Stripe Symmetry of Short-range Charge Density Waves in Cuprate Superconductors

The omnipresence of charge density waves (CDWs) across almost all cuprate families underpins a common organizing principle. However, a longstanding debate of whether its spatial symmetry is stripe or checkerboard remains unresolved. While CDWs in lanthanum- and yttrium-based cuprates possess a stripe symmetry, distinguishing these two scenarios has been challenging for the short-range CDW in bismuth-based cuprates. Here, we employed high-resolution resonant inelastic x-ray scattering to uncover the spatial symmetry of the CDW in Bi$_2$Sr$_{2-x}$La$_{x}$CuO$_{6+\delta}$. Across a wide range of doping and temperature, anisotropic CDW peaks with elliptical shapes were found in reciprocal space. Based on Fourier transform analysis of real-space models, we interpret the results as evidence of unidirectional charge stripes, hosted by mutually 90$^\circ$-rotated anisotropic domains. Our work paves the way for a unified symmetry and microscopic description of CDW order in cuprates.

However there has been a longstanding controversy, that is, whether its spatial symmetry, i.e., the spatial distribution of the CDW ordered state, is bidirectional "checkerboard" or unidirectional "stripe" [22].Although the presence of intertwined spin-charge stripe order is widely perceived, especially in lanthanum (La)based cuprates [4], the formation of mutually 90 • -rotated charge stripe domains due to two equivalent Cu-O bond directions in square-lattice CuO 2 plane leads to experimental results virtually indistinguishable from those of checkerboard CDW.For example, both CDW symmetries commonly generate four-fold electronic diffraction peaks along H and K directions in reciprocal space [23,24].Recent resonant x-ray scattering (RXS) experiments employed uniaxial pressure as an external symmetry-breaking perturbation to lift the degeneracy of two symmetry candidates, and successfully revealed a unidirectional nature of the CDW in La- [25,26] and yttrium (Y)-based cuprates [27,28].For bismuth (Bi)-based compounds, one of three prototypical material classes in cuprates, the CDW symmetry remains elusive.A plethora of scanning tunnelling microscopy (STM) studies reported a short-range checkerboard-like electronic modulation along Cu-O bond directions in real space with a four-unit-cell periodicity [12][13][14][15][16][17][18].Interestingly, several STM studies revealed unidirectional nanodomains of electronic modulation breaking fourfold rotational symmetry [29][30][31].
Elucidating the CDW symmetry in Bi-based cuprates will therefore be indispensable for answering the question: is the unidirectional "stripe" CDW symmetry universal for all different cuprate materials?Since the CDW is intimately related with superconductivity, this question may be essential for understanding the interplay of CDW, SC and other broken-symmetry states.For example, there have been a number of theoretical works suggesting that the fluctuating stripe may play a significant role in the mechanism of SC [32][33][34][35][36].In addition, a unidirectional pair-density-wave order (PDW), where a SC order parameter is intertwined with CDW and periodically modulated, has been proposed as a "primary" order in cuprates [37].The exact answer to the ques- tion of CDW spatial symmetry is pivotal to differentiate microscopic models of PDW order parameter [38].
To address this question, we performed high-resolution resonant inelastic x-ray scattering (RIXS) measurements on Bi 2 Sr 2−x La x CuO 6+δ (La-Bi2201) to map out the CDW patterns in a broad range of reciprocal space.Our systematic studies reveal that the CDW patterns in La-Bi2201 possess an elliptical shape, elongated along the transverse direction.This anisotropic CDW pattern was robustly found in a wide range of temperatures and doping concentrations.By Fourier-transform analysis of real-space models, we concluded that equally populated 90 • -rotated domains of charge stripe orders lead to anisotropic CDW peaks.Our results not only provide evidence for the unidirectional stripe symmetry in Bi-based cuprates, but also suggest that the stripe CDW symmetry is another universal property shared among different cuprate material families.

Anisotropic CDW peaks
The CDW order in La-Bi2201 system manifests itself as quasielastic scattering intensity peaks at four symmetryequivalent positions in reciprocal space, (H, K) = (±δ, 0) and (0, ±δ), where the incommensurability δ ∼ 1/4 reciprocal lattice unit (r.l.u.) is consistent with the previous reports [16,39].By rotating the in-plane azimuthal angle φ with respect to the CDW peak, we obtained momentum-dependent RIXS spectra across the CDW reflections at (δ, 0) and (0, δ), hereafter denoted as H-and K-CDW, with different scan angle α as illustrated in Fig. 1a.Here, α is defined as the angle between the actual scan and the longitudinal propagating direction.The incoming photon energy was tuned to the resonances of Cu L 3 -edge (∼931.6 eV) and O K-edge (∼528.4eV) absorption energy to maximize the sensitivity to CDW.Energy-resolved high-resolution RIXS has an advantage of disentangling a weak quasielastic short-range CDW response from the phonon modes as well as the spin and orbital excitations (Fig. 1b,c), comparing to the energy-integrated RXS [19].An enhancement in the integrated quasielastic scattering intensity is clearly identified near Q = (0.25, 0) and (0, 0.25) at both Cu and O sites, confirming its multiorbital nature [39] (Fig. 1d,e).
To gain an overview of CDW patterns, we show representative CDW scans obtained at Cu L 3 -edge at selected values of α across H-CDW (Fig. 1f) and K-CDW (Fig. 1g) of underdoped La-Bi2201 (x = 0.6, UD23K).The full-width half-maximum (FWHM) of H-CDW in the transverse direction (α = 90 • ) is noticeably larger than that in the longitudinal direction (α = 0 • ), demonstrating an anisotropic CDW peak structure (Fig. 1f).Examination of K-CDW reveals the same anisotropy with comparable FWHMs (Fig. 1g).We also found the elongated CDWs present along both H and K directions at O K-edge (see Supplementary Information).
We now scrutinize the analysis on the shape of the individual CDW peaks.Fig. 1h summarises the extracted FWHM of H-CDW from both Cu L 3 -and O K-edges as well as those of K-CDW as a function of α from -97.5 • to 150 • (Supplementary Figs. 2 and 3).Clearly the data show an oscillatory behavior as a function of α.The least square fit of the data to an elliptical equation (the grey solid line in Fig. 1h, Methods) returns the maximum ∼ 0.118 r.l.u.near α = 90 • and the minimum ∼ 0.066 r.l.u.near α = 0 • .The major and minor axes along 0 • and 90 • indicate that the CDW propagation is locked along two orthogonal Cu-O bond directions.The aspect ratio of 1.8 implies the CDW correlation, i.e., ξ = 2/FWHM, in the longitudinal direction (ξ ∼18.7 Å) is almost twice as long as that in the transverse direction (ξ ⊥ ∼10.4 Å).In Fig. 2c,d, the same set of data are plotted in polar coordinates.The shape of the individual CDW peak is best described as an ellipse elongated along the transverse direction, perpendicular to the direction of CDW propagation.The K-CDW ellipse possesses almost the same size but rotated by 90 • , implying that the CDW pattern in reciprocal space preserves fourfold (C 4 ) symmetry, seemingly reflecting the nature of the underlying CuO 2 square-lattice.

Temperature and doping evolution
We extended the same RIXS study to explore the doping and temperature dependence.length, ξ , and the transverse correlation length, ξ ⊥ , of the samples with different doping concentration x. ξ (x) reaches its maximum at x = 0.6.On the other hand, ξ ⊥ (x) shows a monotonically decreasing trend.The temperature dependence of ξ and ξ ⊥ in OD30K sample are plotted in Fig. 2h.ξ increases much faster below T c , whereas ξ ⊥ displays a continuous linear trend.These results confirm the robust existence of the elongated CDW in a wide range of temperature and doping concentration.

Modelling CDW patterns
The CDW peak patterns in reciprocal space probed by RIXS encode the spatial symmetry of the CDW order in real space.We thus performed theoretical simulations by constructing idealized models, that is, the real-space electronic modulation with either stripe or checkerboard pattern, where both anisotropic and isotropic domains are considered.For each scenario, two individual CDW domains are included with a 90 • rotation from each other to mimic the equally populated orthogonal domains (insets of Fig. 3a-d, see Supplementary Information).The real-space electron density modulation maps ρ(x, y) were built by translating the orthogonal CDW domains in the x and y directions with a random phase shift to represent the actual samples probed by RIXS (Fig. 3a-d).Fourier transforms of the electron density maps ρ(x, y) yield a unique CDW reciprocal-space pattern for each scenario (Fig. 3e-h).
For instance, an anisotropic stripe symmetry produces elliptical CDW diffraction peaks, preserving the C 4 symmetry but elongated in the transverse direction (Fig. 3e).The anisotropic checkerboard domains give rise to star-shaped CDW peaks elongated along both the transverse and longitudinal directions (Fig. 3f).On the other hand, isotropic checkerboard and isotropic stripe phases generate virtually indistinguishable isotropic CDW peaks (Fig. 3g,h).Remarkably, our experimental observations in La-Bi2201 (Fig. 2) coincide with the spatial symmetry of CDW order having unidirectional stripe, rather than bidirectional checkerboard.A consistent conclusion is achieved based on similar Fourier transform analysis on more realistic electron density maps (Supplementary Fig. 6) [40].

DISCUSSION
At the first glance, it may look perplexing that we straightforwardly obtain the unambiguous agreement between experiment and theoretical modeling, while the CDW symmetry is still debatable despite a considerable number of STM studies [11][12][13][14][15].The static checkerboard-like modulation was proposed for Bi-based cuprates from tunneling electron conductance maps.In their Fourier-transformed images, fourfold symmetryequivalent peaks typically appear at an incommensurate wavevector position [12][13][14][15][16][17][18].However, the conservation of C 4 symmetry is not a unique signature of the checkerboard-like modulation pattern as the coexistence of orthogonal domains of unidirectional stripe-like CDW also produces fourfold peak pattern.More sophisticated analysis on tunneling asymmetry at the pseudogap energy scale reveals the presence of four-unit-cell-wide unidirectional nanodomains breaking C 4 rotational symmetry [29][30][31].This observation can be interpreted as a fluctuating "stripe" with short correlation.It is worth noting that the nature of short-range CDW order imposes severe challenges on the assessment of the symmetry based on real-space images which manifest visually checkerboard-like CDW patterns regardless of whether its intrinsic symmetry is stripe or checkerboard, as shown by our modelled density maps (Fig. 3a-d) as well as advanced theoretical studies [41].In this sense, RIXS is more advantageous as it probes the CDW spatial symmetry directly in reciprocal space, despite the short-range nature of the CDW order.Elongated CDW peaks in reciprocal space were also found in Y-based cuprates by RXS and proposed as evidences of unidirectional stripe-like CDW order [23].The interpretation was challenged as the corresponding realspace correlation function C(x, y), i.e., the direct Fourier transform of the reciprocal-space CDW pattern, similar to Fig. 3e, appears checkerboard-like [40].The calculation of C(x, y) via auto-correlation of the real-space electron density maps (Fig. 3a-d) confirmed all four scenarios induce comparable checkerboard-like patterns akin to previous reports (Supplementary Fig. 5) [40].Therefore, it is challenging to directly visualize the symmetry of a short-range CDW from either the electron density map ρ(x, y) or correlation function C(x, y) in real space.Instead of examining the shape of CDW peaks, a recent RIXS study under uniaxial pressure reveals a unidirectional nature of CDW in Y-based cuprates [28].The anisotropic CDW peaks we observed in La-Bi2201 are in stark contrast to isotropic CDW peaks reported in Labased [42][43][44][45] and Hg-based cuprate materials [46].For La 1.88 Sr 0.12 CuO 4 , twinned domains of slanted stripe order with ±3 • away from Cu-O bond directions result in two CDW peaks along transverse direction, eventually merging into a single isotropic peak under mild uniaxial pressure [45].Despite the isotropic peak shape, x-ray diffraction employing uniaxial pressure unambiguously verifies charge stripe order [25].
Alongside the results from La-and Y-based cuprates, our RIXS measurements on La-Bi2201 corroborates unidirectional stripes as the fundamental symmetry of CDW across different cuprate families.This remarkable universality not only provides a perspective on the unified organizing principle of CDW and its role in cuprate phase diagram, but also constrains the symmetry in microscopic models describing CDW and the interplay with other phases.

Sample growth and characterization
High-quality single crystals of Bi 2 Sr 2−x La x CuO 6+δ (La-Bi2201) with hole-doping concentrations of x = 0.8 (non-SC), x = 0.6 (T c = 23 K, UD23K), x = 0.4 (T c = 30 K, OD30K) were synthesized by the travellingsolvent floating-zone method.The as-grown samples were annealed at 650 • C in an oxygen atmosphere for 2 days to improve the homogeneity of the samples.The sample orientation was characterized by Laue diffraction prior our resonant x-ray inelastic scattering (RIXS) experiments.
Superstructure reflections propagating along Cu-Cu bond direction, or equivalently along (H, H) direction, were identified by the Laue diffraction method.All samples were cleaved by mechanical exfoliation, before transferring to a loadlock chamber.
X-ray absorption spectroscopy X-ray absorption spectra of La-Bi2201 samples were collected with a normal incidence geometry (θ = 90 • ) prior to the RIXS experiments.The total electron yield method was employed by measuring the drain current from the samples.Supplementary Figs.1a,b show representative x-ray absorption spectra of La-Bi2201 x = 0.6 (UD23K) near Cu L 3 and O K absorption edges, respectively.The data exhibit a single resonance peak at an energy of 931.6 eV and 528.4 eV.

High-resolution RIXS experiments
We performed high-resolution RIXS experiments on La-Bi2201 at the I21 RIXS beamline of Diamond Light Source in the United Kingdom [47].The experimental geometry is schematically illustrated in Fig. 1a of the main text.Samples were glued on the holder such that its surface normal is the crystallographic c-axis direction.We define a wavevector Q employing pseudo-tetragonal notation Q = Ha * + Kb * + Lc * , or equivalently, Q = (H, K, L) where a * = 2π/a, b * = 2π/b, c * = 2π/c are the reciprocal lattice unit vectors and H, K, L are the indices in reciprocal space spanned by a * , b * , and c * .The lattice parameters of La-Bi2201 are a = b = 3.86 Å and c = 24.69Å.The incoming x-rays were tuned to the resonant energy of Cu L 3 and O K-edge to maximize the signal sensitivity to CDW residing in the CuO 2 plane.Linear vertical, or σ polarization was used throughout the measurements.The outgoing x-rays emitted from the La-Bi2201 sample pass through paraboloidal collecting optics with a large horizontal acceptance angle inside the main sample chamber to enhance the x-ray photon throughput.The spherical grating optics implemented on a spectrometer to vertically monochromatise emitted x-rays and focus to an area detector.The energy resolution of 37.0 meV and 27.3 meV was achieved for Cu L 3 -and O K-edge, respectively.The scattering angle 2θ is fixed to 154 • .

RIXS data fitting
RIXS spectra were firstly normalized to the incident photon flux and the spectral weight of dd excitation integrated over an energy range of 1.2-3 eV.The position of the quasi-elastic peak, or zero-energy position, was determined by measuring the amorphous carbon sample.Compared to energy-integrated resonant elastic x-ray scattering (REXS), the energy-resolved RIXS has the advantage of filtering out unwanted contributions from inelastic scatterings due to various low-energy excitations such as phonon, magnon, and orbital (dd ) excitations.To better disentangle quasielastic diffraction intensity, the low-energy region of RIXS spectra measured at the Cu L 3 -edge was fitted with a sum of quasielastic, phonon, paramagnon, and background contributions as follows: Similarly for the O K-edge, Four terms on the right-hand side of Eq. ( 1) represent a quasielastic, phonon, paramagnon, and background contribution on the RIXS intensity, respectively.The quasielastic peak and phonon excitation are described by Gaussian functions, whereas the paramagnon excitation is modelled by the response function of a damped harmonic oscillator multiplied with the Bose factor [48], as follows: (5) An integer n indicates the number of phonon modes: n = 1 (only bond-stretching mode) for the Cu L 3 -edge and n = 2 (bond-stretching and bond-buckling modes) for the O K-edge [39].

Angular-dependent CDW measurement
Taking advantage of broad accessibility to in-plane rotation angle and accurate control of sample manipulator, we obtained angle-and momentum-dependent RIXS spectra across both H-and K-CDWs.By controlling the polar angle θ, the azimuthal rotation angle φ, and the tilt angle χ using diffcalc, momentum-dependent RIXS scans can be obtained across the CDW peak with different scan angle α.Here, the scan angle α is defined as an angle between the scan direction and the longitudinal scan direction for each CDW.The representative scan directions are schematically described in the inset of Figs.1f,g.For Cu L 3 -edge, the CDW peaks were measured in a range between -97.5 A full-width half-maximum (FWHM) of CDW peak is calculated by 2 √ 2ln2G 3 for each momentum-dependent profile.In Supplementary Figs. 2 and 3, the least-square fitted results are shown with solid lines at different scan angle α.The FWHMs of CDW peak are written in each panel.The angle-dependent FWHM shown in Fig. 1h is fitted with an elliptical equation stated below: where R 1 (R 2 ) is the major (minor) axis of the ellipse and β is an offset angle.The grey-coloured line presented in Fig. 1h is the result of least square fitting which gives R 1 = 0.1184 ± 0.0022 r.l.u., R 2 = 0.0656 ± 0.0013 r.l.u., and β = 1.2 ± 1.9 • .The offset angle β ∼ 0 indicates that the CDW propagation is locked along two mutually-orthogonal Cu-O bond directions.For La-Bi2201 x = 0.6 (UD23K) samples, the data were obtained for three sample pieces from different batches.

FIG. 1 .
FIG. 1. Angle-dependent RIXS experiment on charge-density wave (CDW) in La-Bi2201.a, Schematic illustration of experiment.By changing an in-plane sample rotation φ, H-and K-CDWs are sliced along different directions, defined by azimuthal angle α, in (H, K) plane.b,c, Representative RIXS spectra at Q = (0.25, 0) with Cu L3 (b) and O K resonance (c).Spectral weight from quasielastic, phonon, and paramagnon contribution is highlighted by red or cyan, dark-grey and light-grey color, respectively.d,e, Integrated intensity of quasielastic peaks extracted from RIXS spectra as a function of H at Cu L3and O K-edge.Solid lines are Gaussian fits with polynomial background.f,g, Background-subtracted momentum-dependent profiles of integrated intensity across H-CDW (f ) and K-CDW (g) at Cu L3-edge with different scan angle α.The insets depict scan directions.The number on the right side of each profile corresponds to fitted full-width half-maximum (FWHM).h, FWHMs of H-and K-CDW peaks measured at both absorption edges are plotted as a function of α.The grey solid line indicates the least square fit to elliptical equation (See Methods).All data were obtained at T = 20 K.

FIG. 2 .
FIG.2.Doping and temperature evolution of anisotropic CDW correlation.a-f, Full-width half-maximum of both H-CDW (upper panels) and K-CDW (lower panels) peaks in La-Bi2201 with different doping concentrations as stated.Solid lines are least square fits to elliptical equation (see Methods).g,h, Doping concentration x (g) and temperature T dependence (h) of longitudinal and transverse CDW correlations.The grey dashed line indicates the superconducting transition temperature Tc.The colored trends are guided to the eye.

25 H 25 H 25 H 25 H
FIG. 3. Model CDW modulation in real and reciprocal space.a-d, Real-space electron density modulation map ρ(x, y) calculated by assuming equal population of (a) anisotropic 90 • -rotated stripe domains, (b) 90 • -rotated anisotropic checkerboard domains, (c) 90 • -rotated isotropic stripe domains, and (d) isotropic checkerboard domains.e-h, Corresponding CDW peak patterns in reciprocal space, obtained by direct Fourier transformation of (a-d), respectively.

and C 1 , C 2 , C 3
are the fitting parameters.C 4 was fixed to k B T with the measuring temperature T .

Supplementary Fig. 2 .
Charge density wave measured at Cu L3-edge.a-r, Momentum-dependent profile of integrated quasi-elastic charge-density-wave (CDW) peak intensity measured at different scan angle α as indicated with a bold text.The incident photon energy was fixed to the resonance of Cu L3 absorption edge.The solid line is fit to Gaussian function representing the CDW peak with a composite background (See Methods).The full-width half-maximum (FWHM) of CDW peak is also presented on each panel.

3 .Supplementary Fig. 4 .Supplementary Fig. 5 .
Charge density wave measured at O K -edge.a-h, Momentum-dependent profile of integrated quasi-elastic charge-density-wave (CDW) peak intensity measured at different scan angle α as indicated with a bold text.The incident photon energy was fixed to the resonance of O K absorption edge.The solid line is fit to Gaussian function representing the CDW peak with a polynomial background.The full-width half-maximum (FWHM) of CDW peak is also presented on each panel.individual CDW domains.a,b, Electronic density modulation of an anisotropic stripe domain propagating into horizontal x (a) and vertical y direction (b) with four-unit-cell periodicity.c-g, Same for horizontally (c) and vertically anisotropic checkerboard domains (d), isotropic charge stripe domain propagating into horizontal x (e) and vertical y direction (f ), and istoropic checkerboard domain (g).These domains are used as building blocks for modeling real-space electronic density map ρ(x, y), illustrated in Supplementary Fig. 5. Modeling CDW modulation using simplified electron density maps.a-d, Real-space electronic density modulation map ρ(x, y) generated by equal population of individual charge stripe and checkerboard domains.The insets represent the microscopic structure of individual domains for each scenario.e-h, Structure factor S(H, K) in reciprocal space, calculated by a direct Fourier transform of the corresponding ρ(x, y) in (a-d), respectively.i-l, Densitydensity correlation map C(x, y) directly computed by taking the auto-correlation of real-space map ρ(x, y) in (a-d).
• and 150 • .For O K-edge, α was varied from 0 • to 135 • .Supplementary Figs.2 and 3present the momentum-dependence of integrated quasielastic intensity scans on H-CDW taken at different α as indicated.The momentum-dependence data is fitted with a combination of Gaussian functions, which represents a CDW peak, and a composite background.