Evidence for precursor superconducting pairing above T c in underdoped cuprates from an analysis of the in-plane infrared response

We performed calculations of the in-plane infrared response of underdoped cuprate superconductors to clarify the origin of a characteristic dip feature which occurs in the published experimental spectra of the real part of the in-plane conductivity below an onset temperature T ons considerably higher than T c . We provide several arguments, based on a detailed comparison of our results with the published experimental data, con ﬁ rming that the dip feature and the related features of the memory function (a peak in M 1 and a kink in M 2 ) are due to superconducting pairing correlations that develop below T ons . In particular, we show that (i) the dip feature, the peak and the kink of the low-temperature experimental data can be almost quantitatively reproduced by calculations based on a model of a d -wave superconductor. The formation of the dip feature in the experimental data below T ons is shown to be analogous to the one occurring in the model spetra below T c . (ii) Calculations based on simple models, for which the dip in the temperature range from T c to T ons is unrelated to superconducting pairing, predict a shift of the onset of the dip at the high-energy side upon entering the superconducting state, that is not observed in the experimental data; (iii) the conductivity data in conjunction with the recent photoemission data (Reber et al 2012 Nat. Phys. 8 606, Reber et al 2013 Phys. Rev. B 87 060506) imply the persistence of the coherence factor characteristic of superconducting pairing correlations in a range of temperatures above T c .


Introduction
The possible persistence of some form of superconductivity, at least pairing correlations, many tens of K above the bulk superconducting transition temperature T c in underdoped (UD) cuprate superconductors belongs to the most vividly discussed topics in the field of high-T c superconductivity, for representative examples of related experimental studies, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Surprisingly high (up to about 100 K above T c ) values of the temperature T ons of the onset of an increase of coherence, presumably due to an onset of precursor superconducting pairing correlations, have been deduced from the data of the c-axis infrared response of UD YBa 2 Cu 3 O δ different from and possibly competing with superconductivity rather than by pairing correlations themselves. In this context, it is of high importance to identify manifestations of the increase of coherence below T ons in the inplane response, a quantity revealing aspects of the electronic structure complementary to those manifesting themselves in the c-axis response, and to ascertain their relation to superconducting pairing correlations.
It has been already shown [6] that the real part of the in-plane infrared conductivity of UD Y-123 changes at T ons in a way similar to that of an optimally doped (OPD) superconductor at T c : a dip-like feature with a minimum around − 400 cm 1 begins to form and this is accompanied by a pronounced spectral weight shift from the dip region to very low frequencies (below ca − 200 cm 1 ). The focus, however, has been on qualitative aspects of the relevant spectral weight shifts, and the related spectral structures have not been addressed. Here we concentrate on three prominent spectral features that develop below T ons : the onset of the dip at the high energy side, the corresponding peak in the spectra of the real part of the memory function and the corresponding kink in the spectra of the imaginary part of the memory function. Our analysis involves comparisons of the data of two representative UD cuprates with those of OPD ones and with results of our calculations employing approaches ranging from the Allen's theory to the fully selfconsistent generalized Eliashberg theory. It provides evidence that the features are due to superconducting pairing correlations, and the scale T ons due to a precursor superconducting pairing phase rather than to an ordered state unrelated to superconductivity.
Recall that T ons is significantly lower than T*, the temperature of the onset of the pseudogap as seen, e.g., in the planar copper Knight shift [26], in the c-axis conductivity [27][28][29][30], in the intrinsic tunnelling [31,32], in the angular resolved photoemission spectra of the antinodal region of the Brillouin zone [33] etc. For UD Y-123 with 10% doping > T* 300 K [6,7,28,29,34], while ≈ T 170 K ons [6,7]. The pseudogap region of the phase diagram of the cuprates located between the T c line and the T* line involves two distinct types of electronic correlations: the correlations setting on at T*, on the one hand, cause a depletion of the electronic density of states at low energies and they seem to compete with superconductivity [30]. The correlations setting on at T ons , on the other hand, bring about an increase of coherence of the electronic states manifesting themselves both in the out-of plane and in the in-plane response [6].
The paper is organized as follows. In section 2 we summarize the relevant aspects of the experimental infrared data. As examples we use the published data of UD HgBa 2 CuO δ + 4 (Hg-1201) [35] and Y-123 [6,36]. At the end of the section, the data are compared with the published ones of OPD cuprates. In section 3 we present and discuss results of our calculations aiming to clarify the origin of the spectral structures. First (in subsection 3.1), we focus on the spectra calculated using the Eliashberg theory. We put emphasis on the similarity between the low-temperature (T) data and the low-T calculated spectra and on the similarity between the onset of the spectral structures below T ons in UD cuprates and that below T c in the calculated spectra. In subsection 3.2, the data are discussed in terms of the frequently used extended Allen's theory (EAT) [36][37][38][39]. It is highlighted that the assumption of a superconductivity unrelated gap in the density of states in the temperature range < < T T T c ons leads to inconsistencies with the data. In particular, we show that it leads to a clear shift of the high energy onset of the dip upon entering the superconducting state, that is not observed in the experimental data. This applies also to simple models, where the dip in the temperature range < < T T T c ons is attributed solely to an electron-boson coupling and the temperature dependence of the Fermi and Bose factors. Finally, in subsection 3.3, we present and discuss results of our calculations of the optical spectra, using as inputs recently published properties of the quasiparticle spectral function obtained from the photoemission data by means of analyzing the tomographic density of states [10]. A short summary and conclusions are given in section 4.

Relevant aspects of the published experimental data
The dip feature in the spectra of the real part σ 1 of the infrared conductivity σ will be illustrated with the recently published data of UD HgBa 2 CuO δ + 4 (Hg-1201) with 10% doping and = T 67 K c [35]. Also discussed will be the related structures of the memory function 0 pl 2 [40] and connected to the so called optical selfenergy Here ω pl is the plasma frequency. The data of UD Hg-1201 of [35] are similar to the earlier published ones of UD Y-123 [6,36], but a much more detailed temperature dependence is reported and some spectral features are sharper than in Y-123. Figure 1 shows, in part (a), a selection of the spectra of σ 1 of Hg-1201 from figure 1 of [35], and, in parts (b) and (c), the corresponding spectra of M 1 and M 2 from figure 4 of [35]. The low-temperature spectra of σ 1 contain the δ-peak at ω = 0 (not shown in the figure) corresponding to the loss-free contribution of the superconducting condensate, a narrow low-energy component corresponding to residual quasiparticles, and a deep and broad minimum around 30 meV (see, e.g., the 10 K line in figure 1(a)). This is followed by a broad region of an approximately linear increase terminating at ≈ E 140 meV g , where the slope of the spectrum changes, see the arrow in figures 1(a) and figure 2 . This change of slope will be called the high energy onset of the dip (HEOD). It is associated with a local maximum at a slightly higher energy. The HEOD is further accompanied by a characteristic peak in the spectra of M 1 with an onset feature at the high energy side (around E g ), and a kink in those of M 2 , see the arrows in figures 1(b) and (c). All the three features are well known from earlier infrared studies [28,36,38,39,41]. What has, however,-to the best of our knowledge-not yet been recognized, is the fact that they start to form close to the temperature T ons of [6].  This will be discussed below. We focus on σ 1 first. The 70 K and 120 K spectra in figure 1(a) display a clear HEOD at an energy close to E g , see also figure 2 . The 250 K spectrum, on the other hand, does not display any such feature in the relevant spectral range. It can be seen in figure 1 of [35], see also figure 2 , that the dip feature vanishes at a temperature higher than 160 K but ⩽200 K, close to the maximum T ons of [6] (180 K). Based on this finding and the earlier observations of [6] we assume that the temperature scale of the dip coincides with the earlier established T ons of [6] and T p of [7] and denote it by T ons . The T-dependence of the characteristic peak in the spectra of M 1 is similar: at 70 K and 120 K it is still very pronounced and almost at the same location as at low temperatures, at 250 K a clear maximum is not evident (see also figure 4 of [35]; the remaining very weak band in the 90-130 meV range seems to be of different origin, perhaps related to the sharp structure at ca 80 meV). Note that the onset feature at the high energy side of a similar peak occurring in the case of UD Y-123 has already been reported to vanish around 170 K [36], see discussion of figure 7 in [36]. Next we address the T-dependence of the kink in the spectra of M 2 . At low temperatures the feature is very sharp (note that the 10 K spectrum overshoots the 70 K one). It gets less sharp around T c but persists, at approximately the same energy, up to much higher temperatures, see figure 4 of [35]. In the 120-200 K range, it transforms into a smoother onset at a lower energy due to the presence of the relatively narrow low energy component of σ 1 , that can be seen in figure 2 , rather than to the HEOD. Note finally that the onset of the features is very likely connected with a transition to a purely quadratic T-dependence of the dc resistivity occurring slightly above 200 K [35] and with the appearance below about 200 K of a Fermi-liquid like scaling of ω M ( ) [35]. In OPD materials, a dip feature in σ ω ( ) 1 terminated by a local maximum, a peak in ω M ( ) 1 and a kink in ω M ( ) 2 , similar to those discussed above, set in at T c (or very slightly above T c ). For representative examples, see [42][43][44][45]. These features of OPD materials are clearly caused by superconductivity, since they develop in parallel with the formation of the loss-free contribution to σ ω ( )of the superconducting condensate, involving the δ-peak at ω = 0 in σ ω ( ) 1 : the δ-peak collects the spectral weight lost by the formation of the dip. It appears, that the three features of OPD materials, that set in at T c and are due to superconductivity, continuously transform into those of the UD ones, setting in at T ons . This is a strong phenomenological argument in favor of the superconducting pairing-based interpretation of the three features and the precursor superconducting pairing based interpretation of the T ons scale.

Results and discussion
3.1. Response functions calculated using the Eliashberg theory, comparison with the data Recall that many properties of OPD cuprate superconductors, including superconductivity, can be (at least qualitatively) understood in terms of (various versions of) the model, where charged planar quasiparticles are coupled to spin fluctuations. For reviews, see [46][47][48][49][50][51]. In particular, the low-T spectra of σ 1 in the superconducting state have been well reproduced [52][53][54][55] using a phenomenological version of the model, where the spin fluctuation spectrum χ ν q ( , )is approximated by a narrow (in ν) mode centered at the frequency of the magnetic resonance. By including a continuum contribution to χ ν q ( , )and performing the calculations at the fully selfconsistent generalized Eliashberg level, almost quantitative agreement with the data can be achieved [56], see also [57]. In the following we demonstrate that even the low-T experimental spectra of UD Hg-1201 can be approximately reproduced using the generalized Eliashberg approach formulated in [56]. In order to avoid any misunderstanding, we emphasize here that it is not the purpose of the present paper to advocate a very specific model. There are only two ingredients of the model, that are really essential for our conclusions: the superconducting order of d-wave symmetry and the relatively strong coupling between charge carriers and bosonic excitations (not necessarily spin fluctuations), whose spectral density is peaked around 40-50 meV. Other ingredients-the presence of a tail of the spectrum of the bosonic excitations ranging to high energies, the full selfconsistency etc.-help to reproduce the low temperature data quantitatively.
We begin with a brief description of the computational approach, details can be found in appendix A. The generalized Eliashberg equations (see, e.g., [58]) can be written as where Σ is the matrix selfenergy that can be expressed in terms of the Pauli matrices as 1 , 1 is the renormalized quasiparticle propagator, is the Matsubara counterpart of the the spin susceptibility, g is the coupling constant, ω i n and ν i m are the fermionic and bosonic Matsubara energies, respectively. The equations have been solved on the real axis using spectral representations of Ĝ and χ. We have used the same form of the input spin susceptibility as in our previous studies [55] and [56]. It contains a narrow (in frequency) mode centered at the frequency of the magnetic resonance and a component approximating the spin-fluctuation continuum. The corresponding formulas, the values of all input parameters, providing the values of T c and Δ max of 133 K and 45 meV, respectively, and some intermediate results are given in appendix A. Here Δ max is the maximum value of the superconducting gap of − d x y 2 2symmetry. The high value of T c is not surprising considering the observation of [59] that the value of the spin-fermion coupling constant providing the best fit of the nodal kink in UD Y-123 yields a high T c value of 175 K.
Finally, after obtaining the selfconsistent solution of the generalized Eliashberg equations, the optical conductivity was calculated using the formula Here d is the c-axis lattice parameter (d = 9.52 Å for Hg-1201, d = 11.65 Å for Y-123), N p is the number of CuO 2 planes within a unit cell (N p = 1 for Hg-1201, N p = 2 for Y-123) and ω is here in units of energy. The expressions K xx and Π ω ( ) xx in the numerator provide the diamagnetic and paramagnetic contributions to the optical conductivity, respectively. Both can be expressed in terms of the matrix quasiparticle spectral function The diamagnetic term is given by the following formula (exact) The paramagnetic term is given by the current-current correlation function and can be expressed (in terms of Â ) only approximately. The most frequently used approximation, employed also in our calculations, completely neglects vertex corrections and leads to the following formula for the imaginary part Π″ xx of Π xx : The real part Π′ xx of Π xx was obtained using the Kramers-Kronig transformation. Note that the expression on the right hand side of equation (3) includes the contribution of the condensate, the superfluid plasma energy ω pl,sc is given by In order to assess the validity of the approximation of equation (5), we have also evaluated Π xx using the gauge invariant approach of [56], where an important class of vertex corrections is included. The corrections modify the spectra of Π″ xx only slightly, the only significant problem of the approximation being an underestimation of the total spectral weight at finite frequencies and Π | ′ | (0) xx by ca 8%. For the normal state, this leads to the presence of a small unphysical singular component. In order to avoid the problem, we have replaced . The corresponding value of the (total) plasma energy ω pl is 1.95 eV.
For future reference, we provide here also the formula for the real part σ ω ( ) 1 of σ ω ( ) xx resulting from equation (5): is the usual spectral function. The normal state version of the formula reads show the calculated spectra of σ 1 , M 1 , and M 2 , respectively. Note the agreement between the low-T spectra and the low-T data shown in figures 1(a)-(c). The structures of σ 1 and M 2 , in particular a weak onset around the energy of the resonance mode ω 0 (here 50 meV), and the HEOD of σ 1 and the corresponding kink of M 2 at ω Δ ≈ + 2 0 m a x , can be interpreted along the lines of the earlier theoretical studies [55,56]. For the convenience of the reader, the interpretation is summarized in the following paragraph. Figure 3(a) shows the 30 K spectra of σ 1 and M 2 from figures 1(d) and (f) and the corresponding spectra of the second derivative of ωM 2 , all in arbitrary units. The arrows indicate three important features, occuring also in the experimental spectra of OPD cuprates, whose interpretation was presented in [55,56]: (i) a weak onset of σ 1 and M 2 around the energy ω 0 of the bosonic mode (within the present context the magnetic resonance) [43,44,52,55,56]. (ii) A maximum of the second derivative of ωM 2 [60] at approximately ω Δ + 0 m a x [53,55,56]. (iii) A local maximum of σ 1 at E g with E g approximately equal or slightly above ω Δ + 2 0 m a x [43,44,54,56], associated with the HEOD, and the corresponding kink of M 2 . At the simplest relevant perturbative level (Bogolyubov quasiparticles of a d-wave superconductor coupled to bosons), the basic infrared active final states consist of two Bogolyubov quasiparticles and the bosonic mode, see figure 3(b). Final states consisting of two Bogolyubov quasiparticles are not infrared active in the clean limit. The features (i), (ii), and (iii) are due, as detailed in [55,56], to the appearance above the characteristic energies (ω 0 , ω Δ + 0 m a x , ω Δ + 2 0 m a x ) of final states consisting of (i) two near nodal (Bogolyubov) quasiparticles and the bosonic mode, (ii) a near nodal quasiparticle, an antinodal quasiparticle and the mode, and (iii) two antinodal quasiparticles and the mode. Above ω Δ + 2 0 m a x , the density of available final states saturates. A schematic representation of the three types of final states, associated with the features (i), (ii), and (iii) is presented in figure 3(c).
The relation (Bi-2212) by Homes and coworkers [64]. To conclude, the low-T infrared spectra of UD Hg-1201are consistent with a well established model of a d-wave superconductor.
Next we address the T-dependence of the model spectra. It can be seen in figures 1(d)-(f) that the three important features (the dip of σ 1 , the maximum of M 1 and the M 2 kink) develop below T c in the same fashion as in the data of OPD materials [42][43][44][45]. In particular, the σ 1 dip forms already close to T c . What we would like to highlight here is that not only the T dependence of the data of OPD cuprates but also that of the UD is similar to the model spectra, compare figures 1(a)-(c) with figures 1(d)-(f). Note that the development of the dip in the UD Hg-1201 below ≈ T 180 K ons is analogous to that of the model spectra below T c : somewhat below T ons /T c a characteristic change of slope appears in σ ω ( ) 1 , with decreasing T its energy approaches E g and the minimum at low energies and the characteristic maximum around E g are forming. In the model spectra, the energy scale of the dip develops more gradually than in the data. This difference is most likely due to the mean field character of the model. The similarity between the T-dependence of the data below T ons and that of the model spectra below T c provides a further support for the interpretation of the T ons scale in terms of a precursor superconducting pairing state, for which T ons corresponds to the mean field transition temperature.
Admittedly, the mean field model does not reproduce the low-energy part of the data above T c . The calculated spectra of σ 1 at 70 K, 100 K, and 120 K contain a δ function at ω = 0 (not shown), which is not present in the corresponding experimental spectra. In addition, the calculated spectra of M 2 at 70 K, 100 K, and 120 K display a more gradual increase with increasing frequency than the experimental ones, that satisfy the Fermi-liquid like scaling, 2 and p = 1.5 [35]. In our opinion, these discrepancies are due to the presence of strong fluctuations which suppress the long range phase coherence in the temperature range from T c to T ons . An adequate microscopic description of the spectra in this temperature range must include the fluctuating condensate and/or a contribution of uncondensed pairs, and the coupling between the former and the thermally excited quasiparticles. Note, however, that in the frequency range around the HEOD, the impact of the fluctuations on the spectral features can be expected to be of less importance. For completeness, the T-dependence of the contribution of the condensate/fluctuating condensate in UD cuprates will be qualitatively discussed below. At low temperatures, the response of the condensate is represented by the δ-peak at ω = 0 in σ ω ( ) 1 . When going from T = 0 to T c , the spectral weight of the δ-peak decreases, which is compensated by the filling in of the gap [65,66]. At T c , the δ-peak vanishes but the superfluid does not, merely its long range phase coherence has been lost [67]. Simultaneously, the superfluid acquires a finite phasecorrelation time τ, probably due to a random motion of unbound thermally excited vortices, and the δ-peak is replaced with a Drude peak of finite width τ ∼1 [68,69] that cannot be easily distinguished from the response of thermally excited quasiparticles.
For our considerations on the origin of the three spectral structures, it is very important that the THz experiments [68] have clearly confirmed the presence of a fluctuating condensate at T c and at temperatures slightly higher than T c . For a slightly UD epitaxial film of Bi-2212 with = T 74 K c , the reported value of the fluctuating condensate density at T c ( figure 4 of [68]). Based on this observation only, it can be expected, with a high degree of certainty, that the more strongly UD Hg-1201 sample of [35] will be, at temperatures not much higher than = T 67 K c , e.g., at 70 K, in a fluctuating state with a considerable value of the fluctuating condensate density and the magnitude of the superconducting gap not very different from the low-T one. And, importantly, there is no clear qualitative difference between the 70 K experimental data of [35] and the 80 K, 90 K and 100 K ones.
When going from T c to higher temperatures, the spectral weight of the fluctuating condensate (i.e., that of the corresponding narrow Drude peak) decreases further [68], in parallel with the filling in of the gap. At the same time, τ decreases dramatically [68], so that for T above ca + T 20 K c , the contribution of the fluctuating condensate could not be distinguished from that of normal state electrons [68,69], see also [70]. It has remained an open question [71], whether at higher temperatures the fluctuating condensate is simply absent or merely unobservable by the THz techniques, while manifesting itself in the Nernst and diamagnetic response [1][2][3]. The data reported by Mirzaei and coworkers clearly show that the process of filling in the gap, very similar to that of OPD cuprates or that of a model d-wave superconductor (the similarity has been demonstrated in the present study), continues up to T ons , and the spectral weight comes from the lowest frequencies (ω < 20 meV). For these reasons we suspect that the partially condensed and/or uncondensed pairs persist up to T ons providing a low-energy component of σ ω ( ). This issue will be addressed again at the end of section 3.3. Finally, we note that the precursor phenomena discussed above may be strongly influenced by a spatial inhomogeneity of the superconducting order [4,72].
In our calculations, fluctuations of the condensate have not been included. There is no established way of doing this in conjunction with a retarded interaction, which is needed for a description of the infrared response. Previous attempts are limited to phenomenological models (e.g., [73]) and models involving a nonretarded interaction (e.g., [74]). Manske and coworkers [75] have used the fluctuation exchange approximation combined with results of the Kosterlitz-Thouless theory and thermodynamic considerations based on the Ginzburg-Landau theory to estimate the difference between T c and the mean field transition temperature (T c * ) and have confirmed the claim of [67] that in UD cuprates T c is considerably lower than T c * . This approach, however, cannot be easily extended towards calculations of the optical conductivity in the fluctuating state.
The calculations also do not include the pseudogap setting on at T* ('antinodal pseudogap'). Its opening can be expected to influence the in-plane conductivity in two different ways: (a) directly, by modifying the contribution to σ 1 of the antinodal region, that is approximately determined by the corresponding contribution to the sum over k on the right hand side of equation (8). (b) Indirectly, through a renormalization of near nodal quasiparticles, caused by a nodal-antinodal boson assisted scattering. The former contribution can be expected to be small since in moderately UD cuprates the pseudo-gapped region of the Brillouin zone around the antinode is relatively small [33,76] and, in addition, the magnitude of the corresponding vertex ( small. The mechanism (b) may influence the spectra significantly. The large difference between T* and T ons , however, indicates, that the dip feature in the spectra of σ 1 is not caused by the antinodal pseudogap. In the following subsection we further demonstrate, using the EAT, that the feature is unlikely to be caused by an order setting on at T ons and unrelated to superconducting pairing. (Bi-2212) have been previously successfully fitted and interpreted in terms of the phenomenological EAT [36][37][38][39]. This is based on two implicit assumptions: (i) the in-plane response is dominated by the contribution of near nodal quasiparticles; (ii) the dominant part of the renormalization of these quasiparticles comes from a nodalantinodal boson assisted scattering, the antinodal region being gapped. The essential inputs are a function α ν F ( ) 2 describing bosonic excitations and a density of states ω N ( )displaying the antinodal gap, whose physical origin is not specified. Here we provide two indications that the gap captured by the EAT based fits is likely related to superconducting pairing.
(i) We have fitted the Hwang's data of UD Y-123 reported in [36] using the formulas of the EAT, i.e., the Allen's formula [77,78]  where ω and ω pl are in units of energy, the formula for the imaginary part Σ 2 of the retarded selfenergy Σ proposed in [37] ∫ We have achieved a degree of agreement comparable to that of previous studies (e.g., that of [39]). In the above equations, ν b( ) and ω f( ) are the Fermi and the Bose functions, respectively. Our ansatz for α ν F ( ) 2 , the one for ω N ( ), and a detailed description of our fits can be found in appendix B.
The important findings are: (a) while the presence of the density of states gap is needed to achieve a high quality fit for temperatures well below T ons , it is not essential for temperatures higher than T ons . For the latter temperatures (and not for the former, see appendix C), the simple Allen's theory [77,78], as used in [79,80], appears to be sufficient. The characteristic temperature of the sharp gap of the EAT based fits is thus T ons rather than the pseudogap temperature T*. (b) The opening of the gap in the density of states causes a spectral weight shift from the dip region to low frequencies, the same effect as expected for a precursor superconducting pairing state.
(ii) Assuming that the physics at < < T T T c ons is unrelated to superconducting pairing, we arrive at a contradiction with the experimental data, as outlined below. Starting from the above assumption, the opening of the superconducting gap in the near nodal region below T c can be expected to give rise to a blue shift of the structures established above T c . This blue shift occurs indeed in the calculated spectra, see the discussion below, but is not seen in the experimental data. Figure 1(g) shows the normal (100 K and 67 K) and the superconducting state (67 K) spectra of σ 1 calculated using a version of the hybrid approach [55,81], that is described in the following paragraph.
The hybrid approach uses an approximate expression for the Nambu Green's function, involving the dispersion relation of charged quasiparticles ϵ k ( ), the superconducting gap Δ k , and a selfenergy correction 1 : where ω ω Σ ω = − k ( , ) , . In contrast to [55,81] (see also [49] and references therein), where Σ was obtained using a perturbative nonselfconsistent treatment of the coupling to spin fluctuations, we employ here the selfenergy resulting from our EAT based fits of the Y-123 data of [36]. The superconducting gap is approximated by the common d-wave ansatz cos cos (13) x y k max ⎡ ⎣ ⎤ ⎦ and the components of Σ by where Σ AD is given by equations (10), (11), with α ω F ( ) 2 from our fits of the 100 K and 67 K data and with the value of the gap depth of h = 1 (for the spectra of α F 2 , see figure B1(d), h is defined by equation (B.2)). The construction of Σ (1) ensures that the gap of ν A k ( , )is exactly equal to Δ k for ϵ μ = k ( ) . The in-plane conductivity σ ω ( )has been calculated using equations presented in subsection 3.1. The value of the plasma frequency ω pl,sc , however, has not been obtained by a direct computation of the expression on the right hand side of equation (6), but using the sum rule Here ω pl is the (total) plasma frequency, that has been approximated by In the superconducting state, the feature is clearly shifted to higher energies, and the same applies to the spectra of M 1 and M 2 shown in figures 1(h) and 1(i). This is, however, in contradiction with the experimental data, where there is no comparable shift. The relatively small magnitude of the shift in the calculated spectra as compared to Δ max is a consequence of the Brillouin zone averaging with a major contribution of the near nodal quasiparticles. Note that a similar shift between the above and below T c spectra, not occuring in the data, appears also in results of models, where the dip is solely due to an electron-boson coupling and the temperature dependence of the Fermi and Bose factors, for an example, see figure C1 of appendix C.

Optical conductivity calculated starting from the properties of the quasiparticle spectral function obtained by Reber and coworkers
Reber et al [10,11] have recently analyzed their photoemission data of OPD and UD Bi-2212 using the tomographic density of states. This has led them to the observation that the near-nodal gap evolves smoothly through T c and closes only at a temperature T close , that is for UD samples considerably higher than T c (see [11]) and presumably corresponds to T ons . This is a photoemission data based indication of the precursor superconducting pairing scenario. Here we demonstrate that the energy scales of the infrared data and those of the photoemission ones are consistent with each other. We further argue that the data, taken together, suggest the presence of superconducting pairing correlations in the T-range from T c to T ons . Figure 4 shows the spectra of σ 1 calculated using the type of parametrization of the quasiparticle spectral function, that has been used to fit the photoemission data in [10]: , with Ĝ given by equation (12), Δ k by equation (13), Σ by equation (14), AD ⎧ ⎨ ⎩ and Σ ′ AD by equation (11). The temperature dependence of Γ T ( )has been determined by the condition that , with no temperature dependence for simplicity. The last one is consistent with figure 4(a) of [10]. We have used the same dispersion relation as in section 3.2 and the values of N p and d corresponding to Y-123. The thin lines represent the spectra computed using the standard normal state formula, equation (8), not assuming any specific order. The thick lines have been obtained using the standard superconducting state formula, equation (7), with A (1) corresponding to the Nambu Green's function Ĝ introduced above. In both cases the sum over k entering the expression for σ 1 runs over the restricted region of the Brillouin zone that is shaded in the inset of figure 4 , i.e., over the region addressed in [10].It is clear from the data of [10], that this region is not significantly influenced by the antinodal pseudogap setting on at T*.
Note that the thin (artificial normal state) lines are shown merely to illustrate the role of the off-diagonal component of Â determining the coherence factor.
First of all, it can be seen that the HEOD occurs at an energy very close to that of the experimental infrared data of UD cuprates. Second, from the two blue lines, only the thick one is qualitatively consistent with the data for temperatures slightly above T c . The thin one displays a pronounced shoulder feature in the middle of the dip, that the experimental data do not contain. The energy of the structure is determined by Δ max , here it is located at ca 60 meV. The feature persists to some extent to higher temperatures-note the differences between the onsets at the high energy side of the low energy maximum of σ 1 in the thick lines and those in the thin lines. It corresponds to 'transitions across the gap'. What we mean is the following: the expression on the right hand side of equation (8) ) can be thought of as a sum of contributions of different 'transitions' between ν and ν ω + , for → T 0 between occupied states at negative energies and unoccupied states at positive energies. A pronounced contribution can be expected to originate from transitions between the broad peaks at the negative energy side and those at the positive energy side-the transitions across the gap. For a schematic representation of the transitions, see figure 5. In the superconducting state below T c , the contribution to σ ω ( ) 1 of the transitions across the gap denoted in figure 5 by the solid blue arrow ( → A A) is compensated by the contribution of the transitions denoted by the dashed blue arrow ( → A A (1) (1) , see the second term in the integral of equation (7)). This is a manifestation of the characteristic coherence factor [82] described here by A (1) . In the spectra calculated using the unrealistic normal state model for < < T T T c ons , the contribution of the transitions accross the gap denoted by the solid red arrow in figure 5 , is not compensated and yields the structure in the middle of the dip in figure 4.
The absence of the structure in the above T c experimental data indicates the persistence of the coherence factor of superconducting pairing correlations above T c . In fact, we are not aware of any ordered state not involving superconducting pairing that would be consistent with Reber's observations and at the same time exhibit a coherence factor consistent with the infrared data. For simple charge and spin density wave systems, the relevant coherence factor is such that the contribution of the transitions across the gap is enhanced rather than calculated using the hybrid approach and the same value of the gap magnitude and the same structure of the quasiparticle selfenergy as in [10]. Shown is the contribution of the region of the Brillouin-zone that is shaded in the inset. The thin (thick) lines represent results obtained with the off-diagonal component of the Green's function not included, as in standard normal state calculations (included, as in standard superconducting state calculations). fully suppressed. For a discussion of the coherence factors of density waves, see [82], for examples of conductivity spectra of density wave compounds displaying the Δ 2 peak due to the transitions, see [83][84][85]. A clear Δ 2 peak appears also in the calculated spectra corresponding to the d-density-wave model [86]. Fujita and coworkers reported recently a complex pattern of the charge distribution associated with the density wave occurring in UD cuprates [87]. To the best of our knowledge, possible consequences for the coherence factors have not been investigated thus far.
Finally, we would like to comment on the fate of the spectral weight of the transitions across the gap indicated in figure 5. In the true superconducting state the loss of this spectral weight caused by the presence of A (1) is compensated by the formation of the δ-peak at ω = 0. In the normal state, for < < T T T c ons , there is no long range phase coherence and therefore = A 0 (1) . The persisting pairing correlations, however, must provide a vertex correction to the right hand side of equation (8) playing a similar role as A (1) below T c , i.e., a correction partially reducing the spectral weight of the transitions across the gap. The lost spectral weight can be expected to reappear at low energies, in the contribution to σ 1 of collective degrees of freedom associated with the pairing correlations.

Summary and Conclusions
The published experimental data of σ ω ( ) 1 of UD high-T c cuprate superconductors (HTCS) display a clear dip feature below ca 130 meV, that starts to develop at > T T ons c . This dip feature gives rise to corresponding structures of the memory function. The features are similar to those of OPD HTCS, where they set in very close to T c and are commonly assigned to superconductivity and well understood in terms of the Eliashberg theory. This similarity and the one between the data and our calculated (Eliashberg) spectra strongly suggest that the dip feature of the UD HTCS is also due to superconducting pairing correlations and its persistence in the temperature range from T c to T ons due to the presence of a precursor superconducting pairing phase. In order to support this interpretation, we have further demonstrated that (a) the temperature dependence of the dip feature cannot be simply accounted for in terms of a normal state gap unrelated to superconductivity, or in terms of normal fermions coupled to bosons, and (b) that the infrared data taken together with findings of recent photoemission studies employing the tomographic density of states indicate the persistence of the coherence factor characteristic of superconducting pairing correlations in a range of temperatures above T c . Future studies should focus on clarifying the relation between the superconducting pairing correlations and the observed charge density waves and on the origin of the Fermi-liquid like scaling of ω M ( )in the temperature range from T c to T ons .
involving the renormalized quasiparticle propagator A direct solution of equation (A.1) in Matsubara energies has to be followed by a numerical analytical continuation of the selfenergy to the real axis which is an ill-posed problem. To avoid the numerical difficulties, one can employ the spectral representation of the quasiparticle propagator 2 Im{ˆ( i0, )}and also of the spin susceptibility By inserting the spectral representations in equation (A.1), the following expression for the imaginary part of the retarded selfenergy on the real axis can be obtained where ω f( ) and ν b( ) are the Fermi and the Bose functions, respectively. The corresponding real part of the selfenergy is calculated via the Kramers-Kronig transformation. The evaluation of the expression on the right hand side of equation (A.5) represents the most demanding part of the calculation due to the ν q, -summation for every (ω k , ). The computational effort can be greatly reduced by using the fact that the expression can be written as a difference of two convolutions of the form which can be efficiently evaluated using the fast Fourier transform algorithm. With the above definition, a compact expression for Σ″ reads We have solved the selfconsistent equations for the selfenergy iteratively starting with a BCS spectral function. The bare quasiparticles were described by the tight-binding dispersion with = t 380 meV and ′ = − t 120 meV, = Å a 3.828 . The same form of the model spin susceptibility containing the resonance mode and a continuum as in [55,56] was employed, . The main differences with respect to the input values of [56] are: the energy of the resonance ω 0 is slightly higher (40 meV in [56]), the 'coherence length' ξ C of the continuum is ca three times higher, and the dimensionless spectral weight of the resonance b M is twice as high. With these values of the parameters, the value of the coupling constant of = g 3 eV leads to = T 133 K c and Δ = 45 meV max . After every iteration, the chemical potential μ was adjusted to keep the electron occupancy at = n 0.85 el . Working with the spectral functions, n el is evaluated by using the formula For the sampling of the selfenergy, we have used a grid of 128 × 128 points in the Brillouin zone and the energy axis was discretized using 32768 points covering uniformly the energy range − + ( 4 eV, 4 eV). Figure A1 shows examples of our intermediate results: the real and imaginary parts of the quasiparticle selfenergy at the nodal point as functions of temperature.

Appendix B. EAT with a gap in the density of states
The aim of this section is to explore to what extent we can reproduce the experimental data of the in-plane conductivity in the temperature range < < T T T* c using the EAT. A special attention is paid to the experimental trend occurring below T ons . Our starting point is the fitting procedure developed by Hwang, Sharapov and Carbotte [37,39]. This formalism has been slightly modified so that it is somewhat more rigorous, involves less fitting parameters and also allows one to evaluate all important optical functions.
To describe the model we begin with the boson spectral function α F 2 . The Ansatz consists of a single peak described by two parameters A s and ω s α ω ω grows monotonically and shifts to lower frequencies with decreasing temperature, see figure B1 (d). The gap volume Δ h 2 3 pg , shown in the inset of (d), is equal to zero at 295 K and at lower temperatures displays a weak temperature dependence but with a significant variance of its value.
Closer look at the results for lower and higher temperatures reveals a clear difference in the role of the gap. In figure B2 we compare results obtained by the above described 2+1 parameter model with those obtained using the 2 parameter model with no gap in the density of states. In (a) we see that for higher temperatures the gap does not play any essential role. It only helps to improve the fit quality slightly. On the other hand in (b) we see that for Figure B1. (a), (b) Results of the fitting procedure described in the text applied to the experimental data of underdoped Y-123 reported in [36]. The data (gray lines) are presented for selected temperatures only. We can see that the quality of the fit is good for all temperatures studied. (c) The resulting spectra of σ 1 together with selected data. (d) The obtained temperature dependence of the function α F 2 . We see that with decreasing temperature the maximum of α F 2 grows and shifts to lower frequencies. The inset shows the temperature dependence of the gap volume Δ h 2/3 pg . At 295 K the best fit is with no gap in the density of states and thus the gap volume is zero. Below this temperature the gap volume shows a weak temperature dependence, but with a significant variance. lower temperatures the features in M 1 and M 2 , which are discussed in the main text, can be well reproduced only by the model including the gap in the density of states. This is shown in more detail in figure B3 , where we present the spectra of M 1 calculated using the α F 2 from the fit of the 67 K data (a) without and (b) with the gap in the density of states included. The results confirm that within the studied model the discussed low-temperature feature of M 1 is due to the gap in N.
In figure B4 we review some important characteristics of both types of fits-with and without the gap in N. The measure of the quality of the fit in (a) is given by the integral norm | − | M M num exp described above. We see that for higher temperatures both fits are of a comparable quality. For temperatures closer to T c , starting from ca 200 K, better results are achieved by the approach with a nonzero gap in N. In (b) we show the temperature dependence of the integrated spectral weight, of the experimental data from [36].
Comparison of (a) and (b) reveals that the temperature, below which the gap in N is necessary for achieving a good quality of the fit, approximately coincides with the temperature of the onset of the spectral weight shift from the frequency range 30-200 meV to lower frequencies, see the green and the yellow lines in figure B4(b). This temperature scale was recognized already in [6] and found to be close to T ons of the c-axis data. The temperature dependence of the frequency ω max of the maximum of the boson spectral density in (c) and that of Figure B2. Illustration of the impact of the gap in the density of states on the quality of the fit. (a) 244 K data of UD Y-123 reported in [36] together with the results of the fit involving the gap (red lines) and with those not involving the gap (blue lines). It can be seen that the presence of the gap does not lead to a significant improvement. (b) The same for 67 K. It can be seen that the role of the gap is essential.  In summary, we have fitted the experimental data reported by Hwang et al [36] by the model with and without the gap in the density of states. Comparison of the results shows that at high temperatures the model which uses the boson spectral function α F 2 and an energy independent density of states is sufficient for a reasonable description of the data. On the other hand, for temperatures closer to T c , the gap in N becomes crucial for reproducing the specific features of M 1 and M 2 . The fitting procedure reveals the presence of a characteristic temperature located around 200 K, below which the presence of the gap in the density of states is necessary. This temperature scale is obviously connected to that described in [6] (see the corresponding supporting material).
Appendix C. Remarks on purely normal-state interpretations of the dip feature First, a dip in the in-plane conductivity develops with decreasing temperature naturally, even in the absence of a DOS gap, due to the temperature dependence of the Bose and Fermi factors. Its scale is determined by the characteristic boson energy and it is typically narrower than in the data. An example of a standard normal state temperature dependence of σ 1 is shown in figure C1(a). The spectra have been calculated using the Eliashberg formalism described in appendix A and the same values of the input parameters as in [56], except for = t 0.38 eV, ′ = − t 0.120 eV, n = 0.85 and = g 3.0 eV, for definitions, see [56] and [55]. The onset of superconductivity leads to a different shape of the gap. This is illustrated in figure C1(b) showing the 100 K, 80 K and 60 K normal state spectra from (a), and the 80 K and 60 K superconducting state spectra. For the present values of the input parameters, = T 90 K c . Second, a deepening of the dip for a fixed temperature can be achieved by increasing the spectral weight of the low-energy component of the bosonic spectral function (α F 2 in the Allen's theory, χ ω ″ q ( , )in more realistic models involving spin fluctuations). The latter increase, however, also leads to a reduction of the total optical spectral weight in the infrared, not occurring in the data. This problem is illustrated in figures C1(c) and C2 . Figure C1(c) shows the 100 K and 60 K normal state spectra from (a) and the 100 K and 60 K normal state spectra calculated with a higher value of the dimensionless spectral weight of the resonance of ). The deepening of the minimum can be clearly seen. Figure C2 corresponding to the spectra of figure C1(a), that is qualitatively consistent with the experimental one shown in figure 4 of [36]. With decreasing temperature, ω SW( ) at low frequencies increases. Figure C2(b) shows that the increase of b M , for a fixed temperature, leads to the opposite trend: with increasing b M the spectral weight at low frequencies decreases. It is thus unlikely that the Figure B4. (a) Temperature dependence of the quality of the fit of the UD Y-123 data reported in [36] based on the model with the gap in the density of states (red symbols) and that of the model without the gap (blue symbols). It can be seen that for higher temperatures the performance of the two models is comparable. At temperatures closer to T c the model including the gap in N provides significantly better results. (b) Experimental values (data of [36]) of the optical spectral weight for the three energy ranges formation of the gap in σ 1 could be described as being solely due to a specific temperature dependence of the low-energy component of α F 2 or χ ω ″ q ( , ). Figure C1. (a) Normal state temperature dependence of σ 1 calculated as described in the text. (b) Selected normal state spectra from (a) (100 K, 80 K, 60 K) and the 80 K and 60 K superconducting state spectra. The difference between the normal state gap feature due to the temperature dependence of the Bose and Fermi factors and the superconducting state gap feature due to the opening of the superconducting gap can be clearly seen. (c) Two normal state spectra from (a) (100 K and 60 K) and the 100 K and 60 K normal state spectra calculated with a higher value of the spectral weight of the resonance of