YBCO. CuO 2. the CuO 2. planes is controlled. from deviation from. neutron. , blue star for. Hg12011 (this work) for T c = 72

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1 Supplementary Figure 1 Crystal structures and joint phase diagram of Hg1201 and YBCO. (a) Hg1201 features tetragonal symmetry and one CuO 2 plane per primitive cell. In the superconducting (SC) doping range, YBCO features lower, orthorhombic symmetryy and two CuO 2 planes per primitive cell. In both compounds, the charge-carrier concentration in the CuO 2 planes is controlled through the variation of the interstitial O concentration. In YBCO, these O atoms form Cu-O chains along the b-axis, with various inter-chain ordering patterns [14]. The interstitial O atoms in Hg1201 reside in the Hg-O planes and do not order at the doping level studied here. (b) Joint phase diagram of Hg1201 and YBCO. Solid blue (dashed red) line: superconducting dome of Hg1201 [13] and YBCO [14]). Blue (red) symbols correspond to characteristic temperatures of Hg1201 (YBCO): Pseudogap (PG) temperature T*, determined from deviation from linear-t resistivity in the strange metallic (SM) regime and from neutron scattering experiments of the onset of q = 0 magnetism [11,12]. Yellow area: Fermi-liquid (FL) regime between T* ** and the measureable onset of SC fluctuations at T ρ ρ. Grey shaded area: approximate extentt of antiferromagnetic ( AF) phase for YBCO. Stars indicate T CDW, blue star for Hg12011 (this work) and red stars for YBCO [6,7,15,16,38]. Characteristic transport temperatures for Hg1201 are from ref. [8] and from additional measurements for T c = 72 K and 90 K samples (see Methods). 1

2 Supplementary Figure 2 Geometry of the resonant X-ray scattering experiments. (a) The red arrow represents the in-plane component q // of the momentum transfer vector. The 2θ angle was set to 160 deg and 130 deg in the in RXD and RIXS experiment, respectively. Here, q // is negative. (b) As in a but for q // > 0. (c) Structural (green square) and magnetic (blue square) Brillouin zone in the H-K plane. The red bar represents the accessible q // range: ±0.42 r.l.u. and ±0.38 in the RXD and RIXS experiment, respectively. 2

3 Supplementary Figure 3 Diffuse hard-x-ray performed with a point detector at many L values, with K = 0. scattering for Hg1201 sample with T c = 72 K, collected at 70 K. (a) H-scans Solid vertical lines indicate the momentumm position of the CDW peak as observed with resonant X-ray scattering techniques, and the dashed lines mark the FWHM. The main contribution to the observed intensity is thermal diffuse scattering. (b) Two-dimensional H-L cuts of the Ewald sphere. (c) Two-dimensional H-K cuts of the Ewald sphere. The position of the shadedd stripes in b and c corresponds to the expected H CD DW value. The width of the stripes corresponds to the FWHM expected from RXS measurements. The diffraction patterns were recorded with the two- dimensional MAR345 detector. 3

4 Supplementary Figure 4 Temperature dependence of the CDW peak observed by RXD. (a) Momentum scans at several temperatures, with the spectrum collected at 250 K subtracted. The data were fit with a Gaussian function with a common H CDW position. (b) Temperature dependence of the CDW peak amplitude determinedd from the fits presented in a. (c) FWHM of the Gaussian fit to the CDW peak plotted as a function of temperature. The correlation length, defined from the FWHM of the Gaussian as ξ/a = (1/2π/FWHM), is expressed in the lattice units and presented in Fig. 1e. 4

5 Supplementary Figure 5 CDW wave vector and quantum oscillation period. (a) H CDW for Hg12011 (this work), YBCO [6,7,15,16],, Bi2201 [29] and Bi2212 [32] as a function of hole concentration. This dependence is approximately linear for a given compound. Red open circles are dataa for YBCO from ref. 17. Although the data extend to higher doping, they were not includedd in our analysis due to the lack of QO dataa for these doping levels. (b) Frequency of Shubnikov-de-Haas QO revealed by highh magnetic fields at low temperatures (black squares). The grey solid line is a linear fit to the data. The black star indicates a doping level for which both CDW and QO have been measured in YBCO. Red stars represent the QO frequencies estimated form the linear fit for the doping levels at which only CDW order (but not QO) was studied. (c) QO frequency as a function of the H CD DW for Hg1201 (blue star) and YBCO (red stars). The solid line represents a quadratic fit to the data, and the dashed lines show the fit error range. ( d) Fermi pocket size, expressed as a fraction of the first Brillouin zone, versuss H CDW for Hg12011 (blue) and YBCO (red), with doping as intrinsic parameter. The solid line represents a quadratic fit, whichh extrapolates to H CDW = 0.45(2) r.l.u. in the limit of vanishing pocket size. The dashed lines indicate the fit error range. 5

6 Supplementary Table 1 Summary of YBCO sample information * The hole concentration for this sample has been adopted from the T c versus p relationship established in ref. [14]. The hole concentration reported in the original work is p = ** The CDW modulation wave vector was obtained from a fit of a Gauss function to the published data. The value reported in the original work is H CDW Reference Sample O order T C (K) P H CDW (r.l.u.) ξ/a 16 YBa 2 Cu 3 O 6.54 II YBa 2 Cu 3 O 6.6 VIII * YBa 2 Cu 3 O 6.67 VIII YBa 2 Cu 3 O 6.67 VIII YBa 2 Cu 3 O 6.7 VIII ** YBa 2 Cu 3 O 6.75 III

7 Supplementary Note 1 CDW correlations in YBCO CDW order has been observed in YBCO in the doping range p = to [6,7,15,16,38,39]. Depending on the hole concentration the samples displayed various types of oxygens-chain order (O-I, O-II, O-VII and O-III). A summary of samples in which the CDW correlation length was determined is presented in Supplementary Table 1. Supplementary Note 2 Crystal structures and joint phase diagram of Hg1201 and YBCO The structures of Hg1201 and YBCO exhibit considerable differences (Supplementary Fig. 2). Hg1201 features tetragonal symmetry (space group P4/mmm) and one CuO 2 plane per primitive cell. In the SC doping range, YBCO features lower, orthorhombic symmetry (space group P/mmm) and two CuO 2 planes per primitive cell. In both compounds, the charge carrier concentration in the CuO 2 planes is controlled by the interstitial O concentration. In YBCO, these O atoms form Cu-O chains along the b-axis, with various inter-chain ordering patterns [14]. The interstitial O atoms in Hg1201 reside in the Hg-O planes and do not order in the doping range of interest. Supplementary Fig. 2b shows the combined temperature-doping phase diagram of Hg1201 and YBCO, which features an insulating state with antiferromagnetic order at low holedopant concentrations, unusual translational-symmetry-preserving (q = 0) magnetism [11,12] below the pseudogap temperature T*, a Fermi-liquid regime below T** [8-10], and an approximately parabolic superconducting dome T c (p) [13,14]. The CDW in YBCO has been reported below optimal doping, for hole concentrations that correspond to the plateau in T c (p) [6,7,15,16,38,39]. Supplementary Note 3 Hard X-ray diffraction measurements Hard X-ray diffraction measurements were performed to search for lattice distortions accompanying the CDW on the same sample for which we observed the charge instability with 7

8 the resonant techniques. The experiment was performed with a six-circle diffractometer at the 6- ID-D beam line of the Advanced Photon Source. The incident photon energy was set to 82.7 kev, energy just below Hg K-edge, in order to reduce the fluorescent background. The ratio of the background scattering to the (122) Bragg peak intensity was about 2*10-5, and the detection limit for incommensurate CDW scattering was ~ 10-6 of the strong (122) Bragg reflection. No CDW signal was observed. The 1x1x0.5mm 3 sample was mounted on a cold finger of a standard He closed cycle refrigerator, and the measurement was performed in transmission geometry. An energy dispersive point detector and a two-dimenssional MAR-345 detector were used to collect the diffracted photons. In search of CDW scattering, we explored a wide range of momentum space, using the information gained from the RXD and RIXS experiments. Supplementary Fig. 3a presents a series of theta-2-theta scans along the H-direction, with L varying from 5.5 to 9.9 r.l.u. and K = 0. The temperature was set to 70 K. Broad thermal diffuse scattering intensity is observed between the shoulders of the Bragg peaks in the neighborhood of integer values of H. With the intensity of the (102) reflection 6 orders of magnitude above the background level, there is no indication of the CDW peaks at the H positions (marked by solid vertical lines) where CDW peaks were detected in the resonant scattering experiments. The twodimensional cuts of the Ewald sphere shown in Supplementary Fig. 3b,c do not reveal any CDW superstructure peaks. Observation of the CDW order via RXD and lack of its signature in the hard X-ray diffuse scattering of La 2 x Sr x CuO 4 have been initially attributed to the surface nature of the electronic correlations [40]. However, it has been demonstrated shortly after that the absence of CDW peak in the diffuse scattering was caused by the experimental routine and not due to lowdimensionality of the order [41,42]. In order to detect CDW peak via hard X-ray diffraction in Hg1201 further improvement of the sensitivity is required. Supplementary Note 4 CDW and QO in Hg1201 and YBCO To make a connection between the CDW and QO results for Hg1201 and YBCO, the frequency of the oscillations is plotted as a function of the CDW modulation vector H CDW (see Supplementary Fig. 5 and inset of Fig. 3c). Although the dependence appears to be linear (due to relatively narrow H CDW range) a similarly good fit has been achieved when a quadratic function 8

9 has been used: the adjusted for the number of predictors R-Squared is for linear fit and for quadratic function of the form F = a + b/h CDW, where a = -415(60) and b = 95(6). The form of the function connecting F and H CDW has been chosen based on the experimental data in connection with the prediction of the expected H CDW in the limit of vanishing Fermi pocket size (F = 0 T). The fit of the quadratic function to the experimental data gives H CDW = 0.45(2) r.l.u. in the limit of vanishing Fermi pocket which should occur for very low hole concentration and result in H CDW spanned between the nodes. This dependence suggests decreasing of the CDW modulation vector with expansion of the FS that gives rise to QO. As frequency is directly proportional to the FS area (F [T]= ħ/2πe S, where; ħ is Planck s constant, e is elementary charge and S is FS area in Angstroms), the size of the FS expressed as a fraction of BZ (S ab/(2π) 2 ) is a quadratic function of H CDW. This is in agreement with scenario in which the electron pockets at nodal points of the BZ give rise to QO with frequency proportional to the area of the electron pocket and with H CDW connecting parts of reconstructed Fermi surface [22]. A simple schematic model is shown in Fig. 3d. An increase of the area of the electron pocket, spanned on the Fermi arc, is related to a decrease of H CDW connecting two parts of FS. Supplementary References 38. Blanco-Canosa, S. et al. Resonant X-ray Scattering Study of Charge Density Wave Correlations in YBa 2 Cu 3 O 6+x. Phys. Rev. B 90, (2014). 39. Hücker M. et al., Competing charge, spin, and superconducting orders in underdoped YBa 2 Cu 3 O y, Phys. Rev. B 90, (2014). 40. Wu H.-H. et al., Charge stripe order near the surface of 12-percent doped La 2 x Sr x CuO 4. Nat. Comm. 3, 1023 (2012) 41. Croft T. P. et al., Charge density wave fluctuations in La 2 x Sr x CuO 4 and their competition with superconductivity. Phys. Rev. B 89, (2014) 42. Thampy V. et al., Rotated stripe order and its competition with superconductivity in La 1.88 Sr 0.12 CuO 4. Phys. Rev. B 90, (R) (2014) 9