Supplementary Information for “Exploring Quasiparticles in High-Tc Cuprates Through Photoemission, Tunneling, and X-ray Scattering Experiments”

One of the key challenges in the field of high-temperature superconductivity is understanding the nature of fermionic quasiparticles. Experiments consistently demonstrate the existence of a second energy scale, distinct from the d-wave superconducting gap, that persists above the transition temperature into the"pseudogap"phase. One common class of models relates this energy scale to the quasiparticle gap due to a competing order, such as the incommensurate"checkerboard"order observed in scanning tunneling microscopy (STM) and resonant elastic X-ray scattering (REXS). In this paper we show that these experiments are better described by identifying the second energy scale with the inverse lifetime of quasiparticles. We develop a minimal phenomenological model that allows us to quantitatively describe STM and REXS experiments and discuss their relation with photoemission spectroscopy. Our study refocuses questions about the nature of the pseudogap phase to the study of the origin of inelastic scattering.


SI-1. ARPES SPECTRA AND FERMI ARCS
In the section we discuss implications of our model for ARPES experiments. Previous works [1][2][3][4] have already pointed out that a finite quasiparticle lifetime provides a natural explanation for the ARPES spectra, including the emergence of Fermi arcs in underdoped samples. Here we review their arguments and relate them to the Green's function formalism used in this paper. At low temperatures ARPES probes the spectral function, defined as the imaginary part of the diagonal elements of G(q, ω) [5]. For momenta on the Fermi surface, k = µ, the (symmetrized) ARPES signal is then given by In Fig. S1a, c we directly compare the imaginary part of G(k, ω) with the symmetrized spectrum observed in ARPES experiments and find a very good agreement. Eq. (S1) behaves differently depending on the ratio Γ/∆ k F . For Γ/∆ k < √ 3, it has two maxima at In Fig. S1b, d we show that this expression qualitatively reproduces the evolution of the "Fermi arcs", provided that Γ is assumed to be temperature dependent. For Γ/∆ k > √ 3 the same curve has a single maximum at E = 0. As a consequence, "Fermi arcs" are expected to be observed in the vicinity of the nodes for all momenta satisfying ∆ k = ∆ 0 (cos(k x ) − cos(k y ))/2 < Γ/ √ 3. The growth of the Fermi arcs with increasing temperature [6] and underdoping [7] can be explained in terms of a growth of Γ, rather than a closing gap.
Because ARPES directly probes the nodal quasiparticles, while STM is mostly sensitive to antinodal quasiparticles, a systematic comparison of these two methods on the same materials and temperatures will deliver valuable information about the anisotropy of the inelastic scattering, and help to understand its physical origin.

SI-2. THEORETICAL DESCRIPTION OF STM MEASUREMENTS
In this Appendix we present the derivation of Fig. (2), describing the Fourier transformed STM signal induced by a single time-independent impurity. As mentioned in the text, STM measures the differential conductance with ∆ 0 = 50meV, and µ = 30meV (p = 0.08). c-d, Experimental measurements of underdoped samples of Bi2212 with respectively c T c = 65K [4] (measured at T = 75K) and d T c = 55K [7] (measured at T = 11K − 60K). Subplot c is reprinted by permission from Macmillan Publishers Ltd: Nature Physics, Ref. [4], copyright (2012).
where G(r, r , V ) is the dressed Green's function including the effects of disorder. In the case of a time-independent scatterer at position r 0 , first-order perturbation theory gives: Here G(r − r , V ) = G(r, r , V ) is the translational-invariant bare Green's function (1), which includes the effects of interactions. Introducing its Fourier transform G(k, V ) = dr e ikr G(r, V ) we obtain where T 0 (k 1 , k 2 ) = dr 1 dr 2 e ik 1 r 1 −ik 2 r 2 T 0 (r 1 , r 2 ).
For a local impurity T 0 (k 1 , k 2 ) = T 0 (k 1 ) + T 0 (k 2 ) and If both the bare Green's function and the impurity scattering are invariant under inversion symmetry k → −k, only the cosine component contributes to the integral and The Fourier transformed STM signal is then Using the identity dr e iqr cos(q (r − r 0 )) = e iqr 0 (δ q+q + δ q−q ) /2 we find (S10) Due to the above-mentioned symmetry (k ↔ −k) the contributions from terms with k −k = q and k−k = −q are identical, and the finite-q components of Eq. (S10) are given by Eq. (2).

CLES
In the text we explained that the non-dispersive peaks observed in STM originate from enhanced scattering at the antinodes (see also Ref. [8]). This observation is in contradiction with the well-known "octet model" [9], which predicts that antinodal quasiparticles should contribute only at a specific voltage, V ≈ ∆ 0 , due to energy conservation. Accordingly, for small voltages V ∆ 0 only nodal quasiparticles are expected to contribute. We now show that this picture is dramatically changed when the appropriate matrix elements ("coherence factors") are taken into account. For convenience, we define the integrand of Eq. (2) as Eq. (S11) determines the contribution to the differential conductance originating from the scattering of a quasiparticle from momentum k to momentum k + q. The experimental observable g(q, V ) is obtained by integrating S over all momenta k.
Let us first consider modulations of the chemical potential (α = 1), in the limiting case of zero voltage and Γ → 0. In this limit Eq. (S11) simplifies to The denominator of Eq. (S12) vanishes if both k and k + q correspond to the nodal points (where k = k+q = µ and ∆ k = 0) in agreement with the octet picture. However, precisely at this point the numerator vanishes as well and the contribution to the differential conductance is zero. Because on the two sides of the nodal point Eq. (S12) has opposite sign (dependending on whether k is larger or smaller than µ), the integral over k gives an almost-vanishing contribution. In this case, the peak predicted by the octet model is completely washed out.
We now consider the role of the coherence factors (S11) in generating the non-dispersive peak at q ≈ (0.2, 0) × 2π. Fig. S2b presents a colorplot of S ( k, k + q, V ), associated with the modulations of the chemical potential. In agreement with our previous argument, we find that the coherence factors change sign across the Fermi surface, and are strongly suppressed when the sum over all k is taken into account. In contrast, the coherence factors due to modulations of the pairing gap (subplot c) do not change sign and therefore dominate the predicted STM signal at this wavevector. It is interesting to compare our results with the octet model. This model predicts contributions from quasiparticles with a specific momentum, given by the intersection between the Fermi surface and the line k x = q x /2. For q = q * this momentum is approximately half-way between the nodal and antinodal points.
In contrast to the octet model, our approach shows that the STM signal is determined by quasiparticles with a broad range of momenta, close to the antinodal points (blue regions in  To further highlight the predominance of antinodal scattering in STM signal we now consider a momentum-dependent quasiparticle lifetime of the form where Γ n and Γ a are, respectively, the quasiparticle lifetime at the nodes and at the antinodes. The resulting predictions for the Fourier-transformed STM signal is shown in Fig. S3.
We find that the predicted differential conductance is strongly dependent on Γ a and almost FIG. S3. STM signal with a momentum-dependent quasiparticle lifetime. Same as

SI-5. HOMOGENEOUS COMPONENT OF THE STM SIGNAL
In this section we consider the spatially-homogenous conductance dI/dV . In actual experiments, this component can be measured by averaging the STM signal over a large area Ω: The theoretical predictions and experimental observation of this component are compared in Fig. S5a (2), whose actual maximum varies in space. The sum of these two terms is expected to give rise to a "kink", which was indeed universally observed in experiments (see for example Ref. [14]).
An alternative method to extract the homogeneous component of the STM signal has been proposed in Ref. [15]. Assuming that in the normal phase g(x, V ) = const, the homogenous component of the differential conductance can be obtained from g(x, V, T )/g(x, V, T norm ), where T norm > T c is an arbitrary temperature. The experimental signal is reproduced in Fig. S5c for an overdoped sample of Pb-Bi2201 with T c = 15K, using T norm = 17K. The position of the peaks coincides with our identification of the superconducting gap for this sample, ∆ 0 = 8meV. As the temperature increases, the distance between the peaks does not significantly vary, but their visibility rapidly diminishes. In Fig. S5c we reproduce this result by assuming a linear dependence between Γ and the temperature. (In our case, we have established that, at T = 6K, the inverse quasiparticle lifetime Γ = 6meV, leading to the simple relation Γ/T ≈ 1meV/K, see also Appendix SI-9). Using this assumption and normalizing the theoretical predictions with respect to the value at Γ = 17meV, we obtain a good agreement between theory and experiment, as shown in Fig. S5c-d. SI-6. ANALYSIS OF THE NON-DISPERSIVE PEAK AT q π,π = (0.5, 0.5) × 2π Figure S6 shows a two-dimensional cut of the data at fixed voltage V = 5meV, for wavevectors inside the first Brillouin zone. Comparing subplots a and c we find that the theory quantitatively reproduces the experiment, with one important exception: the experiment shows a broad peak around q π,π ≡ (±0.5, ±0.5) × 2π, while the theoretical predictions exactly vanishes there (due to the symmetry of the coherence factors appearing in Eq. (2)).
To identify the nature of the q π,π peak, we study its voltage dependence (Fig.S7) and find it to be anti-symmetric with r espect to the V → −V . As explained above, this behavior is characteristic of the scattering from local modulations of the chemical potential. Fig. S6c shows that, indeed, this type of perturbation leads to a g-map that is peaked around q π,π .
Our findings may also explain the experimental observations of Ref. [8], who showed that the peak at q π,π responds to magnetic field and temperature in the opposite way than the rest of the map, highlighting its different physical origin.

SI-7. EFFECTS OF THE BAND STRUCTURE ON THE REXS SIGNAL
In the main body of the article we found that the predicted width of the REXS peak in Y123 is larger than the one observed in experiments [16]. One interesting possibility is that this discrepancy is due to an enhancement of the CDW order caused by electronelectron interactions. This effect can be described using an random-phase approximation (RPA) [17,18] and, in general, acts to sharpen the predicted peak. Here we follow a simpler interpretation and relate the observed discrepancy in the REXS signal to a deviation of the actual band structure from the phenomenological model obtained in Ref. [19]. Unlike the k . b, Same as before, for local modulations of the chemical potential, T k = T (1) k . c, Experimental signal g(q, V ) for the an underdoped sample (UD32K). The intensity peak at q π,π = (0.5, 0.5) × 2π observed in the experiment is due to modulations of the chemical potential. case of Bi2212, the band structure of Y123 is known less accurately, due to surface effects and to the presence of CuO chains [20]. In Fig. S8 we compare calculations for the REXS signal using two different phenomenological band structures with similar Fermi surfaces.
The position of the REXS peak q = 0.31 is uniquely determined by the doping, and is largely model independent. In contrast, the widths of the predicted signals significantly differ between the two models and vary from ξ = 0.1 to ξ = 0.07. We note that the phenomenological model with a larger number of parameters (N = 5) displays a sharper peak and offers a better agreement with experiments. In general, a sharp peak in the REXS signal requires a nested band-structure, whose characterization involves many fitting parameters. Accordingly, we observe that the band structure of Ref. [21], obtained using one single fitting parameter, does not generate any significant REXS peak. To further explore this point we consider the effects of an additional momentum-dependent term in the band structure of Ref. [19], with approximately the same amplitude as the previous ones (see last column of Table S1). We find that this term leads to a further sharpening of the REXS peak and an excellent agreement with experiments. We therefore propose that REXS experiments  Table S1). The model N = 6 is obtained by adding an additional arbitrary term to the band structure of Ref. [19] (and adjusting the chemical potential to keep the density fixed). b, Predicted (continuous curves) and measured (symbols and dotted lines) REXS signal. The discrepancy between theory (see text) and experiment ( [16]) is far below the combined uncertainty of both. The theoretical curves where obtained using the same parameters as in Fig. 4. To allow a comparison between the different models and experiments, we have renormalized each curve by subtracting the value at q = 0.22 and dividing by the maximal intensity.
can be used to probe the band structure of the antinodal regions of cuprates.  T (4) should best reproduce the physical situation, as shown in Fig. 4c. In addition to the peaks at q a = (0.25, 0) × 2π and q b = (0, 0.25) × 2π, we predict a pronounced peak at q a ± q b , whose maximal intensity is larger than the one predicted for q a and q b . In contrast, if the REXS intensity is dominated by local impurities with d-wave symmetry (T(3)), no peak is predicted in the (q, q) direction.

SI-9. IMPLICATIONS FOR THE PHASE DIAGRAM
Our analysis indicates that a finite quasiparticle lifetime is fundamental for understanding the single-particle properties of cuprates. Here we explore the possibility that Γ may also play an important role in determining the critical temperature T c . We observe that, in Pb-Bi2201, T c seems to correspond to the point were antinodal quasiparticles become overdamped, i.e. where their inverse lifetime equals to twice their gap: Γ(T c ) = 2∆ 0 . To obtain this result, we assume a linear dependence of Γ on the temperature, Γ(T ) = αT , found in both theoretical calculations [23][24][25] and experiments [26]. Starting from the observed values of ∆ 0 and Γ (see Table I