−(𝜋,𝜋) in the compounds La2−𝑥Sr𝑥CuO4 (LSCO), Bi2Sr2CuO6 (Bi2201), and Bi2Sr2CaCu2O8 (Bi2212) at different temperatures and dopings. Panel (f) shows the value 1+𝜆′, evaluated by the change of the dispersion angle in the weak coupling theory for noninteracting electrons in metal, after Lanzara et al. [64].

Kinks were explained by interaction with phonons [90, 91, 131, 132] and by interaction with magnetic resonance mode [129, 133–139] which is observed in the magnetic neutron scattering on the cuprates [140–143]. Actually, it is very difficult to discern theoretically whether kink is governed by magnetic or lattice subsystem. Detailed study [144] has led to conclusion that it is difficult to state which modes give the origin to the kink.

However, the nature of the kink can be revealed by experiments. There are two unambiguous evidences of the phononic nature of the kink. The first proof is obtained in [145]. As it is noted in [145] the optical phonons have energies 40 meV and 70 meV [146, 147] while the energy of the magnetic resonance mode in the hole doped compounds is 40 meV [140, 148]. Therefore, it was very difficult to distinguish between phononic and magnetic scenarios because of the similar energy scales of magnetic and phononic excitations. However, recently found magnetic resonance mode in the electron doped compounds [149, 150] is on the energy of 10 meV and does not correspond to the energy of the kink whose energy is the same as it was in the hole doped compounds.

Another unambiguous evidence of the phononic nature of the kink follows from the isotope effect on the kink [113]. The change of the ARPES spectra occurs only in the close vicinity of the kink and the characteristic energy of the shift is 3.4±0.5 meV. Theoretical estimate for the isotopic shift of the breathing mode at the energy 70 meV is 3.9 meV. This value is in the excellent agreement with the measured isotopic shift.

3.2. Phonons

EPI decreases the frequencies of phonons and leads to decay of the phonons when the line observed in the inelastic neutrons scattering broadens. To determine which of the phonons are anomalous one makes an attempt to fit the experimental phonon branches by some standard, say, shell model. The phonons laying considerably lower than those predicted by the shell model are considered to be anomalous. From this point of view the breathing and half-breathing phonons are anomalous in doped systems [94, 101, 108, 147, 151–156].

Another characteristic feature of the anomalous phonon is the short lifetime manifested in the broad peak with large linewidth which sometimes reaches 5 meV [101] which is an order of magnitude larger than those of another lines whose linewidth is limited by experimental resolution. For example, breathing and half-breathing phonons connected strongly to Zhang-Rice singlet have large linewidth whereas the quadrupole ones which are not coupled to the singlet are narrow [108]. The phonon 𝑂^𝑍_𝑍 associated with the movement of the oxygen perpendicular to the CuO2 plain has large width of 16 meV [94, 95]. One more anomalous phonon is 𝐵_1𝑔 one changing its width with temperature [146, 157–159]. There is one more phonon anomaly with wave vector 𝐪=(0.25,0,0) which was believed to be associated with stripe-like inhomogeneities of charge distribution [104] at the doping level 𝑥≈1/8. However, the same anomaly is observed at 𝑥≈0.08 [160]. Moreover, it was noticed that the energy and momentum of the kink in ARPES match the corresponding parameters of the phonons [161].

In one of the first theoretical papers considering lattice vibrations in cuprates the phonon softening was found by exact diagonalization technique in the model where effective Hamiltonian was obtained from the three-band model [162]. Softening of the breathing 𝐪=(𝜋,𝜋) and half-breathing 𝐪=(𝜋,0) phonons was calculated in [83, 84, 87, 92]. The calculation predicted dependence of the softening on the doping concentration which was later confirmed experimentally [95]. Also, theoretical data on the phonon broadening [84] showed, in agreement with experiment, that the peak width is maximal at 𝐪=(𝜋,0) and much smaller at 𝐪=(𝜋,𝜋).

The authors of the paper [88] studied the dependence of the EPI matrix element on the wave vectors of the electron and phonon and concluded that the interaction is strongly anisotropic. The authors found that the maximum of the broadening of the longitudinal optical phonons is associated with effective coupling of the 𝑑-symmetry pairing. Papers [163, 164] considered the charge response of the high-temperature superconductors arising from the ionic nature of the compounds. These papers predicted strong softening of the phonons with displacement along 𝑐-axis prior to experiment.

Using the first principle calculations and three-band model the paper [61] introduces the t-J-H model with derived from the first principles parameters. The exact diagonalization used to obtain the phonon spectral function gives the phonon dispersion which is in a good agreement with experimental data (see Figure 2) at the doping concentration 𝑥≈0.125.

Figure 2 

Phonon dispersion in the (1,0) and (1,1) directions. Experimental results are shown by dotted lines for 𝑥=0 and 𝑥=0.15. Theoretical results (full curve) are shown for 𝑥=0.125. The bars in theoretical results show the spread due to different boundary conditions, after Rösch and Gunnarsson [61].

3.3. Ghost Particles in the ARPES

The spectral function of the t-J-H model (3)–(7) was previously calculated by exact diagonalization (ED) of small clusters [165] and in the Non-Crossing Approximation (NCA) where all crossings of propagators are neglected [166, 167]. However, small size of the clusters in ED gives essentially discrete spectrum and prevent from studying of the lineshape [165, 168]. On the other hand the NCA is valid, strictly speaking, only for the weak coupling to phonons [42]. The NCA is good for magnons since the spin 1/2 cannot be flipped more than one time and, hence, the multiple accumulation of the bosons on one site is impossible. However, there is an accumulation of multiple bosons in the EPI channel and, hence, one has to avoid NCA for correct treatment of the EPI.

The implementation of the DMC method in [42] takes into account the mutual crossing of the phonon propagators but neglects both crossing of the phonon and magnon lines and mutual crossing of magnon lines. The NCA in the pure t-J model, as it was shown by comparison of the NCA results with the ED data [67, 68, 71, 169, 170], is good for 𝐽/𝑡≤0.4. Similar conclusion is drawn for the t-J-H model [171]. Also, recent calculations of the spectral function of the t-J-H model [172] by variational method [173, 174], which does not use NCA, confirmed the approximation made in [42].

Figures 3(a)–3(e) show the low-energy part of the spectral function in the ground state at 𝐤=(𝜋/2,𝜋/2) in the regime of weak, intermediate and strong EPI. 𝜆-dependence of the energies of the peaks (see Figure 3(f)), 𝑍^{𝐤=(𝜋/2,𝜋/2)}-factor of the lowest peak (see Figure 3(g)), and average number of phonons in the phonon cloud (see Figure 3(h)) is typical for the self-trapping phenomenon [40, 175, 176].

Figure 3 

(a) Lehmann spectral function L of the ground state at 𝐤=(𝜋/2,𝜋,2) for 𝐽/𝑡=0.3 and 𝜆=0. Low-energy spectral function L in the ground state 𝐤=(𝜋/2,𝜋,2) at 𝐽/𝑡=0.3: (b) 𝜆=0; (c) 𝜆=0.3; (d) 𝜆=0.4; (e) 𝜆=0.46. 𝜆-dependence at 𝐽/𝑡=0.3: (f) energies of the lowest resonances; (g) 𝑍-factor of the lowest resonance; (h) average number of phonons.

Momentum dependence of the spectral function is shown in Figures 4(a)–4(d). Energy of the lowest peak with small spectral weight (shown by vertical arrows in Figures 4(a)–4(d)) does not demonstrate any visible momentum dependence. To the contrary, dispersion of the broad peak is perfectly reproduced by the relation (see Figure 4(e)) 𝜀_𝐤=𝜀_min+𝑊_𝐽/𝑡5×{[cos𝑘_𝑥+cos𝑘_𝑦]²+[cos(𝑘_𝑥+𝑘_𝑦)+cos(𝑘_𝑥−𝑘_𝑦)]²4}, (8) which perfectly describes the dispersion of the pure t-J model in the wide range of parameters 0.1<𝐽/𝑡<0.9 [71]. Note that such behavior of the broad Franck-Condon shake-off peak is robust for the whole strong coupling regime of the EPI.

Figure 4 

Low-energy part of the spectral function of the hole at 𝐽/𝑡=0.3 and 𝜆=0.46 (a)–(d). The vertical arrows in (a)–(d) denote the position of the quasiparticle invisible in the scale of the figure because of low weight. (e) Dispersion of resonances at 𝐽/𝑡=0.3: broad resonance (circles) and the quasiparticle with small weight (squares) at 𝜆=0.46. The solid line is the dispersion of the hole (8) in the pure t-J model at 𝐽/𝑡=0.3 (𝑊𝐽/𝑡=0.3=0.6) and 𝜀min=−2.52. Panel (f) shows the potential of ground state 𝑄2/2 (solid line), potential of the excited state without the lattice relaxation (dashed line), and potential of the excited state after the lattice relaxation 𝐷+(𝑄−𝜆)2/2−𝜆2/2 (dotted line).

At low temperature the spectral function in the adiabatic approximation is the sum of transitions between the lower 𝐸_low(𝑄) and the upper 𝐸_up(𝑄) sheet of the adiabatic potential. The transitions are weighted by the wave function of the lower sheet |𝜓_low(𝑄)|² [58]. If EPI is absent both in the initial 𝐸_low(𝑄)=𝑄²/2 and final 𝐸_up(𝑄)=𝒟+𝑄²/2 states of transition, the spectral peak has maximum at the energy 𝒟. However, if the EPI Δ𝐸_up(𝑄)=−𝜆𝑄 is present only in the final state Δ𝐸_up(𝑄)=−𝜆𝑄, the upper sheet of the adiabatic potential 𝐸_up(𝑄)=𝒟−𝜆²/2+(𝑄−𝜆)² has the same energy at 𝑄=0. Hence, since the maximum of the probability in the initial state |𝜓_low(𝑄)|² is at 𝑄=0, the energy of the peak of the spectral function of the transition is still at the same energy 𝐷 (see Figure 4(f)). Note that the situation described above is the same as in the ARPES experiment in undoped cuprates. There is no EPI in the half-filled system (initial state) but there is EPI in the system with hole (final state).

The lowest dispersionless peak with small weight is too small to be easily observed in experiment. To the contrary one can easily observe wide Franck-Condon shake-off peak which mimics the dispersion of the pure t-J model. This theoretical picture suggests that the chemical potential of the weakly doped cuprates should not be associated with the broad peak but must be pinned by invisible real quasiparticle. The above conclusion was lately confirmed in experiment [59] (see Figure 5). Chemical potential is pinned to the peak B while the energy of the broad peak A is far from the peak B.

Figure 5 

(a) ARPES spectrum of the undoped Ca2CuO2Cl2 in the nodal point 𝐤=(𝜋/2,𝜋/2) fitted by Lorentzian (dashed line) and Gaussian (thick solid line). The notations A and B define the maximum of the peak and the point where spectral density arises, respectively. Narrow peak in the vicinity of zero energy is the ARPES spectrum of SrRuO4. (b) Dispersion of A and B. The positions of the chemical potentials for different samples are shown by the horizontal lines. (c) Experimental ARPES spectra in the vicinity of the nodal point, after Shen et al. [59].

3.4. The Values of λ

One of the most detailed study of the strength of the EPI was undertaken by several methods in [62]. For the explanation of the large linewidth of the ARPES spectra in the undoped La2CuO4 the authors of [62] use the Hamiltonian obtained from the reduction of the three-band model. The parameters of the model were such that they give correct description of the phonon spectra in the realistic model of the La2CuO4 with 21 phonon branches [61]. The EPI in the realistic model with 21 phonon branches reads 𝐻_ep=1√𝑁_𝐪𝜈𝐢𝑔_𝐪𝜈(1−𝑛_𝐢)2𝜔_𝐪𝜈𝑄_𝐪𝜈𝑒^{𝑖𝐪𝐑𝐢}. (9) This Hamiltonian describes the interaction with the empty places forming singlet in the t-J model. Interaction is linear with respect to the lattice coordinates 𝑄_𝐪𝜈 and coupling constants 𝑔_𝐪𝜈. The phonon mode of frequency 𝜔_𝐪𝜈 is defined by its wave vector 𝐪 and phonon branch index 𝜈. 𝑛_𝐢 is the electron filling of the site 𝐑_𝐢 and the total number of sites is 𝑁.

Effective EPI constant is defined in [62] as 1𝜆≡𝑁_𝐪𝜈|𝑔_𝐪𝜈|²4𝑡𝜔_𝐪𝜈. (10) The value of 𝜆 at 𝑡=0.4 eV is estimated as 𝜆=1.2. Authors of [62] calculated the differential spectral distribution of the effective EPI 1𝛾(𝜔)=𝑁_𝐪𝜈|𝑔_𝐪𝜈|²𝜔_𝐪𝜈𝛿(𝜔−𝜔_𝐪𝜈). (11) The spectral distribution (11) is compared with the fine structure of renormalization of the dispersion in the vicinity of kink measured in [128]. Good agreement of these two data encourages to believe into the phononic nature of the kinks and ensures that the estimate for the effective 𝜆 is correct.

To estimate the value of effective EPI from the difference of the energy of the real quasiparticle and Franck-Condon peak authors of [62] introduced the scaling 𝑔_𝐪𝜈→Λ𝑔_𝐪𝜈 of the coupling constant in the (10). The theoretical binding energy for Λ=1 is 1.2 eV which is considerably larger than the experimental value 0.5 eV. The binding energy is proportional to Λ² and even small overestimate of the coupling constants, originating from the underestimate of the screening in the shell model, can lead to strong overestimate of the binding energy. Reasonable values of the binding energy and linewidth can be obtained for Λ=0.8 which corresponds to 𝜆=0.75. Thus, one can conclude that EPI in the undoped cuprates ranges like 0.75<𝜆<1.2. These values are larger than 𝜆_𝑐=0.6 [46] which is required for transition of the tt′t′′-J-H model into the strong coupling regime.

Another source of the information about the strength of the EPI is the temperature dependence of the ARPES spectra. Comparison of the experimentally measured temperature dependence [177] and that calculated from the t-J-H model [50] gives the estimate 0.5<𝜆<1.0. Comparison of the experimental and theoretical data for high-energy part of the ARPES spectra gives the same values. The waterfalls, observed in the range 1 to 2 meV [178–182], were reproduced by artificial broadening of the string resonances of the t-J model [183, 184]. Consecutive search for the physical mechanism for the artificial broadening led to conclusion that the broadening caused solely by nonzero temperature is not enough to explain the linewidth and has to take into account rather considerable EPI [63] with 𝜆=0.65.

Another method to get the value of 𝜆 is to compare the theoretical growth of the linewidth when the energy of quasiparticle exceeds the frequency of the dispersionless phonon with that measured above the kink energy. Such analysis for experimentally measured ARPES spectra in the electron doped compounds Sm2−𝑥Ce𝑥CuO4 (𝑥=0.1,0.15,0.18), Ng2−𝑥Ce𝑥CuO4 (𝑥=0.15), and Eu2−𝑥Ce𝑥CuO4 [145] gives the estimate 𝜆=0.8.

Recently developed method of time resolved electron diffraction is capable of studying the time evolution of the excited states [185]. Comparison of the experimental data with the results obtained in the framework of the theory of relaxation in metals [186] led to conclusion that anisotropic 𝜆 is in the range 0.08<𝜆<0.55. And last, comparison of the isotope shift of the OC of the undoped compounds [187] with theoretical results [188] gave the estimate 𝜆≥1 [189].

3.5. Isotope Effect in ARPES

Experimentally, isotope effect (IE) was studied only for highly doped compounds and the main emphasis was made on studies of the kinks. The ARPES spectra data were measured in compounds where oxygen was represented by 16O and 18O isotopes. The first results for IE on the kinks in ARPES spectra [110, 190] received a lot of skepticism caused by the structure of the isotope shift. The shift was observed in the high-energy region beyond the phonon energy.

Further experiments were in contradiction with the results obtained in [110, 190]. Measurements [111] did not observe the IE at large energies. It was also noted that even tiny sample misalignment of 0.1 degree can lead to considerable shifts at large energies. Also, measurements presented in [111] noted the general shift of the spectral features by 3 meV which is consistent with results of tunneling experiments [191, 192].

The most detailed and precise analysis of the IE on ARPES spectra was presented in [113]. IE on ARPES was observed only in the vicinity of the kink and the shift of spectral feature around 3.4±0.5 meV (see Figure 6) was found. Extremely high precision of the measurement done in [113] arises from the novel method of low-energy ARPES with resolution much higher [193] than that typical for conventional ARPES technique. Calculations show that the breathing mode at the energy 70 meV has to be shifted by the IE by 3.9 meV which is in perfect agreement with experimental data.

Figure 6 

(a) Real part of the self-energy ReΣ(𝜔) from five samples both for 16𝑂 (lower curves) and 18𝑂 (upper curves). All ReΣ(𝜔) are deduced by subtracting a bare band dispersion from the experimental one. (b) Imaginary part of the self-energy ImΣ(𝜔) determined from the full width of the momentum distribution curves. (c), (d) Obtained kink energy as a function of sample numbers both for 16𝑂 (to the left) and 18𝑂 (to the right) from ReΣ(𝜔) and ImΣ(𝜔), after Iwasawa et al. [113].

There are a lot of studies of the IE on the polaronic systems. Holstein model was studied in [194–197] by the path integral Monte Carlo method [198] and Dynamical Mean Field Theory (DMFT) method [199]. The change of the effective mass by the isotope substitution is the largest in the intermediate coupling regime. The IE in the Holstein-Hubbard systems was studied in [200–202].

The IE on the ARPES spectra for three-band model was studied in [132]. The interplay of EPI and electronic correlation is neglected in this paper but, however, the usage of the realistic model leads to many new and interesting conclusions. In agreement with experiment [113] the maximal IE is observed in the vicinity of the phonon frequency. It was also concluded that the kink cannot be caused by the electronic subsystem alone.

Nowadays it is important to attempt experiments to study IE on the ARPES spectra of undoped compounds. In addition to the high-temperature superconductors, there are other systems where broadening of the ARPES peak by EPI can be considered as a possible scenario. The list of such systems includes diatomic molecules [203], manganites with colossal magnetoresistance [204], quasi-one-dimensional Peierls conductors [205, 206], and Verwey magnetites [207].

Theoretical study of the IE on ARPES spectra of undoped system, specific for high-temperature superconductors because of choice of the tt′t′′-J-H model, was done in [46]. Dimensionless coupling constant 𝜆=𝛾²/4𝑡Ω does not depend on the isotope mass in case when the relation between phonon frequency and mass Ω∼1/√𝑀 holds. The isotopic factor is defined as 𝜅_iso=Ω/Ω₀=𝑀₀/𝑀. Parameters for tt′t′′-J-H model were chosen to reproduce the experimental ARPES spectra dispersion [6]: 𝐽/𝑡=0.4, 𝑡^′/𝑡=−0.34, and 𝑡^′′/𝑡=0.23. Phonon frequency [5] is Ω₀/𝑡=0.2 and the isotopic factor for change from 16O to 18O is 𝜅_iso=16/18.

To avoid errors the ARPES spectra in [46] were calculated for nonsubstituted compound (𝜅_nor=1), actual isotope substituted compound (𝜅_iso=16/18), and imaginary anti-isotope substituted compound (𝜅_ant=18/16). Monotonic dependence of the spectral function and other parameters on 𝜅 ensures in the absence of instability. The errorbar of a number 𝒜 can be evaluated using values 𝒜_iso−𝒜_nor, 𝒜_nor−𝒜_ant, and (𝒜_iso−𝒜_ant)/2.

Figure 7 shows the IE on the low-energy part of ARPES spectra for different EPI couplings in the nodal and antinodal points. All spectral features are shifted to larger energies for larger mass (𝜅<1). The shift of the broad Franck-Condon peak is considerably larger than the shift of the narrow peak corresponding to the real quasiparticle. Moreover, for larger values of 𝜆 the shift of quasiparticle energy goes to zero and the only influence of the IE is the decrease of the spectral weight 𝑍 with the increase of the isotope mass. To the contrary, shift of the Franck-Condon peak is not suppressed by the increase of EPI. In all cases, except the spectral function in the nodal point at 𝜆=0.62 (see Figures 7(a) and 7(b)) where weight 𝑍 of the quasiparticle is still large, there is general feature of the IE on the broad Franck-Condon peak. The height of this peak increases with increase of the isotope mass. Taking into account the sum rule ∫^+∞_−∞𝐿_𝐤(𝜔)=1, very small weight 𝑍 of the quasiparticle peak, and the complete absence of IE on the hight energy spectrum [46] one can conclude that the Franck-Condon peak decreases its width with the increase of the isotope mass.

Figure 7 

Low-energy part of the spectral function. Spectra of normal, isotope substituted, and “anti-isotope’’ substituted compounds are drawn by solid, dotted, and dashed lines, respectively. (a), (c), (e) and (g), (i), (l) correspond to nodal (𝜋/2,𝜋/2) and antinodal (𝜋,0) points. Insets (b), (d), (f), (h), and (k) show the peak corresponding to real quasiparticle.

The spectral function of the independent oscillator model is the Poisson distribution [208]: 𝜉𝐿(𝜔)=exp[−⁰𝜅]^∞_𝑙=0[𝜉₀/𝜅]^𝑙𝒢𝑙!^𝜅,𝑙(𝜔). (12) Here 𝜉₀=𝛾²₀/Ω²₀=4𝑡𝜆/Ω₀ is dimensionless coupling constant for initial system, and 𝒢_𝜅,𝑙(𝜔)=𝛿[𝜔+4𝑡𝜆−Ω₀𝜅𝑙] is a 𝛿-function. For strong EPI limit the exactly solvable independent oscillator model is very helpful to describe the IE on ARPES spectra because the most of properties of the numeric solution are very close to the properties of the spectrum described by (12).

3.6. Temperature Dependence of ARPES

The unique feature of the undoped and underdoped compounds with high-temperature superconductivity is the strong interplay of the lattice and magnetic degrees of freedom. One of the most vivid consequence of this interplay is the anomalous temperature dependence of the ARPES spectra. Really, it was realized long ago that the very scale of the experimental temperature dependence is considerably larger than that predicted by the conventional polaron theory [60]. The magnetic subsystem alone is also a wrong candidate [114] to describe the experimental anomaly since the typical energy scale of the magnon energy is ∼2𝐽≈0.2 eV which is even larger then the typical phonon energies ∼Ω₀≈0.04 eV. Recent studies revealed one more puzzling feature questioning the polaronic scenario in general. The temperature dependence of the width of the broad Franck-Condon peak is linear in the range 200K<𝑇<400K [177] and extrapolates to zero at zero temperature.

Previously, the ARPES spectra of the t-J-H model were studied by exact diagonalization method [165, 209, 210], in the noncrossing approximation [166, 167], and by DMC method [42, 46]. All these calculations were done for the case of zero temperature. All the above methods have their own difficulties preventing their reliable and fast generalization to finite temperatures.

A novel method capable of obtaining reliable data on the ARPES spectra of the t-J-H model at finite temperature was developed in [50, 211]. The main difficulty for studying of the t-J-H model lies in the profound difference between interactions of a hole with phonons and magnons. The Hybrid Dynamical Momentum Average (HDMA) method, developed in [50], just using difference of two interactions treating them with different techniques. The HDMA method unites advantages of classical Momentum Average (MA) method [212–221], which keeps information on the dispersion of the bare quasiparticle, and self-consistent Dynamical Mean Field Theory approximation (DMFT) [222–232], which is capable of treating strong but local interactions nonperturbatively.

Although the energy scales of bosons and phonons are basically the same, the nature of interactions of the hole with magnons and phonons is considerably different. The interaction with magnons is essentially momentum dependent and always weak. Really, spin 1/2 cannot be flipped more than once limiting, thus, the maximal number of magnons on one site to one [233]. Hence, for small enough values of 𝐽/𝑡 [68, 171] NCA is a good approximation. To the contrary, interaction with phonons is local and can be very strong. Thus, one cannot use NCA for phonons since it fails already for intermediate couplings [42]. Hence, one is to treat magnons by a weak coupling method which is able to handle the momentum dependence carefully and sum the phonon variables by a local nonperturbative method.

Nonperturbative approaches which neglect the momentum dependence of self energy on momentum are MA and DMFT methods. In both cases, the hole self-energy is expressed in terms of continuous fraction: Σ_h-ph𝛾[𝛼(𝜔)]=²𝛼(𝜔−Ω₀)1−2𝛾²𝛼(𝜔−𝜔₀)𝛼(𝜔−2Ω₀)1−3𝛾²𝛼(𝜔−2Ω₀)𝛼(𝜔−3Ω₀).1−⋯ (13) Then, DMFT and MA differ in the definition of the function 𝛼(𝜔) which is determined from the self-consistent procedure in DMFT and obtained from the momentum average of the Green function in MA. Suggested in [50] approach introduces the self-energy as a sum of contribution form magnetic and phononic subsystems: Σ_tJH(𝐤,𝜔)=Σ^SCBA_h-mag(𝐤,𝜔)+Σ_h-ph[𝛼_tJH(𝜔)]. (14) Weak and anisotropic interaction with magnons is treated in NCA: Σ^SCBA_h-mag(𝐤,𝜔)=_𝐪𝑀²_𝐤,𝐪𝜔−𝜔_𝐪−Σ_tJH(𝐤−𝐪,𝜔−𝜔_𝐪,)+𝑖𝜀 (15) while the function 𝛼(𝜔) for the phonon part of the hole self-energy 𝛼_tJH1(𝜔)=𝑁_𝐤1𝜔−Σ^SCBA_h-mag(𝐤,𝜔)+𝑖𝜀 (16) is expressed in terms of momentum-averaged bare Green function whose momentum dependence is determined by magnetic self-energy (16) obtained in NCA. Usage of MA instead of DMFT in (16) is crucial since the DMFT even does not distinguish even between t-J𝑧 and t-J model [223]. To the contrary, results obtained from (13)–(16), as it is shown by comparison with DMC data [42], correctly reproduce not only the ground state properties but the spectral function too.

The advantage of the suggested in [50] approach is that it is easily generalized to finite temperatures. Transforming magnetic self-energy Σ^SCBA_h-mag(𝐤,𝜔) [208, 234] to the Matsubara form, one gets Σ^SCBA_h-mag(𝐤𝜔)=_𝐪𝑀²_𝐤,𝐪(1+𝑛_𝑏(𝜔_𝐪))𝜔−𝜔_𝐪−Σ_tJH(𝐤−𝐪,𝜔−𝜔_𝐪)+𝑖𝜀+_𝐪𝑀²_{𝐤+𝐪,𝐪}(𝑛_𝑏(𝜔_𝐪))𝜔+𝜔_𝐪−Σ_tJH(𝐤+𝐪,𝜔+𝜔_𝐪,)+𝑖𝜀 (17) where 𝑛_𝑏(𝜔) is the Bose distribution. To generalize the phonon self-energy (13) to finite temperatures one uses the momentum-independent relation [225, 226] Σ_h-ph[𝛼(𝜔)]=𝛼⁻¹(𝜔)−^∞_𝑛=0(1−𝑥)𝑥^𝑛𝛼⁻¹(𝜔)−𝐴_𝑛(𝜔)−𝐵_𝑛.(𝜔) (18) Here 𝑥=exp(−𝛽Ω₀), and 𝐴_𝑛(𝜔) and 𝐵_𝑛(𝜔) are known functions depending on the coupling constant and frequency [50]. Also, (14), (16), (17), and (18) are solved self-consistently. The spectral functions obtained from these equations obey three first sum rules for any coupling constants and temperatures [212, 220]. Hence, the peak position and its width should be reliable.

Temperature dependence of the spectral function in the ground state 𝑘=(𝜋/2,𝜋/2) for different 𝜆s is shown in Figure 8. The general trends are in agreement with experiments [60, 109, 177, 235]. Distance of the Franck-Condon peak from the chemical potential and its width increases with the increase of temperature. Note that peak width is a constant up to the temperature 𝑇≃Ω₀/2≈200 K and then show a linear temperature dependence (see Figure 8). This linear dependence, if no data for low temperatures, can be indeed extrapolated to wrong low temperature values. However, in accordance with recent experiments [60, 236], the linewidth saturates at temperatures 𝑇≤Ω₀/2 to a constant value. Note that the slope of the temperature dependence for 𝑇>Ω₀/2 does not depend on 𝜆 whereas the saturation values of the linewidth for 𝑇<Ω₀/2 are very sensitive to the strength of EPI.

(a)

(b)

Figure 8 

(a) Spectral functions for different 𝜆 and temperatures 𝛽=𝑡/𝑇. (b) Peak width Δ𝜔 as a function of temperature 𝑇 for the t-J model (𝜆=0) and t-J-H model in the strong coupling EPI limit (𝜆=0.462 and 𝜆=1). Temperature is in Kelvins under assumption that 𝑡=0.4 eV.

Hence, temperature dependence of the Franck-Condon peaks in the low-energy (0.3–0.6 eV) part of the ARPES spectra manifests strong EPI. The evidence of the strong EPI was recently found in the high-energy ARPES too. It follows from the temperature dependence of high-energy “waterfall’’ features in ARPES [63]. The “waterfalls’’ at the energies 1-2 eV are the parts of the ARPES spectra where dispersion is parallel to the energy axis in the energy-momentum reference frame [178–182]. At first, the structure of “waterfalls’’ was reproduced by artificial broadening of the spectra of the t-J model [183, 184]. Then, an attempt to explain the artificial broadening by finite temperature and EPI was made in [63]. The method of self-consistent treatment of Dyson equation [68, 69, 237] was generalized to finite temperature and EPI is added. It is concluded [63] that the finite temperature alone is not enough to explain the effect and strong EPI is compulsory to explain the data.

Another explanation of the “waterfalls’’ was suggested in [238] where spectral density at large energies is attributed to contribution of localized states.

3.7. Multipeak Structure of the Optical Conductivity Mediated by EPI

The optical conductivity (OC) of the weakly doped cuprates has the following characteristic features [23, 24, 70]. The high-energy part of OC in undoped systems has the peak at 𝜔≅1.5 eV which corresponds to charge transition between p-orbitals of O and d-orbitals of Cu. Low-energy part of OC, caused by movement of holes, arises with doping. The Drude contribution at the lowest energy in proportional to the holes concentration 𝑥 and the relaxation rate of the Drude theory is proportional to temperature. Puzzling part of the OC is the mid infrared (MIR) peak with doping-dependent frequency around 𝜔_MIR≅0.5 eV whose interpretation is still far from adopted [25]. Theoretically the OC of t-J model was studied in the low density limit and a peak at 2𝐽≈0.28 eV was found [9–22] (for typical 𝑡=0.4 eV and 𝐽=0.35𝑡). The above energy is almost two times smaller than the low-doping experimental value 0.5 eV. Hence, to shift theoretical peak of the t-J model to higher energies, one can try to add EPI and consider t-J-H model.

However, OC of the t-J-H model was studied by ED [165], using NCA both for phonons and magnons [167], and by DMFT method which is exact in the infinite dimension limit [224]. NCA is not valid for strong EPI and small systems in ED method give too discrete spectrum. Beside, it is not known whether infinite dimension is a good approximation for the two-dimensional (2D) system. To circumvent the above difficulties, the OC was calculated in [51] by DMC method for 2D infinite system and without NCA in the phonon channel. Theoretical data were compared with experimental ones obtained by ellipsometry, Figure 9 shows OC (a) in Holstein model, (b) t-J model and (c) t-J-H model. Experimental result is in Figure 9(d). No model, except the t-J-H one, reproduces even the gross features of the experiment. The most striking feature is the two-peak structure of MIR band. There is an MIR peak at 𝜔_MIR=4600 cm−1 and a low-energy peak at 𝜔=1000 cm−1 which is situated just above the phonon lines at around 800 cm−1. Note that in the t-J model, the MIR peak is at 𝜔_t-J≅2𝐽≅2000 cm−1. However, EPI of t-J-H model shifts the peak to its experimentally observed frequencies.

(a)

(b)

(c)

(d)

Figure 9 

Typical OC of different models in two-dimensions and experimental data in weakly doped system: (a) Holstein model at 𝜆=0.44; (b) t-J model at 𝐽=0.3; (c) t-J-H at 𝐽=0.3 and 𝜆=0.39; (d) in-plane OC of doped by 𝑥=1.5% holes Eu1−𝑥Ca𝑥Ba2Cu3O6 at 𝑇=10 K. Theoretical energy scale assumes 𝑡=0.3 eV (1 eV = 8065.5 cm−1). The absolute values of the theoretical 𝜎1 estimated assuming that hopping distance in plane is 𝑎=3.86 Å and the bulk concentration of holes is 𝑛ℎ=1.72×10−23 cm−3.

It is shown in [51] that the lower peak of OC originates from the phonon-mediated scattering of the hole between the states located in the coherent t-J band and the threshold of the optical absorption corresponds to the phonon energy. The higher energy peak is a magnetic satellite of the lower peak. Note that its energy is higher in comparison with system where EPI is set to zero. Assuming strong coupling regime, when one can take advantage of the Franck-Condon scheme [48], one can explain this shift to higher energies as follows. The fluctuations of the energies on different sites of the lattice are around the Franck-Condon relaxation energy. On the other hand, these fluctuations are frozen during the characteristic time of optical transition. Hence, the energy of transition of the hole from the ground to excited state is the sum of the Franck-Condon energy and that of excited magnon.

Figure 10 shows how OC changes with increase of 𝜆. OC at weak coupling shows low-energy peak which intensity starts just above the phonon energy and there is apparent two-peak structure of the absorption. Actually, the two-peak structure was observed in experiment even in early measurements [23, 27, 115–120]. Especially apparent two-peak structure is seen in OC measured by ellipsometry in the weakly doped Eu1−𝑥Ca𝑥Ba2Cu3O6 (see Figure 9(d)). The low-energy peak is close to the phonon energy up to 𝜆≈0.4 when the system comes into the strong coupling regime [42] and for 𝜆>0.4 all peaks quickly enhance their energies (see Figure 11(a)). To reveal the nature of the low-energy peak the OC of the effective Holstein model was studied. In this model, the hopping integral was chosen 𝑡̃=0.4𝑡 to reproduce the enhancement of the mass in the t-J model. Besides, the crossover to the strong coupling regime occurs in the effective model at 𝜆≈0.4. The OC of such model well reproduces the low-energy part of the t-J-H model (see Figure 11(b)).

Figure 10 

Optical conductivity of a one hole in the t-J-H model calculated by DMC method at 𝐽/𝑡=0.3 for different 𝜆. Vertical line at 𝜔/𝑡=0.1 shows phonon energy.

(a)

(b)

(c)

(d)

Figure 11 

(a) Energies of the dominant MIR peak (solid line with circles) and low-energy peak (dashed line with squares) of the t-J-H model versus 𝜆. Energies of the dominant peaks of the Holstein model for 𝑡=1 (dashed line) and for 𝑡̃=0.4 (solid line). (b) OC of the t-J-H model (solid line) and effective Holstein model with 𝑡̃=0.4 (dashed line). Both OC are for 𝜆=0.1. (c) and (d) Ratio of the effective coupling constants on doping 𝑥 (or real concentration 𝑝 of the hole in plain for YBCO [244]) and that at almost zero doping. Effective 𝜆(𝑥) is determined by the fit of the MIR position at given doping.

Doping dependence of the kink angles in ARPES [64] suggests that the EPI strength decreases with doping and, hence, experimentally observed softening of the MIR energy with doping [25, 26, 28] is because of decrease of EPI. Theoretically, the decrease of the effective EPI with doping was found for a gas of Fröhlich polarons [239–242]. Comparison of the experimental position of the MIR peak with the results of the t-J-H model gives the estimate [51] for change of the effective EPI with doping (Figures 11(c) and 11(d)). The figures show the ratio of the coupling constant at the given doping to that at zero doping. However, since zero doping is characterized by the coupling constant 𝜆(𝑥=0)≈1 (see Section 3.4), one can think of the data in Figures 11(c) and 11(d) as showing the absolute value of 𝜆(𝑥). Note very similar behavior of 𝜆(𝑥) for different compounds. It is clear that the compounds become superconducting when the system is not already in the strong coupling regime of the EPI. However, the value of the EPI strength 𝜆≈0.5 is still considerable.

Alternative explanation of the doping dependence of the MIR energy by the doping dependence of the exchange integral 𝐽(𝑥) was suggested in [224, 243]. One cannot distinguish between 𝐽(𝑥)- and 𝜆(𝑥)-scenario just looking on OC. However, 𝜆(𝑥)-scenario easily explains experimentally observed strong dependence of the kink angle on concentration [64, 97, 121, 128]. To the contrary, 𝐽(𝑥)-approach [224, 243] does not give an explanation. Experimental data in electron doped Nd2−𝑥Ce𝑥CuO4 were interpreted as indicating that 𝜆 does not show doping dependence [145]. However, the range where 𝜆 is determined in [145] is too narrow to reach the final conclusion.

The polaronic scenario was used to interpret OC in cuprates many times [115, 242, 245–249]. However, all these cited papers discussed the one-peak structure and did not describe the complex two-peak structure of the experimental spectra. One of exceptions where two-peak structure is discussed is [250]. It is pointed out that the two-peak structure of OC occurs at some coupling constants in the Fröhlich model [41, 48]. There is also some range of coupling constants where two-peak structure of OC is seen in the Holstein model [251]. However, one needs fine tuning of the EPI strength to get the two-peak OC in the Fröhlich or Holstein model. To the contrary, the two-peak structure is the robust property of the experimental data and the OC of the t-J-H model. The authors of [250] used similar approaches [252, 253] to calculate the OC and ARPES spectra and found clear relation between the position of the peaks in ARPES and OC spectra.

Another interpretation of the low-energy peak of OC is given in [254], where the peak is considered to be of the purely magnetic origin. It seems that in the framework of the interpretation given in [254] the low-energy peak is connected with local magnetic excitations. Therefore, the nature of the low-energy peak is still under debate theoretically although experimentally it is a vivid feature of the OC of many cuprates with different doping levels [23, 27, 115–120].

3.8. Nonlocal EPI

The simplest model (3)–(7) does not reproduce all peculiarities of cuprates. The reason is that actually EPI vertex depends on momenta of both hole and phonon [61, 82, 85–90, 92, 93]. As a result, some properties of the t-J-H model do not match experimental data. For example, effective mass of a hole in the strong coupling regime is very large which is in contradiction with rather moderate masses of carriers observed in experiment [25]. Even more profound disagreement with experiment was found in [210]. It was shown there that in the strong coupling regime of the t-J-H model the mobile hole is transformed into the localized one with four broken bonds around it. In such case, the percolation model predicts that the antiferromagnetic model survives up to hole concentration 𝑥=0.5 which is in severe disagreement with experiment where antiferromagnetic phase is limited by the doping concentrations 𝑥≤0.02–0.04. Similar trend was noted in [255] where it was demonstrated that EPI helps to survival of the antiferromagnetism since EPI suppresses the motion of the holes which is necessary to suppress antiferromagnetism.

The minimal Hamiltonian of the t-t′-t′′-J model with nonlocal EPI consists of the sum of the Hamiltonian of the t-t′-t′′-J model, phonon Hamiltonian with phonon frequency 𝜔₀, and the Hamiltonian of nonlocal EPI 𝐻_h-ph=𝜔_0𝑙𝑔(𝑙)_𝑖𝜖𝐴𝑓^†_𝑖𝑓_𝑖(𝑐^†_𝑖+𝑙+𝑐_𝑖+𝑙)+𝜔_0𝑙𝑔(𝑙)_𝑖𝜖𝐵ℎ^†_𝑖ℎ_𝑖(𝑐^†_𝑖+𝑙+𝑐_𝑖+𝑙), (19) which is defined in terms of local coupling constant 𝑔(0)=𝑔 and nonlocal coupling to the displacements of near neighbors 𝑔(𝛿)=𝑔₁. However, 𝑓 and ℎ denote annihilation operators of two sublattices of the two-dimensional antiferromagnetic lattice.

The problem of nonlocal EPI in t-t′-t′′-J model was solved in [49] with a novel approach. In that approach, starting from the state of a hole in an antiferromagnet, one defines the states of the basis |ℎ⟩_𝑗[∏_𝑖|𝜇_𝑖⟩]|𝑞₁,…,𝑞_𝑙,𝑙⟩. Here |ℎ⟩_𝑗 is a hole at site 𝑗 and 𝑖 runs through the whole lattice. The set of states |𝑞₁,…,𝑞_𝑙,𝑙⟩ is limited by magnons which component of the 𝑙th order is enough to reproduce the results of NCA [256]. As it is shown in [49], 𝑙=4 is enough for 𝐽/𝑡≥0.3 since diagonalization of this 𝑙≤4 basis in the retraceable path approximation [257] gives results reproducing those of the NCA.

However, the hardest problem for the numeric solution comes from the phonon basis. Exponential growth of the phonon basis with coupling previously limited the system sizes to 10 sites [165]. The problem is circumvented by usage of the coherent states (CS) [258, 259] which are the canonical transformations of the phonon basis: |ℎ,𝑖⟩=𝑒^{𝑔ℎ(𝑏𝑖−𝑏†𝑖)}|0⟩^(𝑝ℎ)_𝑖=𝑒^{−𝑔2ℎ2∞/2}_𝑛=0(−𝑔ℎ)^𝑛√𝑛!|𝑛⟩_𝑖 (20) with free parameter ℎ. Such approach can treat 8×8 lattices which are the largest systems ever treated for t-J and related models which include coupling to phonons. CS with ℎ=0 is the bare state and for ℎ=1 it is the exact solution to the independent oscillators model with local EPI coupling 𝑔. A comparison with DMC data showed [49] that four CSs are enough for reliable treatment of t-t′-t′′-J model in all coupling regimes.

The tt′t′′J-H model was compared in [49] with the model where interaction with the displacements on near neighbors 𝑔₁=𝑔/2 (see (19)) was added. Figure 12 shows dependence on 𝜆/𝜆_𝑐 of the ground state spectral weight 𝑍, spin deviation SD, and magnon-mediated kinetic energy 𝐾_𝑡. Spin deviation SD=(𝑆_AFM−⟨𝑆_NN⟩)/𝑆_AFM is a measure of how the spin 𝑆_NN on the neighboring to the hole site deviates from the value of spin 𝑆_AFM in the ideal antiferromagnet. The decrease of 𝑍 is a measure of suppression of coherent motion of the hole whereas the absolute values of 𝐾_𝑡 and SD are the measures of intensity of the near neighbor hoppings. Decrease of 𝐾_𝑡 and SD indicates suppression by EPI of the movement of a hole on near neighbors.

(a)

(b)

(c)

Figure 12 

(a) Spectral weight, (b) spin deviation (b), and (c) average kinetic energy of mediated by magnons near neighbor transitions for 𝑔1=0 (diamonds) and 𝑔1=0.5𝑔 (crosses). Other parameters are 𝐽/𝑡=0.4, 𝜔0/𝑡=0.2, 𝑡′=−0.5𝑡, 𝑡′′=0.4𝑡, and 𝑁=4×4=16. Critical 𝜆𝑐 is set at such coupling where the spectral weight of the ground state reaches 0.3 of the unrenormalized by EPI weight.

Nonlocality of the EPI is manifested in the following features. Coherent motion is suppressed stronger at 𝜆<𝜆_𝑐 and weaker at 𝜆>𝜆_𝑐 (see Figure 12(a)). Also, intensity of the near neighbors hoppings is considerably weaker suppressed by nonlocal EPI than by the local one. The last statement is easy to illustrate in the strong coupling regime where one can think in terms of the adiabatic potential. The adiabatic potential in the strong coupling regime is a 𝛿-function giving no possibility of a motion of the hole over the near neighbors. To the contrary, for long-range EPI, the adiabatic potential is not steep and, thus, the motion of the hole over the near neighbors is not suppressed even in the strong coupling limit. Such mobility over the near neighbor gives possibility to destroy the antiferromagnetic state even in the strong coupling regime. Besides, the polaron with nonlocal EPI is lighter in the strong coupling regime. For example, for 𝜆=1.1𝜆_𝑐 the diagonal 𝑚_𝑑 (𝑘_𝑥=𝑘_𝑦) and transverse 𝑚_𝑡 (𝑘_𝑥=−𝑘_𝑦) masses for the local EPI are 𝑚_𝑑=200 and 𝑚_𝑡=88 whereas for nonlocal EPI the masses 𝑚_𝑑=20 and 𝑚_𝑡=10 are considerably lighter. Lighter mass for nonlocal EPI [98, 195, 260–265] or dispersive phonons [266–269] was observed in many models.

Common influence of long-range EPI and on site repulsion binds polarons into mobile inter-site bipolarons [270, 271]. For peculiar lattice structures such polarons can be very light [272].

Manifestation of the nonlocality of the EPI can be seen in the OC [4, 273]. For example, the peak in the OC of the Holstein model in the strong coupling regime is seen on the energy 2𝜖_𝑝, where 𝜖_𝑝 is the polaron binding energy. On the other hand, the peak of OC in the models with long-range EPI is on the considerably smaller energies [273]. It can be explained by the fact that OC is defined in terms of the current-current correlation function. The current operator in the polaronic models with near neighbor hopping corresponds to the transition of a quasiparticle to the neighboring site. The shifted electron in the Holstein model looses the energy 𝜖_𝑝 and leaves the excited phonons with the same energy on the initial site. For the long-range EPI the phonons are excited on the polaron site and on the neighboring sites which leads to considerably weaker lattice relaxation during transition to the neighboring site. Hence, the energy of the peak in OC for nonlocal EPI is smaller.

The peculiarities of the influence of nonlocal EPI is determined not only by the range but also by its fine structure. Comparison of buckling and breathing phonons is made in [274]. In the Hamiltonian: 𝐻_Br-Bu=𝛾_𝐢,𝜹(𝑏_𝐢,𝜹+𝑏^†_𝐢,𝜹)(𝑛_𝐢±𝑛_𝐢+𝜹), (21) where𝐢 is position of cooper and 𝜹=𝑥,𝑦 is the link direction. Plus (minus) corresponds to buckling (breathing) mode. It is shown [274] that the breathing mode suppresses the kinetic energy considerably stronger than it is done by buckling mode. Breathing phonon decreases the energy of one site and increases it on the neighboring one. To the contrary, the buckling mode changes the energy of the neighboring sites in the same direction and, hence, does not suppress the kinetic energy effectively. Studies of influence of the EPI structure on the properties of polarons can be found in [98, 194, 195, 262, 275].

4. Electronic Correlations and EPI

Manifestations of the EPI in the properties of particles and phonons are profoundly different although the first naive but correct impressions is that both elementary excitations become softer and broader. Let a hole (phonon) in a system without EPI has dispersion 𝜀(𝐤)=𝜖(𝐤) (𝜀(𝐤)=𝜔(𝐤)). In such system, experimentally observed response in ARPES (neutron scattering) is expressed in terms of spectral function 𝑆(𝐤,𝜔) having the form of a delta function 𝑆(𝐤,𝜔)=𝛿(𝜔−𝜀(𝐤)). EPI leads to additional self-energy part of the Green function of a hole (phonon) Θ(𝐤,𝜔)=Σ(𝐤,𝜔) (Θ(𝐤,𝜔)=Π(𝐤,𝜔)). With this self-energy, the spectral function, measured in experiment, reads 1𝑆(𝐤,𝜔)=𝜋|ImΘ(𝐤,𝜔)|[𝜔−𝜀(𝐤)−ReΘ(𝐤,𝜔)]²+[|ImΘ(𝐤,𝜔)|]². (22) Imaginary part |ImΘ(𝐤,𝜔)| determines the EPI-driven broadening.

There is a sum rule for phonon self-energy in the t-J model (in the 𝑈→∞ limit of Hubbard model) when charge fluctuations in the half-filled system are completely suppressed [83, 276]: 1𝜋𝑁_𝐤^∞_−∞|ImΠ(𝐤,𝜔)|𝑑𝜔≈𝛾²[2𝛿(1−𝛿)]. (23) Here 𝑁 is number of sites and 𝛿 is concentration of holes which is zero in half-filled system. Naturally, with no empty spaces in half-filled system there is no charge fluctuations at all and the phonons are left untouched. Increase of 𝛿 adds empty states and the charge fluctuations renormalize and broaden phonons. To the contrary, one does not need doping to see the manifestations of EPI in ARPES spectra. For 𝛿=0 the sum rule for the hole self-energy reads [276] 1𝜋⁰_−∞ImΣ(𝐤,𝜔−𝑖0⁺)𝑑𝜔=𝛾². (24) Thus, half-filled system does not suppresses EPI in the ARPES channel. The reason is that ARPES creates its own hole even in the half filled system. This hole is a charge whose fluctuations are not suppressed (see also [277]).

In general, 𝑈 suppresses EPI. Also, DMFT calculations show [278–280] that EPI in paramagnetic system is severely depressed by correlations. Antiferromagnetic state considerably enhances the role of EPI in comparison with paramagnetic system [281]. Doping suppresses the EPI contribution to the electronic properties. To the contrary, contribution of EPI into the phonon properties increases with doping (cf. (23)).

In any case, for small filling of the t-J model, one can conclude that in comparison with the Holstein model with small filling the influence of the EPI is larger for the t-J model. This result was verified by a number of calculations [42, 165, 282–284]. Comparing critical coupling of the t-J-H (3)–(7) model 𝜆^𝑐_t−J≈0.4 [42] with the critical coupling of the Holstein model with the same hopping 𝑡𝜆^𝑐_𝐻≈1, one concludes that the interaction of the hole with magnons makes the transition into the strong coupling regime faster.

For a hole in the bottom of the band the ratio of effective 𝜆 for the t-J model and 𝜆₀ for the Holstein model with bare mass 𝑚₀=1/(2𝑡) depends on numerous factors [4, 166, 171]: 𝜆𝜆₀≈4𝑍²₀𝑚_‖𝑚_⟂𝑚₀. (25) The decreasing factor 𝑍₀<1 arises because of the shift of the spectral density to higher frequencies. On the other hand, larger effective masses of the t-J model 𝑚_‖>𝑚₀ and 𝑚_⟂>𝑚₀ enhance the influence of EPI. The factor 4 arising because of 4-fold degeneracy of the ground state on the wave vectors (±𝜋/2,±𝜋/2) becomes sometimes decisive. At 𝐽/𝑡=0.2, one has 𝑍²₀=0.05 and 𝑚_‖𝑚_⟂=10𝑚₀. In this case there is doubling of the EPI, partly due to the factor 4. For large 𝐽/𝑡=2, the enhancement of EPI 𝜆/𝜆₀≈16 can be very large. One can conclude that the change of the role of EPI usually has no universal trends and very often determined by fine features of the model, dimensionality, filling, and so forth. For example, the role of EPI is suppressed by Coulomb repulsion in the ground state of one-dimensional Mott insulator [285] whereas its role in formation of the exciton spectrum is enhanced by the same Coulomb repulsion [286].

One of the factors influencing the role off EPI in cuprates is typical for cuprates inhomogeneity [287–290]. Moderate and even weak EPI can lead to dramatic changes of the properties of inhomogeneous electronic gas [291]. Interfaces and surfaces are another inhomogeneities enhancing the role of EPI [292]. Similar conclusions can be drawn from many numeric calculations [53, 293–298].

Current review is restricted to theoretical approaches considering strong correlations. The significant number of important results of band structure calculations, the incomplete list is [299–310], revealing large 𝜆≥1 value of EPI, are not discussed here because of space limitation.

5. Conclusions

This review presents a lot of evidences for the important role of the EPI in formation of the spectral properties of underdoped cuprates. Theoretical efforts to reveal the fingerprints of the EPI in spectral response went through the sequence of models with increasing complexity. The simplest models with short range EPI in the ideal lattice at zero temperature were the systems to start with. Then, the progress went through generalizations to finite temperatures, to nonlocal EPI, and to systems with imperfections.

The ultimate goal of theory is to describe the realistic situation in cuprates. Such goal requites further development of the numeric approaches since the realistic description of cuprates requires methods which are capable of describing an infinite systems with imperfections at finite temperature.

Acknowledgments

This review is supported by RFBR 07-02-0067a. The author acknowledges stimulating discussions with N. Nagaosa, N. V. Prokof'ev, B. V. Svistunov, Yu. Kagan, A. S. Alexandrov, G. De Filippis, V. Cataudella, A. Sakamoto, Z.-X. Shen, T. P. Devereaux, C. Bernhard, J. Zaanen, G. Sawatzky, K. M. Shen, J. Bonča, E. Cappelluti, S. Ciuchi, M. Berciu, T. Egami, J. T. Devreese, T. Tohyama, O. Gunnarsson, G. Sangiovanni, and C. Kim.