Low-energy quasi-circular electron correlations with charge order wavelength in Bi₂Sr₂CaCu₂O_8+δ

Abstract

Most resonant inelastic x-ray scattering (RIXS) studies of dynamic charge order correlations in the cuprates have focused on the high-symmetry directions of the copper oxide plane. However, scattering along other in-plane directions should not be ignored as it may help understand, for example, the origin of charge order correlations or the isotropic scattering resulting in strange metal behavior. Our RIXS experiments reveal dynamic charge correlations over the q_x-q_y scattering plane in underdoped Bi₂Sr₂CaCu₂O_8+δ. Tracking the softening of the RIXS-measured bond-stretching phonon, we show that these dynamic correlations exist at energies below approximately 70 meV and are centered around a quasi-circular manifold in the q_x-q_y scattering plane with radius equal to the magnitude of the charge order wave vector, q_CO. This phonon-tracking procedure also allows us to rule out fluctuations of short-range directional charge order (i.e., centered around [q_x = ±q_CO, q_y = 0] and [q_x = 0, q_y = ±q_CO]) as the origin of the observed correlations.

In cuprate superconductor, temporal fluctuations of 2D charge patterns along x/y rotate into all directions of the 2D plane.

INTRODUCTION

Dynamic fluctuations from periodic charge order (CO) pervade the phase diagram of cuprate superconductors, perhaps even more than superconductivity itself (1). The detection of these fluctuations over energy and momentum was enabled by several recent advances in the energy resolution of resonant inelastic x-ray scattering (RIXS) instruments operating in the soft x-ray regime. In the case of YBa₂Cu₃O_6+δ, Cu-L₃ RIXS detects dynamic correlations at the CO wave vector, q_CO, with a characteristic energy scale of approximately 20 meV (2). It has been proposed that these low-energy short-range dynamic CO correlations are a key ingredient to the strange metal behavior (3, 4) characterized by linear-in- temperature resistivity (5, 6). On one hand, this temperature behavior is often associated with an isotropic scattering rate that depends only on temperature in units of energy and Planck’s constant (i.e., ∝k_BT/ℏ, sometimes called the Planckian regime) (7–11), as supported by recent angle-dependent magnetoresistance measurements of La_1.6−xNd_0.4Sr_xCuO₄ (12). On the other hand, combined transport and RIXS studies have recently shown an unexpected link between linear-in-temperature resistivity and CO in YBa₂Cu₃O_6+δ (13, 14). Combined, these latest results suggest that fluctuations of the CO should somehow result in an effective isotropic scattering. Still, high-resolution RIXS experiments have largely focused on the fluctuations along the high-symmetry crystallographic directions only, leaving the full structure of electron correlations within the copper oxide plane unknown.

Recently, in Bi₂Sr₂CaCu₂O_8+δ (Bi-2212), RIXS measurements found the existence of a quasi-circular pattern in the q_x-q_y plane at finite energies and with the same wave vector magnitude as that of the observed static CO peak at q = [q_x = ±q_CO, q_y = 0] and [q_x = 0, q_y = ±q_CO]—i.e., dynamic correlations with CO wavelength along all direction in the CuO₂ plane (15). Although the medium energy resolution of those measurements (ΔE ≈ 0.8 eV) precluded a more precise determination of their energy profile, the results suggested that these quasi-circular dynamic correlations (QCDCs) appear broad over the mid-infrared ranges (defined approximately as 100 to 900 meV). This scattering manifold, which may result from combined short- and long-range Coulomb interactions (15–17), would provide a large variety of wave vectors for connecting all points of the Fermi surface (i.e., an effective isotropic scattering). However, it is not yet experimentally known whether this manifold extends to electron scattering at lower energies, in the quasi-elastic regime. To experimentally investigate this scenario, we used high energy resolution (≈37 meV) Cu-L₃ RIXS q_x-q_y mapping of the electronic correlations in Bi-2212. Using the softening of the bond-stretching (BS) phonon in RIXS as a marker of CO correlations, our measurements reveal the presence of low-energy quasi-circular dynamic electronic correlations with ∣q∣ ≈ q_CO.

RESULTS

High-resolution RIXS mapping of dynamic correlations in the q_x-q_y plane

We performed Cu-L₃ RIXS measurements (approximately 931.5 eV incoming photon energy) at ϕ = 0°, 25°, 30°, 35°,  and 45°, where ϕ is defined as the azimuthal angle from the q_x axis. For each ϕ, we acquired RIXS spectra at different values of in-plane momentum transfer q = ∣q∣ by varying the incident angle on the sample. Throughout the paper, values of q are reported in reciprocal lattice units (r.l.u.), where 1 r.l.u. is defined as 2π/a and a = 3.82 Å (the lattice constant along ϕ = 0°). The measurements were performed at T = 60 K, which is the superconducting transition temperature (T_c) for the measured sample, except for one measurement performed at T = 25 K below T_c. Supporting measurements on a similar sample (T_c = 54 K), shown in the Supplementary Materials, were performed at 54 K, except for one measurement at 300 K (Supplementary Materials, figs. S2 and S5). In Fig. 1 (A and B), we show representative spectra obtained at q near q_CO for ϕ = 0° and 30°, and energies below 1.1 eV. In these two cases, the minimal model that fits the data includes five contributions: a quasi-elastic peak, a BS phonon peak at ≈70 meV, a peak at ≈135 meV (likely from a two-phonon process), a broad paramagnon, and a broad background feature of unknown origin. A similar assessment can be made regarding all other high-resolution spectra acquired in this work. It may be the case that high-energy QCDCs (15) are somehow encompassed by the phenomenological background term of the fitting function. However, in this type of fitting analysis, it is generally difficult to disentangle overlapping contributions to the RIXS spectra using a fitting model with so many parameters, thus precluding the extraction of the exact spectral profile of the QCDCs with any reasonable confidence. Still, we note that these high-resolution data are consistent with the previously reported medium-resolution data (15), which can be verified by integration of the high-resolution data (see fig. S9).

Fig. 1. RIXS spectra and fitting.

(A and B) Examples of spectra at q = 0.27 r.l.u. for ϕ = 0° and 30°, respectively (open circles). The red lines are fits to the spectra, composed of a quasi-elastic peak (pink), a BS phonon peak at ≈70 meV (blue), a peak at ≈135 meV (likely from a two-phonon process) (purple), a broad paramagnon (orange), and a broad background feature of unknown origin (brown). (C and D) RIXS measured BS phonon peak for various values of q measured for ϕ = 0° and 30°, respectively (black circles). The blue lines are the fits to the spectra. The vertical orange dashed lines, indicating the lowest phonon peak position at each ϕ, are shown to help the reader observe the phonon dispersions in the raw data. r.l.u., reciprocal lattice units.

From the above discussion, we consider the possibility that the spectral intensity of QCDCs in Bi-2212 is so dilute over energy as to preclude the reliable use of curve fitting algorithms for the extraction of their spectral structure amid stronger paramagnon and phonon signals. Instead, here we develop a different method to detect QCDCs at lower energies, by tracking the evolution of the BS phonon over the q_x-q_y plane. This method is based on the phenomenology revealed by several recent RIXS measurements of the cuprates along q_x and q_y, which indicate an apparent softening of the BS phonon peak at the momentum location of the static CO peak (18–23). The BS phonon is also an ideal probe of QCDCs because both its dispersion and electron-phonon coupling vary weakly as a function of ϕ in the absence of CO correlations (24, 25). We call the procedure where we probe QCDCs with the BS phonon a phonon-tracking method and note that the softening of the BS phonon can be caused by either static or dynamic correlations. In the case of Bi-2212, it has been proposed that the apparent softening of the BS phonon in RIXS is due to an interplay between low-energy dynamic fluctuations of the CO and BS phonons that result in a Fano-like interference (18, 21, 23, 26). Another possibility is that the apparent softening is simply the result of the phonon peak and a low-energy CO peak overlapping, as recently suggested by measurements of both YBa₂Cu₃O_6+δ and Bi-2212 (27). In either interpretation, the location of the phonon softening can be used as a marker for low-energy dynamic CO correlations.

Figure 1 (C and D) shows the spectra acquired as a function of q for ϕ = 0° and 30°, respectively, focusing on the region of the BS phonon. At ϕ = 0°, it is clear that the phonon peak position softens to its lowest energy value at q = q_CO ≈ 0.29 r.l.u. (Fig. 1C). Careful observation of the spectra taken along ϕ = 30° shows a similar softening effect with the lowest phonon energy position occurring for q ≈ q_CO (Fig. 1D). Figure 2A shows the mapping of the BS phonon mode at ϕ = 0° and 30° obtained after subtraction of the fitted elastic line, once again showing the softening of the RIXS phonon even at ϕ = 30°. To precisely determine the locations of the softening in the q_x-q_y plane, we fit the spectra to extract the dispersion of the BS phonon for each ϕ (Fig. 2B). We observe a softening of the RIXS measured phonon line for all ϕ, except for ϕ = 45°. All observed softening occurs at a value of q ≈ q_CO, precisely as expected for QCDCs at low energies.

Fig. 2. Location of low-energy dynamic correlations extracted from the phonon dispersion.

(A) Energy-momentum structure of the excitations at ϕ = 0° and 30° after subtraction of the elastic line. The image is constructed from RIXS spectra deconvoluted from the energy resolution. (B) Location of the phonon peak obtained by fitting the RIXS spectra deconvoluted from energy resolution for different ϕ (see Materials and Methods and also fig. S3). The solid lines are obtained by fitting the q dependence of the phonon peak (circles) with a negative Lorentzian function plus a linear background. The shaded regions around the solid lines are generated from the 95% confidence interval obtained for the various fits to the spectra (see Materials and Methods for details). The solid lines for ϕ = 0° and 30° in (B) appear as dashed white lines in (A). r.l.u., reciprocal lattice units.

Discriminating QCDCs from short-range directional order

Dynamic correlations emanating from short-range order are bound to be broad in q. It is therefore reasonable to ask whether the measured q_x-q_y profile of the BS phonon could simply be the result of diffuse scattering from short-range directional order. The fundamental difference between QCDCs and short-range directional order is that the former forms a manifold of dynamic correlations centered at q ≈ q_CO [similar to Brazovskii-type fluctuations (15, 28)], while the latter results in dynamic correlations around q = [q_x = ±q_CO, q_y = 0] and q= [q_x = 0, q_y = ±q_CO]. To contrast these scenarios, we consider two simple toy models (more details on the toy models are provided in Materials and Methods). In both cases, we start with a flat ∣q∣-independent phonon mode at 72 meV, which is a reasonable approximation given the small dispersion of the BS phonon in the absence of CO (25). In the first model (M1), we construct the QCDCs scenario, where the q cuts for various ϕ always have a minimum located at q = q_CO (Fig. 3, A and B). In the second model (M2), we consider the case where dynamic CO correlations emerge isotropically from static peaks at [q_x = ±q_CO, q_y = 0] and [q_x = 0, q_y = ±q_CO]. The corresponding phonon profile is shown in Fig. 3 (C and D). To roughly emulate the data, we also introduce a ϕ-dependent softening magnitude in M1, which increases from ϕ = 0° to 45° (Fig. 3A). However, note that the magnitude of the softening depends on the amount of CO correlations and on the ϕ structure of the electron-phonon coupling, which is not precisely known experimentally or necessary for discerning the two scenarios. The q cuts show a qualitatively similar behavior in both models: a clear phonon softening at ϕ = 0° that continues to exist even as ϕ approaches 45°. However, in M2, the q location of the phonon minima clearly decreases with increasing ϕ from 0° to 45°. This comparison explains our selection of ϕ values for these studies: The experimental ability to differentiate between M1 and M2 is largest in the ϕ = 25° to 45° range. The polar plot in Fig. 3E summarizes the analysis, comparing the q location of the minima for both models to the minima obtained from experiments (red markers with error bars). The experimental data are clearly consistent with M1, although deviations of M1 from a perfectly circular manifold (i.e., quasi-circular correlations) are possible within error bars. The comparison in Fig. 3E clearly rules out M2, the short-range directional order scenario, and instead indicates the quasi-circular nature of the low-energy correlations associated with the CO.

Fig. 3. Models of phonon softening for QCDCs and directional order.

(A and C) Phonon dispersion for M1 and M2, as described in the text. (B and D) Momentum q cuts of the phonon dispersion at different ϕ for the simulated data in (A) and (C), respectively. The dashed orange and green lines in (A) to (D) identify the location of the phonon softening in the q_x-q_y plane. (E) Polar plot contrasting M1 and M2 models (orange and green solid lines) and the experimental data (red symbols). The error bars in (E) are obtained from the fits to the phonon dispersion in Fig. 2B. See Materials and Methods for more details. r.l.u., reciprocal lattice units.

DISCUSSION

The experiments presented here provide evidence for the existence of QCDCs at low energies in underdoped Bi-2212, which could be a key ingredient for models that connect CO to an effective isotropic scattering. Long-range translational symmetry breaking cannot be responsible for this isotropy due to the characteristic length scale and directionality of the ordered state. Although short-range electron correlations from directional order, occupying a much larger region of momentum space, could, in principle, emulate isotropic scattering (3, 4, 29), the QCDCs revealed by our experiments offer a different scenario. Extending not only around the static CO wave vectors but also in the azimuthal direction, QCDCs might be a more viable platform for isotropic scattering. These QCDCs may also be related to the featureless continuum in the charge susceptibility observed by inelastic electron scattering, which has been discussed in connection to the marginal Fermi liquid hypothesis (7, 30). To fully understand this possible relation and the impact of QCDCs to electronic properties of the cuprates, one requires knowledge of the energy structure of these correlations. Although this might still be beyond the current experimental capabilities, our experiments provide some constraints to the low-energy structure of the QCDCs. In particular, for the ϕ values where a softening is detected, QCDCs must exist below ≈70 meV (i.e., the approximate energy of the bare phonon). Given the weak ϕ dependence of the BS phonon coupling to electrons [see fig. S7 and (24)], the variations in the amount of energy softening observed from 0° to 45° likely reflect changes in the underlying CO correlations. Unfortunately, the amount of energy softening at q = q_CO by itself, without precise knowledge of the ϕ-dependence of the electron-phonon interaction, does not provide more information about the energy structure of the QCDCs. Therefore, it remains possible that QCDCs at ϕ = 45° exist below 70 meV but do not substantially interact with the BS phonon.

To test for the presence of CO correlations even at ϕ = 45°, we analyzed the momentum dependence of the spectrum integrated over different energy windows (fig. S6, A and B). For ϕ ≥ 25°, including ϕ = 45°, we found a featureless momentum dependence when integrating around the elastic line, [ − 10,10] meV. On the other hand, integration in a window between the elastic and phonon lines, [25,40] meV, reveals a broad maximum around q_CO even at ϕ = 45°. Given the 37 meV energy resolution of our RIXS measurements, we performed the same analysis over energy deconvoluted spectra (fig. S6, C and D) and obtained similar results. Although consistent with our phonon-tracking analysis, the energy integration analysis should be taken carefully because the integrated signal may also include contributions from low-energy phonons. For this reason, it is also a less accurate indicator of the momentum structure of QCDCs when compared to the phonon-tracking method. Perhaps q_x − q_y mapping by oxygen K edge RIXS measurements, which can reach resolutions better than 20 meV, may better resolve the low-energy structure of QCDCs and investigate their relation to the Planckian regime, akin to what has been done for La-based cuprates along ϕ = 0° (31). Together, the phonon-tracking analysis and the absence of a peak in the elastic channel near q_CO for ϕ ≥ 25° establish the dynamic nature of the quasi-circular correlations.

Our experiments clearly reveal different manifolds for the static and low-energy dynamic correlations. While the static signal is confined to peaks centered around [q_x = ±q_CO, q_y = 0] and [q_x = 0, q_y = ±q_CO], the low-energy dynamic correlations form a quasi-circle in the q_x-q_y plane. This difference suggests that while, in principle, the QCDCs could condense into static order along any direction, the lattice constrains the static signal to a high-symmetry direction of the lattice. The quasi-circular shape of the low-energy correlations is also similar to the shape obtained from the analysis of higher-energy correlations [figs. S8 and S9 and (15)]. This similarity raises the possibility that the QCDCs exist up to much higher energies of the order of 1 eV. As discussed in (15), the quasi-circular correlations cannot be explained by an instability of the Fermi surface. Instead, it was proposed that the location of the dynamic CO correlations in q-space is determined by the minima of the effective Coulomb interaction, which becomes nonmonotonic in q due to the inclusion of a long-range Coulomb interaction. However, this nonmonotonic Coulomb interaction by itself failed to capture the intensity anisotropy observed at q ≈ q_CO. Likewise, here the same proposed Coulomb interaction could also explain the most salient feature of our data, namely, the quasi-circular shape of the low-energy correlations. Recently, a more complete theoretical description based on a t-J model with long-range Coulomb interaction shows the presence of ring-like charge correlations with the correct intensity anisotropy (17). The results presented here can serve as a guide for future theoretical investigations that also account for the apparent decrease of the phonon softening from ϕ = 0° to 45°.

Beyond the fact that both the energy-integrated correlations (15) and the low-energy dynamic correlations appear to occupy the same quasi-circular scattering manifold, the current RIXS measurements do not provide further experimental evidence to connect the two phenomena. Comparing the two results, note that the analysis of the medium-resolution RIXS spectra of (15) relies on an energy window that also encompasses spin excitations known as paramagnons, which are centered around 300 meV near q = q_CO. Nonetheless, the paramagnons do not influence the dispersion anomaly of the BS phonon below ≈70 meV. An understanding of exactly how the low-energy QCDCs reported here connect to the high-energy correlations may come from polarimetric RIXS experiments that are able to decompose charge and spin excitations in the mid-infrared range, as has been done for electron-doped cuprates (32). In addition, compared to the energy-integration procedure, the phonon-tracking method provides higher precision for mapping CO correlations in the q_x − q_y plane, because the large integration ranges required for the former result in very broad features in q-space. We have already performed medium-resolution RIXS measurements that detect the presence of similar quasi-circular scattering manifolds in the energy-integrated spectrum of optimally and overdoped samples, but the investigation of their doping dependence is hindered by the large experimental uncertainty associated with the integration method (see fig. S8). Instead, our procedure to track QCDCs using measurements of the RIXS BS phonon goes beyond demonstrating the existence of QCDCs in underdoped Bi-2212 at low energies. It is also a methodology that can be used to detect QCDCs in other cuprates and understand related phenomena such as the electron-doped cuprates, which also show a quasi-circular scattering (33). Last, the application of this method to multiple cuprate families at different dopings and/or temperatures will help unveil whether and how QCDCs and the strange metal are related.

MATERIALS AND METHODS

RIXS experiments

High-resolution RIXS experiments were performed at the I21 beamline (34) at Diamond Light Source, UK, and at the 2-ID beamline (35) at the National Synchrotron Light Source II, Brookhaven National Laboratory, USA. The orientation of the crystal axes of the underdoped Bi₂Sr₂CaCu₂O_8+δ samples was obtained by x-ray diffraction before the RIXS experiment. The samples were cleaved in air just moments before inserting them into the ultrahigh-vacuum chambers. For experiments at I21, the crystal was aligned to the scattering geometry in situfrom measurements of the 002 Bragg reflection and the b-axis superstructure peak. The scattering angle, 2θ, was fixed at 154° (I21) and 153° (2-ID). The fixed scattering angle setup, where q_z varies as a function of sample angle, is commonly used for Cu-L₃ RIXS on cuprates given the broad nature of CO correlations along q_z. Using c = 30.8 Å as the out-of-plane lattice constant, we obtain the L = 3.83 × 2π/c for the out-of-plane Miller index at an in-plane q = 0.29 × 2π/a. The incoming light was set to vertical polarization (σ geometry) at the Cu-L₃ edge (≈931.5 eV). The combined energy resolution (full width at half maximum) was about 37 meV (I21) and 40 meV (2-ID), with small variations (±3 meV) over the course of multiple days. In both cases, the energy resolution was relaxed in a trade-off for intensity. The projection of the momentum transfer, q, in the q_x-q_y plane was obtained by varying the incident angle on the sample (θ). The momentum resolution depends on the geometry of the beamline and on the particular values of θ and 2θ. For the θ scans in this work, we find that the in-plane momentum resolution was better than ±0.006 × 2π/a (±0.009 Å⁻¹) (34) at I21 and ±0.002 × 2π/a (±0.004 Å⁻¹) (35), approximately. All the measurements were performed at T = 60 K (I21) or T = 54 K (2-ID), which are the superconducting transition temperatures for the measured samples, except for one measurement performed at T = 25 K below T_c (Fig. 3E; T_c = 60 K sample) and one measurement at 300 K (figs. S2 and S5; T_c = 54 K sample).

Analysis of RIXS spectra

To ensure the robustness of the extraction of phonon dispersion from the RIXS spectra, we analyzed the data using multiple methods. Although the overall RIXS cross section may depend on ϕ, we did not perform any normalization or intensity correction procedure to the spectra because the energy location of the phonon does not depend on the overall intensity. A comparison between the results for different methods is available in fig. S4.

Method 1

In an effort to maintain an agnostic approach and to not assume particular functional forms of the different contributions to the RIXS spectra, we extracted the dispersion by simply tracking the energy positions of the phonon peak maximum in the RIXS spectra deconvoluted from the energy resolution. See below for details of the deconvolution procedure.

Method 2

The phonon dispersion shown in Fig. 2B was extracted by fitting the deconvoluted RIXS spectra (see Fig. 2A and fig. S3) in the [−30,130] meV range to a double Gaussian function plus a second-order polynomial background, keeping all parameters free. The shaded regions around the solid lines in Fig. 2B were generated by fitting the 95% confidence intervals (obtained from the fits) to a polynomial function of q.

Method 3

Following previous works (23, 36), the raw RIXS spectra were fit to a five-component model that includes a Gaussian (elastic peak of amplitude A_el, position ω_el, and width w_el), two anti-Lorentzians (phonon and double-phonon peaks of different amplitude A_i and position ω_i, and sharing width w_ph and Fano parameter width q_F—note that i = 1,2 indicates the first and second phonon, respectively), a damped harmonic oscillator line shape (paramagnon of amplitude A_pm, position ω_pm, and damping parameter γ_pm), and an error function (smooth background described by an error function with amplitude A_BG, position ω_BG, and width w_BG)

$$\begin{array}{c}f(\mathrm{\omega })=A_{\mathrm{el}}{e}^{-\frac{{(\mathrm{\omega }-{\mathrm{\omega }}_{\mathrm{el}})}^2}{w_{\mathrm{el}}^2}}+{\displaystyle \sum _{\mathrm{i}=1,2}}{A}_{\mathrm{i}}\frac{2(\mathrm{\omega }-{\mathrm{\omega }}_{\mathrm{i}})/w_{\mathrm{ph}}+q_{\mathrm{F}}}{{[2(\mathrm{\omega }-{\mathrm{\omega }}_{\mathrm{i}})/w_{\mathrm{ph}}]}^2+1}+\\ +A_{\mathrm{pm}}\frac{{\mathrm{\gamma }}_{\mathrm{pm}}\mathrm{\omega }}{{({\mathrm{\omega }}^2-{\mathrm{\omega }}_{\mathrm{pm}}^2)}^2+4{\mathrm{\gamma }}_{\mathrm{pm}}^2{\mathrm{\omega }}^2}+A_{\mathrm{BG}}[\mathrm{erf}\left(\frac{\mathrm{\omega }-{\mathrm{\omega }}_{\mathrm{BG}}}{w_{\mathrm{BG}}}\right)+1]\end{array}$$ (1)

The fitting model is convoluted with the RIXS energy resolution (∼37 meV). From this analysis, we extracted the phonon dispersion for each ϕ that quantitatively matches the phonon dispersion shown in Fig. 2B. All parameters are kept free, except for ω_BG, which is constrained within a range of [0.2,0.6] eV.

Fitting the phonon dispersion

To obtain a phenomenological form to the dispersions in Fig. 2B, the extracted peak locations as a function of q were fit to a linear background plus a negative Lorentzian function. From this fit, we obtain the q location of the softening (red markers in Fig. 3E). To obtain the error bars in Fig. 3E, we follow a conservative approach by taking the average of the two q intercepts of the fitted curve at E_min + 2 meV, where E_min is the lowest energy of the dispersion and ±2 meV is the typical amount of scatter observed in the data. For ϕ = 45°, the data are fit to a line.

Deconvolution procedure

We used the Lucy-Richardson deconvolution procedure (37) to deconvolve the energy resolution (∼37 meV) from the RIXS spectra (deconvoluted curves for all azimuths are displayed in fig. S3). The number of iterations and region of interest of the deconvolution procedure were optimized by ensuring that the convolution of the deconvoluted curves reproduced the raw data.

Model simulations

The toy models M1 and M2 are purely phenomenological. For M1, the phonon dispersion was modeled as

$$E=E_0-\mathrm{\xi }(\mathrm{\varphi })\frac{\text{Δ}}{{\left(\frac{q-q_0}{\text{Γ}}\right)}^2+1}$$ (2)

where E₀ = 72 meV, Δ = 30 meV, Γ = 0.065 r.l.u., q₀ = 0.29 r.l.u., and ξ(ϕ) = (∣cos (2ϕ)∣ + 0.08)/(1.08). For M2, the phonon dispersion was modeled as

$$E=E_0-{\displaystyle \sum _{i=1}^4}\frac{\text{Δ}}{{\left(\frac{\mathbf{q}-{\mathbf{q}}_i}{\text{Γ}}\right)}^2+1}$$ (3)

where E₀ = 74 meV, Δ = 30 meV, Γ = 0.065 r.l.u., and q_i are the four peaks located at [q_x = ±q_CO, q_y = 0] and [q_x = 0, q_y = ±q_CO] (q_CO = 0.29 r.l.u.).

Supplementary Materials

This PDF file includes:

Figs. S1 to S11

Connection between low- and high-energy QCDCs