Commensurate 4a₀-period charge density modulations throughout the Bi₂Sr₂CaCu₂O_8+x pseudogap regime

Significance

Strong Coulomb interactions between electrons on adjacent Cu atoms result in charge localization in the cuprate Mott-insulator state. When a few percent of electrons are removed, both high-temperature superconductivity and exotic charge density modulations appear. Identifying the correct fundamental theory for superconductivity requires confidence on whether a particle-like or a wave-like concept of electrons describes this physics. To address this issue, here we take the approach of using the phase of charge modulations, available only from atomic-scale imaging. It reveals a universal periodicity of the charge modulations of four crystal unit cells. These results indicate that the particle-like concept of strong interactions in real-space provides the intrinsic organizational principle for cuprate charge modulations, implying the equivalent for the superconductivity.

Abstract

Theories based upon strong real space (r-space) electron–electron interactions have long predicted that unidirectional charge density modulations (CDMs) with four-unit-cell (4a₀) periodicity should occur in the hole-doped cuprate Mott insulator (MI). Experimentally, however, increasing the hole density p is reported to cause the conventionally defined wavevector Q_A of the CDM to evolve continuously as if driven primarily by momentum-space (k-space) effects. Here we introduce phase-resolved electronic structure visualization for determination of the cuprate CDM wavevector. Remarkably, this technique reveals a virtually doping-independent locking of the local CDM wavevector at $\left|{\mathit{Q}}_0\right|=2\mathit{\pi }/4{\mathit{a}}_0$ throughout the underdoped phase diagram of the canonical cuprate Bi₂Sr₂CaCu₂O₈. These observations have significant fundamental consequences because they are orthogonal to a k-space (Fermi-surface)–based picture of the cuprate CDMs but are consistent with strong-coupling r-space–based theories. Our findings imply that it is the latter that provides the intrinsic organizational principle for the cuprate CDM state.

Strong Coulomb interactions between electrons on adjacent Cu sites result in complete charge localization in the cuprate Mott insulator (MI) state (1). When holes are introduced, theories based upon the same strong r-space interactions have long predicted a state of unidirectional modulation of spin and charge (2–11), with lattice-commensurate periodicity for the charge component. Experimentally, it is known that even the lightest hole doping of the MI state produces nanoscale clusters of charge density modulations (CDMs) (12, 13), which implies immediately that r-space interactions predominate. However, with increasing hole density p, the conventionally defined wavevector Q_A of the CDM is reported to increase (14) or diminish (15) continuously as if driven primarily by k-space (Fermi surface) effects. Distinguishing between the r-space and k-space theoretical perspectives is critical to identifying the correct fundamental theories for the phase diagram and Cooper pairing mechanism in underdoped cuprates. Here we introduce an approach to this challenge by applying phase-resolved electronic structure visualization (16–18) in combination with the technique of phase demodulation–residue minimization, to explore the CDM wavevector. Using these methods, we visualize the phase discommensurations (19) and their influence on the doping dependence of both the conventionally defined Q_A and the fundamental local wavevector Q₀ of the underdoped cuprate CDM state.

CDMs in the Pseudogap Phase

As holes are introduced into the CuO₂ plane of the MI, the first nonmagnetic state to appear is the pseudogap (PG) phase (Fig. 1A). It contains nanoscale CDM clusters (12, 13) even at lowest hole-density p; near p = 0.06 these CDM clusters percolate and the superconducting state appears (12). X-ray scattering experiments (15) now report a robust CDM state throughout the range 0.07 < p < 0.17 spanning the pseudogap regime. Both the PG and the CDM states terminate somewhere near p ∼ 0.19 and give way to a simple d-wave superconductor. A fundamental reason for the great difficulty in understanding this complex phase diagram has been the inability to discern the correct theoretical starting point. Should one focus on the intense r-space electron–electron interactions that form the basis for the parent MI state? Or should one focus upon a Fermi surface of momentum space eigenstates representing delocalized electrons?

Fig. 1.

(A) Schematic phase diagram of hole-doped cuprates. The high-temperature superconductivity coexists with d-symmetry form factor charge modulations (compare blue and pink “domes,” respectively) through most of the underdoped regime. (B) Typical measured ${\psi }_R\left(\mathit{r},150\hspace{0.17em}\text{meV}\right)$ image of Bi₂Sr₂CaCu₂O_8+x in the charge modulation phase. The subatomic resolution image shows a typical charge modulation pattern of d-symmetry form factor, spanning eight lattice constants horizontally (compare with E and F), and with an overlay showing how the d-symmetry form factor affects each oxygen site. The green crosses mark positions of Cu atoms in the underlying CuO₂ plane. (C) Typical ${\psi }_R\left(\mathit{r},150\hspace{0.17em}\text{meV}\right)$ of underdoped Bi₂Sr₂CaCu₂O_8+x. The short-range nature of the charge modulations is clear. (D) The d-symmetry form factor Fourier amplitudes $\left|{\stackrel{\sim }{\psi }}_R\left(\mathit{q},150\text{meV}\right)\right|$ calculated using complex Fourier transforms of sublattice-resolved images O_x(r);O_y(r) derived from C. (E) Modeled d-form factor charge density wave that represents an incommensurate modulation by having a horizontal wavevector of length $Q=0.311\times 2\pi /a,$ where $a$ is the lattice unit. The density values are sampled and color coded at each Cu site (green crosses), O site, and center of CuO₂ plaquette, to emphasize the modulation pattern and relation to the underlying lattice. The initial phase is chosen so that the modulation on the leftmost line of Cu sites matches in value the commensurate modulation in F. Incommensurate modulations such as the one shown naturally arise from Fermi surface instabilities and therefore have the Fermi surface nesting wavevector $Q=2k_F$ and period $2\pi /Q=\pi /k_F.$ The dashed line is the profile of the density wave along the horizontal direction, without imposing a d-form factor intraunit-cell structure, and the period is marked by the length of the double arrow. (F) Modeled d-form factor charge density wave that is commensurate, having wavevector $Q=1/4\times 2\pi /a$ and period $2\pi /Q=4a$ with $a$ the lattice unit. The density values are sampled and color coded at each Cu site (green crosses), O site, and center of CuO₂ plaquette, to emphasize the modulation pattern and relation to the underlying lattice. The initial wave phase is chosen so that the modulation maximum occurs on a horizontal O site. The dashed line is the profile of the density wave along the horizontal direction and the period is marked by the length of the double arrow.

A new opportunity to address these questions has emerged recently, through studies of the CDM phenomena now widely observed in underdoped cuprates (15, 20, 21). Pioneering studies of CDMs in La_2-xBa_xCuO₄ and La_2-x-yNd_ySr_xCuO₄ near p = 0.125 discovered charge modulations of period 4a₀ or Q = ((1/4,0);(0,1/4))2π/a₀ (14, 21). The initial intuitive explanation for a periodicity that was half that predicted from a Hartree–Fock momentum space treatment invoked an r-space model involving local magnetic moments whose antiferromagnetic order becomes frustrated upon hole doping. A variety of powerful theoretical techniques (2–11) support this strongly interacting r-space viewpoint. In the interim, however, CDM phenomena have been discovered within the pseudogap regime of many other underdoped cuprates (15, 20, 21). In these studies, the magnitude of the conventionally determined CDM wavevector Q is reported to increase/diminish with increasing p, as if evolution of momentum space electronic structure with carrier density is the cause. Thus, distinguishing between an r-space–based and a Fermi-surface–based theoretical approach to the cuprate CDM remains an outstanding and fundamental challenge and one that is key to the larger issue of controlling the balance between different electronic phases. The reason is that, in the k-space context (22–25), competition for spectral weight at the Fermi surface between different electronic states including the superconductivity is a zero sum game: Suppressing one state amplifies another and vice versa. By contrast, in the strong interaction r-space context (2–11), the physics of holes doped into an antiferromagnetic MI yield “intertwined” states (4, 8–11, 26, 27) including superconductivity, where closely related ordered states are generated simultaneously by the same microscopic interactions.

CDMs and Phase Discommensurations

Understanding the cuprate CDM phenomenology has proved challenging (28–30) because its q-space peaks are typically broad with spectral weight distributed over many wavevectors (15–21) and also because of the form factor symmetry (17, 31–33). For example, Fig. 1 B and C shows a typical image of the electronic structure of underdoped Bi₂Sr₂CaCu₂O_8+x ${\psi }_R\left(\mathit{r}\right);$ here ${\psi }_R\left(\mathit{r},150\hspace{0.17em}\text{meV}\right)\equiv I\left(\mathit{r},150\hspace{0.17em}\text{meV}\right)/I\left(\mathit{r},-150\hspace{0.17em}\text{meV}\right)$ and $I\left(\mathit{r},V\right)$ is the measured tunnel current at position r for bias voltage V. The CDM exhibits a d-symmetry form factor meaning that modulations at the x-axis–oriented planar oxygen sites O_x(r) are π out of phase with those at y-axis–oriented oxygen sites O_y(r), as shown schematically by the overlay in Fig. 1B. Thus, the complex Fourier transforms ${\stackrel{\sim }{O}}_x\left(\mathit{q}\right);{\stackrel{\sim }{O}}_y\left(\mathit{q}\right)$ of sublattice-resolved images O_x(r);O_y(r) that are derived (17, 33) from ${\psi }_R\left(\mathit{r}\right)$ yield the d-symmetry form factor Fourier amplitudes $\left|{\stackrel{\sim }{\psi }}_R\left(\mathit{q}\right)\right|$ = $\left|{\stackrel{\sim }{O}}_x\left(\mathit{q}\right)-{\stackrel{\sim }{O}}_y\left(\mathit{q}\right)\right|$ shown in Fig. 1D (SI Text, d-Symmetry Form Factor of CDMs). One sees directly the wide range of q values that exist under each CDM peak in ${\stackrel{\sim }{\psi }}_R\left(\mathit{q}\right)$ (dashed boxes, Fig. 1D). Such broad peaks indicate quenched disorder of the CDM but with two quite distinct possibilities for the identity of the fundamental ordered state: (i) an incommensurate CDM state whose wavevector Q evolves continuously along with the Fermi wavevector k_F (e.g., Fig. 1E) but is perturbed by disorder or (ii) a commensurate CDM with constant fundamental wavevector Q₀ driven by strong-coupling r-space effects (e.g., Fig. 1F), but whose wavevector defined at the maximum of the fitted Fourier amplitudes, ${\mathit{Q}}_A,$ evolves due to changing arrangements of discommensurations (DC).

McMillan (19) defined a “discommensuration” as a defect in a commensurate CDM state where the phase of the CDM jumps between discrete lattice-locked values. For example, consider a sinusoidal modulation in one spatial dimension with a commensurate period 4a₀:

$$\psi \left(x\right)\equiv A\exp \left[i\mathrm{\Phi }\left(x\right)\right]=A\exp \left[i\left(Q_{0}x+\phi \right)\right].$$ [1]

Here $Q_0=2\pi /4a_0$ is the commensurate wavevector, A is the amplitude, and $\mathrm{\Phi }\left(x\right)=Q_0\hspace{0.17em}x+\phi$ is the position-dependent phase argument of the function $\psi \left(x\right).$ To form phase-locked regions, the phase offset can take one of four discrete values $\frac{\phi }{2\pi }=0,\hspace{0.17em}\frac{1}{4},\hspace{0.17em}\frac{2}{4},\frac{3}{4}.$ The DCs then form boundaries between these regions as indicated by different colors in Fig. 2 A, i and ii. When a commensurate CDM (Eq. 1) is frustrated by Fermi-surface–based tendencies, a regular DC array allows more/fewer modulations to be accommodated through successive jumps in phase (Fig. 2 A, ii and ref. 19). The result is a new phase-averaged wavevector $\overline{Q}$ that depends on the profile of the DC array $\phi \left(x\right)$ through $Q_{0}x+\phi \left(x\right)\equiv \overline{Q}x+\stackrel{\sim }{\phi }\left(x\right),$ where the slope $\overline{Q}$ is chosen so that the residual phase fluctuations $\stackrel{\sim }{\phi }\left(x\right)$ average to zero $\left(\overline{\stackrel{\sim }{\phi }\left(x\right)}=0\right).$ Graphically, finding the $\overline{Q}$ is then equivalent to finding the best linear function for $\mathrm{\Phi }\left(x\right)$ as shown in Fig. 2 A, ii. In this case, the difference in slope between the commensurate and phase-averaged wavevectors, $Q_0$ and $\overline{Q},$ is called the incommensurability $\delta \equiv \overline{Q}-Q_0$ (Fig. 2 A, iii) of such a nominally incommensurate phase. Note that such a DC array does not affect the correlation length of the CDM even though it does shift the Fourier amplitude-defined wavevector $Q_A$ to value $\overline{Q}$ from the fundamental commensurate wavevector $Q_0$ (Fig. 2 A, iii). In contrast, when the combined phase jumps of all of the DCs average to zero (Fig. 2 B, i and ii), the phase-averaged wavevector $\overline{Q}$ equals the local commensurate wavevector; i.e., $\overline{Q}=Q_0.$ Here, in the absence of additional amplitude disorder, the $Q_A$ should also peak at $\overline{Q}=Q_0$ (Fig. 2 B, iii). However, in the most realistic case where disorder in the CDM amplitude and the random spatial arrangement of DCs coexist, $Q_A$ is demonstrably a poor measure of the fundamental commensurate wavevector $\overline{Q}=Q_0$ (Fig. 2 C, i and ii; SI Text, Heterogeneity and Demodulation Residue; and SI Text, Statistical Analysis: Two-Dimensional Fitting).

Fig. 2.

A, i–iii shows a DC model in a situation that may apply to YBa₂Cu₃O_7-x. The model in B and C corresponds to our findings in Bi₂Sr₂CaCu₂O_8+x (BSCCO). (A, i) Modulation (blue, thick line) is the real part of complex wave $\psi \left(x\right)=e^{i(Q_0\hspace{0.17em}x+\phi \left(x\right))}$ having commensurate domains with local wavevector $Q_0=1/4\times 2\pi /a_0$ (period $4a_0$). Colors (see key on top) label the modulation phase within the domains, determining the alignment of modulation maxima (labeled 1…10) and underlying lattice. Phase slips, occurring at DCs between domains, each of size $\pi ,$ add up to give an average $\overline{Q}=0.3\times 2\pi /a_0,$ so that 10 modulation maxima are squeezed into $31a_0.$ (A, ii) The phase argument $\mathrm{\Phi }\left(x\right)=Q_0\hspace{0.17em}x+\phi \left(x\right)$ of $\psi \left(x\right)$ in A, i. Commensurate domains occur in regions (colored) where $\mathrm{\Phi }\left(x\right)$ is parallel to line $Q_0\hspace{0.17em}x$ (red dashed line). The average $\overline{Q}=0.3\times 2\pi /a_0$ is seen as a slope of the best-fit line to $\mathrm{\Phi }\left(x\right)$ (blue dashed line). The difference in slope gives the incommensurability $\delta =\overline{Q}-Q_0.$ (A, iii) Fourier amplitudes $\left|\stackrel{\sim }{\psi }\left(q\right)\right|$ of the modulation $\psi \left(x\right)$ in A, i (blue line) show singular peaks starting at $\overline{Q}=0.3\times 2\pi /a_0\ne Q_0$ with satellites separated by $2\delta ,$ because DCs of size $\pi$ form a periodic array. The satellites depend on DC profile and are sensitive to disorder (SI Text, Heterogeneity and Demodulation Residue). The phase-sensitive figure of merit, demodulation residue $\left|R_q\right|$ (red dashed line), as a function of $q$ has the minimum exactly at the average $\overline{Q}.$ By definition its minimum corresponds to the slope of the best-fit line through $\mathrm{\Phi }\left(x\right)$ (A, ii). (B, i) Modulation (blue, thick line) is the real part of complex wave $\psi \left(x\right)=e^{i(Q_0\hspace{0.17em}x+\phi \left(x\right))}$ having commensurate domains with fundamental local wavevector $Q_0=1/4\times 2\pi /a_0$ (period $4a_0$). Colors label the modulation phase within the domains, determining the alignment of modulation maxima (labeled 1…8) and underlying lattice. All of the phase slips, of sizes $+\pi ,-\left(3\pi /2\right),+\left(\pi /2\right),$ occurring at DCs between domains, cancel to give an average $\overline{Q}=Q_0,$ seen in preserving the eight modulation maxima within $31a_0.$ (B, ii) The phase argument $\mathrm{\Phi }\left(x\right)=Q_0\hspace{0.17em}x+\phi \left(x\right)$ of $\psi \left(x\right)$ in B, i. Commensurate domains occur in regions (colored) where $\mathrm{\Phi }\left(x\right)$ is parallel to line $Q_0\hspace{0.17em}x$ (blue dashed line). The average $\overline{Q}=Q_0$ is seen as the slope of the best-fit line to $\mathrm{\Phi }\left(x\right),$ which coincides with the dashed line. (B, iii) Fourier amplitudes $\left|\stackrel{\sim }{\psi }\left(q\right)\right|$ of the modulation $\psi \left(x\right)$ in B, i (blue line) have a sharp peak at $\overline{Q}=Q_0$ and additional irregularly distributed weight due to disorder in DC’s position (SI Text, Heterogeneity and Demodulation Residue). The calculated phase-sensitive figure of merit, demodulation residue $\left|R_q\right|$ (red dashed line), as a function of $q$ has the minimum exactly at the average $\overline{Q}.$ By definition its minimum corresponds to the slope of the best-fit line through $\mathrm{\Phi }\left(x\right)$ (B, ii). (C, i) Modulation (blue, thick line) is the real part of complex wave $\psi \left(x\right)=A\left(x\right)e^{i(Q_0\hspace{0.17em}x+\phi \left(x\right))}$ having commensurate domains with local wavevector $Q_0=1/4\times 2\pi /a_0,$ additional smooth disorder in phase of up to $\pi /10,$ and smooth disorder in amplitude $A\left(x\right)$ (details in SI Text, Heterogeneity and Demodulation Residue). All random phase slips cancel to give a $\overline{Q}=Q_0,$ akin to the case in B. (C, ii) Fourier amplitudes $\left|\stackrel{\sim }{\psi }\left(q\right)\right|$ of modulation $\psi \left(x\right)$ over 125a₀ range (exemplified in C, i). The broad asymmetric amplitude peak makes the $Q_A$ (orange vertical line), at maximum of Lorentzian fit to amplitude (orange dashed line, multiplied by 1.5 for visibility), different from phase-averaged $\overline{Q}$ (red vertical line) at minimum of demodulation residue $\left|R_q\right|$ (red dashed line) (SI Text, Heterogeneity and Demodulation Residue).

How then can one correctly determine the fundamental $Q_0$ of the CDMs in underdoped cuprates? The spatial arrangement of DCs is inaccessible to diffraction probes designed to yield the Fourier amplitude of the CDM, although the situation in Fig. 2 A, iii may be detectable through the observation of satellite peaks at $\overline{Q}\pm \delta$ (SI Text, Heterogeneity and Demodulation Residue; and SI Text, CDM Commensurability in Underdoped BSCCO). Phase-sensitive transmission electron microscopy can achieve DC visualization (34), whereas coherent X-ray diffraction might (35), but neither one has been used on cuprates. Instead, we consider CDM visualization using spectroscopic imaging scanning tunneling microscopy (20) because it offers full access to both the amplitude and phase of $\stackrel{\sim }{\psi }\left(q\right),$ with the definition of phase referenced to the underlying atomic lattice (33). Then, based on such phase visualization capabilities, we introduce an approach for identifying the fundamental wavevector of the cuprate CDM state. To achieve what is graphically represented in Fig. 2 as the dashed linear fit to a measured phase profile $\mathrm{\Phi }\left(x\right),$ we use an algorithmic procedure that evaluates the demodulation of measured CDM image $\psi \left(\mathit{r}\right)$ at each possible wavevector q, using the demodulation residue

$$R_q\left[\psi \right]\equiv {\displaystyle \int }\frac{dx}{L}Re\left[{\psi }_q^{\ast }\left(x\right)\left(-i{\partial }_x\right){\psi }_q\left(x\right)\right].$$ [2]

Identifying the value of wavevector $q$ for which $\left|R_q\left[\psi \right]\right|$ is a minimum is the 2D equivalent to finding the best-fit slope to $\mathrm{\Phi }\left(x\right)$ in Fig. 2 A and B. The resulting wavevector $q=\overline{Q}$ with high accuracy (SI Text, Heterogeneity and Demodulation Residue), and determination of this $\overline{Q}$ is the general objective and utility of this technique.

SI Text

d-Symmetry Form Factor of CDMs.

For a given BSCCO sample spectroscopic imaging scanning tunneling microscopy (SI-STM) measurement of the Z map, where Z(r, E) = g(r, +E)/g(r, −E), we calculate the dFF $\stackrel{\sim }{\psi }\left(\mathit{q}\right)$ directly in Fourier space, which is equivalent to the procedure introduced in ref. 17, as

$$\stackrel{\sim }{\psi }\left(\mathit{q}\right)={\displaystyle \sum }_{\mathit{G}}\stackrel{\sim }{Z}\left(\mathit{q}-\mathit{G}\right)f\left(\mathit{G}\right),$$ [S1]

where $\mathit{G}$ are reciprocal lattice vectors of Z-map Fourier transform $\stackrel{\sim }{Z}\left(\mathit{q}\right),$ and $f\left(\mathit{q}\right)$ is the Fourier transform of an arbitrary smooth function on the sublattice scale that is used to sample the Z-map data at oxygen sites. We choose $f\left(\mathit{G}\right)$ to be 1 for the nine $\mathit{G} \text{s}$ closest to the origin and zero for the others.

Heterogeneity and Demodulation Residue.

The CDMs observed in cuprates are short ranged, and disorder in the modulations challenges the measurement of their orientation, symmetry, and wavevector. In Fig. S1 we demonstrate the effect of disorder on two CDM scenarios from Fig. 2. To generate DCs themselves in our simulations, e.g., in Fig. 2, we rely on a Ginzburg–Landau description of CDMs detailed in SI Text, CDM Commensurability in Underdoped BSCCO.

Fig. S1.

DC model in A–C is for a situation that may apply to YBa₂Cu₃O_7-x. D–G are models of our findings in BSCCO. (A) Modulation is the real part of complex wave $\psi \left(x\right)=A\left(x\right)e^{i(Q_0\hspace{0.17em}x+\phi \left(x\right))}$ having commensurate domains with local wavevector $Q_0=1/4\times 2\pi /a_0$ (period $4a_0$). The amplitude $A\left(x\right)\ge 0$ varies smoothly around value 1, seen as the envelope of modulation. Phase slips are incorporated in $\phi \left(x\right)$ (B). The average $\overline{Q}=0.3\times 2\pi /a_0.$ (B) The local phase $\phi \left(x\right)$ of $\psi \left(x\right)$ in A, constructed as a DC array in the phase argument $\mathrm{\Phi }\left(x\right)=Q_0\hspace{0.17em}x+\phi \left(x\right).$ Phase slips of all DCs are set to $+\pi .$ The distances between neighboring DCs vary randomly around the average distance set by value of incommensurability $\delta =\overline{Q}-Q_0=0.05\times 2\pi /a_0$ (SI Text, CDM Commensurability in Underdoped BSCCO). (C) Fourier amplitudes $\left|\stackrel{\sim }{\psi }\left(q\right)\right|$ of the modulation $\psi \left(x\right)$ in A (blue line) show a narrow peak at $\overline{Q}=0.3\times 2\pi /a_0.$ The satellites depend on the spatial profile of DCs, but here they are strongly suppressed by disorder. The calculated phase-sensitive figure of merit, demodulation residue $\left|R_q\right|$ (red dashed line), as a function of q has the minimum exactly at the average $\overline{Q}.$ (D) Modulation is the real part of complex wave $\psi \left(x\right)=A\left(x\right)e^{i(Q_0\hspace{0.17em}x+\phi \left(x\right))}$ having commensurate domains with local wavevector $Q_0=1/4\times 2\pi /a_0$ (period $4a_0$). The amplitude $A\left(x\right)\ge 0$ varies smoothly around value 1, seen as the envelope of modulation. Phase slips are incorporated in $\phi \left(x\right)$ (E). (E) The local phase $\phi \left(x\right)$ of $\psi \left(x\right)$ in D, constructed as a DC array in the phase argument $\mathrm{\Phi }\left(x\right)=Q_0\hspace{0.17em}x+\phi \left(x\right).$ Phase slips of DCs are random integer multiples of $\pi /2,$ and they all cancel across the x axis to retain the average $\overline{Q}=Q_0.$ Distances between neighboring DCs vary randomly. The spatial profile of each DC is obtained as variational solution of Ginzburg–Landau theory for CDM phase, using 20 harmonics for $\delta =0.01$ (SI Text, CDM Commensurability in Underdoped BSCCO). (F) Fourier amplitudes $\left|\stackrel{\sim }{\psi }\left(q\right)\right|$ of the modulation $\psi \left(x\right)$ in D (blue line) have a broad irregular intensity peak at $\overline{Q}=Q_0.$ The calculated phase-sensitive figure of merit, demodulation residue $\left|R_q\right|$ (red dashed line), as a function of q has the minimum exactly at the average $\overline{Q}.$ (G) The local phase $\phi \left(x\right)$ used to create Fig. 2 C, i and ii, constructed as a random DC array in the phase argument $\mathrm{\Phi }\left(x\right)=Q_0\hspace{0.17em}x+\phi \left(x\right).$ The array is generated in the same way as in E, except that it has double density of DCs, and the random distribution of phase-slip signs has a long-range correlation. (H) The complex phase of d-form factor order parameter ${\mathrm{\Psi }}_{\overline{\mathit{Q}}}\left(\mathit{x}\right)$ for CDMs along the x axis, obtained at dopings 0.06 (Left) and 0.14 (Right). The demodulation is by phase-averaged $\overline{\mathit{Q}}$ of length $\left(1/4\right)\left(2\pi /a_0\right)$ in both cases. The complex phase of ${\mathrm{\Psi }}_{\overline{\mathit{Q}}}\left(\mathit{x}\right)$ is color coded (key below), whereas complex amplitude varies the color saturation from white (zero amplitude) to full (maximal amplitude in image). H, Left and H, Right show a field-of-view of ∼32 × 32 unit cells, and smoothing parameter $\Lambda$ is the same. Commensurate domains seem to be more disordered at lower dopings.

In the case of the DC lattice (Fig. S1A) we consider variation in sizes of commensurate domains, keeping the incommensurability fixed (Fig. S1B), and we add smooth variation of CDM amplitude (Fig. S1A). The initially singular Fourier amplitude satellite peaks (Fig. 2 A, iii) can completely disappear in the generated noise (Fig. S1C). Observation of the DC lattice by amplitude-measuring scattering probes can therefore be hindered by disorder, as is demonstrably the case in layered dichalcogenides. In the case of randomly positioned DCs having random phase slips (Fig. 2 B, i and Fig. S1E), we add disorder in amplitude, using smooth variation (Fig. S1D). The resulting Fourier amplitudes (Fig. S1F) show a broad irregular peak with high values at multiple wavevectors. This situation is characteristic of BSCCO SI-STM data and can be generated in the case of a periodic DC array (Fig. S1B), e.g., by adding random DC with random phase slips. To avoid ambiguities in determining the average wavevector from Fourier amplitudes (below and SI Text, Statistical Analysis: Two-Dimensional Fitting), we introduce a phase-sensitive measurement based on demodulation residue, which robustly determines the average wavevector with high precision (Fig. 2 C, ii and Fig. S1F).

To describe the phase-sensitive measurement of the average wavevector, we consider a generally short-ranged 1D smooth modulation $\psi \left(x\right),$ defined to spatially average to zero. The case in Fig. 2 B, ii is an example representing BSCCO; however, the following analysis is relevant for any disordered $\psi \left(x\right),$ not necessarily a case of a commensurate wave dominated by DCs. The demodulation ${\psi }_q\left(x\right)=\exp \left[-i\hspace{0.17em}q\hspace{0.17em}x\right]\psi \left(x\right)$ has the wavevector $\overline{Q}-q,$ where $\overline{Q}$ is the wavevector of $\psi \left(x\right).$ The ${\psi }_q\left(x\right)$ has a phase that is, by definition, a fluctuating function of x with zero average value.

Generally, any measurement samples the modulation in some finite region $0<x<L,$ so that the randomly fluctuating phase might accidentally have a small average value, i.e., have a component linear in $x;$ this effect, however, rapidly decays with L for a typical Gaussian disorder.

According to Eq. 1, we can interpret ${\psi }_q\left(x\right)$ as having zero wavevector whereas its phase argument ${\mathrm{\Phi }}_q\left(x\right)$ fluctuates around a linear in x term that is proportional to $\overline{Q}-q.$ The value $q=\overline{Q}$ corresponds to a vanishing linear term,

$$S_{q=\overline{Q}}\equiv \underset{0}{\overset{L}{{\displaystyle \int }}}\frac{dx}{L}{\partial }_x\left({\mathrm{\Phi }}_{q=\overline{Q}}\left(x\right)\right)\equiv 0.$$ [S2]

The $S_q$ tracks the accumulation of phase error in the (smooth) periodic function such as the ${\mathrm{\Phi }}_q\left(x\right)$ might be. The phase argument function itself cannot be unambiguously extracted from the data, suggesting we replace it with the well-defined periodic factor $\exp \left[i\hspace{0.17em}{\mathrm{\Phi }}_q\left(x\right)\right],$ which gives the equivalent ${\displaystyle {\int }_0^L\left(dx/L\right)}\exp \left[-i\hspace{0.17em}{\mathrm{\Phi }}_q\left(x\right)\right]{\partial }_x\left(\exp \left[i\hspace{0.17em}{\mathrm{\Phi }}_q\left(x\right)\right]\right)\equiv 0.$ However, the derivative of the exponential factor can be singular due to sharp phase slips. The physically meaningful expression must use the full, smooth, complex function ${\psi }_q\left(x\right),$ which gives the equation

$$R_{q=Q}\equiv \underset{0}{\overset{L}{{\displaystyle \int }}}\frac{dx}{L}{\left|{\psi }_{q=Q}\left(x\right)\right|}^2{\partial }_x{\mathrm{\Phi }}_{q=Q}\left(x\right)\equiv 0,$$ [S3]

i.e., the demodulation residue figure of merit introduced in the main text. The demodulated pattern with the phase-optimized choice Q therefore has the minimal amount of spatial modulation. In an example of uniform modulation (long-range CDW with no disorder), the optimally demodulated function is strictly a constant equal to the amplitude of the modulation, and $Q=\overline{Q}.$

In the presence of disorder, phase-optimized Q rigorously equals $\overline{Q}$ in either of two limits: (i) when spatial fluctuations of complex amplitude $\left|{\psi }_Q\left(x\right)\right|$ are negligible compared with its spatial average value and (ii) when the complex phase spatially fluctuates on shorter scales than the complex amplitude does, even if the amplitude fluctuates strongly. The CDMs observed in SI-STM images of BSCCO have primarily phase disorder, putting them close to the first limit.

We emphasize that even if the randomly disordered modulation strongly violates conditions for both limits i and ii, the error $\left|Q-\overline{Q}\right|$ is still bounded by the characteristic wavevector, $k,$ of amplitude fluctuations. Namely, consider $R_{Q+q}:$ As q grows the complex phase of demodulation varies faster in space (due to the $q\cdot x$ term), so that at values $\left|q|>|k\right|$ the condition of limit ii is satisfied, making $R_{Q+q}\approx S_{Q+q}.$ Therefore, the functions S and R can deviate from each other only within a region of width $k,$ and hence

$$\left|Q-\overline{Q}\right|\mathrm{\lesssim }k.$$ [S4]

Note that by its definition $S_{\overline{Q}+q}\approx q$ through a large portion of the Brillouin zone, and it is strictly periodic in $q.$ The linear behavior is also observed in demodulation residue R (Fig. 2 and Fig. S1).

To determine $\overline{Q}$ of short-ranged modulations, conventionally one seeks the maximum position, $Q_A,$ of a fit to the broad and irregular peak of Fourier amplitudes, although the fit can be very poor (SI Text, Statistical Analysis: Two-Dimensional Fitting). The use of phase-optimized Q is especially necessary when the amplitude peak is asymmetric, in which case $Q_A$ deviates significantly from $\overline{Q},$ e.g., in Fig. 2 C, ii. With a roughly symmetric peak the $Q_A$ might be close to Q and give a good estimate of $\overline{Q},$ e.g., in Fig. S1F. One way to generate the former case with large mismatch of $Q_A$ and $\overline{Q}$ is by having long-range fluctuations in phase slips, illustrated by the DC arrangement in Fig. S1G, used to obtain Fig. 2 C, ii, compared with the case in Fig. S1E, used to obtain Fig. S1F. In BSCCO SI-STM data, the mismatch of commensurate Q and doping-dependent (33) $Q_A$ is largest at lowest doping, and we suspect this is caused by the propensity for smaller random commensurate domains at lower doping, as illustrated in Fig. S1H.

Additionally, in two dimensions there are topological point-like defects, dislocations (Fig. S2A), where the considered demodulation ${\psi }_{\mathit{q}}\left(\mathit{x}\right)$ has a vortex in its phase argument. These are especially relevant for SI-STM data on BSCCO. A singularity of the phase argument function can occur only at the point where the complex amplitude $\left|{\psi }_{\mathit{q}}\left(\mathit{x}\right)\right|$ vanishes. The density of 2D demodulation residue ${\mathit{R}}_{\mathit{q}}^{\alpha }\left(\mathit{x}\right)$ (SI Text, Demodulation in Two Dimensions, Smoothing, and Optimization Error) is therefore smooth at the position of dislocation, even though it contains derivatives. Crucially, the local disturbance in ${\mathit{R}}_{\mathit{q}}^{\alpha }\left(\mathit{x}\right)$ created by a dislocation spatially averages to zero (Fig. S2 B and C). The dislocations therefore generally do not influence the demodulation measurement of the phase-optimized wavevector.

Fig. S2.

(A) A spatial modulation in two dimensions, with a dislocation created in the center of the image by a $2\pi$ vortex in the complex phase and a suppression of complex amplitude by a Gaussian of width one modulation period: $\mathrm{Re}\left[\psi \left(x,y\right)\right]$ with $\psi \left(x,y\right)=\exp \left(-\frac{1}{2}\left(x^2+y^2\right)\right)\exp \left(i\hspace{0.17em}Q\hspace{0.17em}x+i\hspace{0.17em}\text{atan}\left(\frac{y}{x}\right)+i\frac{\pi }{2}\right).$ (B) Demodulation residue density along the x axis evaluated for modulation $\psi \left(x,y\right)$ in A at wavevector $Q\widehat{\mathit{x}},$ which is the phase-optimized wavevector. Dislocation position is marked by a cross. (C) Demodulation residue density along the y axis evaluated for modulation $\psi \left(x,y\right)$ in A at wavevector $Q\widehat{\mathit{x}},$ which is the phase-optimized wavevector. Dislocation position is marked by a cross.

Statistical Analysis: Two-Dimensional Fitting.

Given the dFF data, we focus on a single characteristic intensity peak (out of two, $X$ or $Y,$ centered on the x and the y axis, respectively) by restricting $\mathit{q}$ to a square covering one-quarter of the first Brillouin zone (e.g., dashed square in Fig. 3B). These restricted data are simply referred to as “dFF data” and labeled $\stackrel{\sim }{\psi }\left(\mathit{q}\right)$ within this section.

Fig. 3.

(A) Typical measured $\psi \left(\mathit{r}\right)$ of Bi₂Sr₂CaCu₂O_8+x in the charge modulation phase at hole-doping level p = 0.06. The subatomic resolution image shows charge modulations at pseudogap energy. Coordinate axes $x,y$ correspond to copper-oxide lattice principal axes. (B) The Fourier transform amplitudes of d-symmetry form factor, $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|,$ extracted from the measurement in A. Four broad intensity distributions appear due to CDMs, and one of them (along the x axis) is marked by the dashed square. The unit-cell constant, $a_0,$ is determined by Bragg peaks (red crosses). (C) Measured d-form factor Fourier amplitudes $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|$ (solid circles) along the $q_x$ axis in the dashed square in B, i.e., surrounding the ${\mathit{Q}}_X$ CDM peak, from the origin toward the Bragg peak, showing the cut through the best-fitting smooth 2D peak function (SI Text, d-Symmetry Form Factor of CDMs; and SI Text, Statistical Analysis: Two-Dimensional Fitting). The fit residual at each wavevector (vertical drop from solid circle to fit function) is color coded. The integers on the horizontal axis count the pixels in the Fourier transform image; i.e., wavevector length on the horizontal axis is measured in units $2\pi /L,$ with $L$ the field-of-view size in lattice units. (D) Close-up of the Fourier amplitudes of $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|$ from B marked by the dashed square. The discrete set of wavevectors in $\mathit{q}$ -space area of B, i.e., surrounding the ${\mathit{Q}}_Y$ CDM peak, and their d-form factor Fourier amplitudes are shown. Each pixel at which $\left|{\mathit{R}}_{\mathit{q}}\right|$ is calculated (E) is color-symbol coded. Commensurate value $\mathit{q}=(\mathrm{0,1}/4)$ is marked by a cross. (E) Demodulation residue $\left|{\mathit{R}}_{\mathit{q}}\right|$ vs. Fourier amplitude used as two figures of merit in CDM period analysis, for d-symmetry form factor component measured in Bi₂Sr₂CaCu₂O₈ at doping p = 0.06 (data presented in A and B). A set of wavevectors (the color-symbol–coded pixels in D) is used in this plot, showing the value of demodulation residue $\left|{\mathit{R}}_{\mathit{q}}\right|=\sqrt{{\left(R_{\mathit{q}}^x\right)}^2+{\left(R_{\mathit{q}}^y\right)}^2},$ calculated using cutoff $\mathrm{\Lambda }=0.08\times 2\pi /a_0$ (SI Text, Demodulation in Two Dimensions, Smoothing, and Optimization Error), vs. the Fourier amplitude $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|,$ for each wavevector in the chosen set. The boxed data point is for the discrete wavevector value that is identified as minimizing the demodulation residue and therefore we define it as the phase-optimized wavevector. Its length is $(0.24\pm 0.03)\times 2\pi /a_0,$ where the error value follows from spatial variation of residue (SI Text, Demodulation in Two Dimensions, Smoothing, and Optimization Error). The residue is significantly closer to zero at the phase-optimized wavevector than for others, even though there are many wavevectors having higher Fourier amplitudes. [Note that conventional analysis by fitting a broad peak to Fourier amplitudes of these data would identify wavevector length $Q=0.29\times 2\pi /a_0$]. (F) Measured $\psi \left(\mathit{r}\right)$ of Bi₂Sr₂CaCu₂O_8+x in the charge modulation phase at hole-doping level p = 0.08. The overlay is the x-axis CDM demodulated by phase-optimized commensurate wavevector $\left|{\mathit{Q}}_X\right|=1/4\times 2\pi /a_0,$ with high amplitude having high color saturation. Phase-locked domains with phase an integer multiple of $2\pi /4$ (color key) are visible. Across the field of view, e.g., following the black dashed line, the phase slips between domains average to zero, evoking Fig. 2B.

The Fourier amplitude of the dFF, $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|,$ typically shows a broad intensity distribution with peaks at multiple $\mathit{q}$ values, as presented for all samples in Fig. S3. The standard procedure for describing the modulation in this situation involves a 2D fit of $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|$ with a single-peaked smooth function, reporting the position of that peak. We apply the 2D fit of several smooth functions: the offset Gaussian distribution $c+a\hspace{0.17em}\exp \left[-{\left(\mathit{q}-{\mathit{q}}_0\right)}^2/2s^2\right],$ offset Lorentz distribution $c+a/\left[{\left(\mathit{q}-{\mathit{q}}_0\right)}^2+s^2\right],$ two-variate Cauchy distribution $c+a/{\left[{\left(\mathit{q}-{\mathit{q}}_0\right)}^2+s^2\right]}^{3/2},$ and their versions with different widths s in orthogonal directions. For example, the data presented in Fig. S3A are best fitted by the shifted Lorentzian centered at ${\mathit{q}}_0=(0.21,\hspace{0.17em}0.22)\times 2\pi /a,$ with width $s=0.11\times 2\pi /a.$

Fig. S3.

The Fourier amplitude of dFF, $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|,$ with wavevector $\mathit{q}$ restricted to a square area with a corner at the Fourier space origin (black square) and a center at ${\mathit{Q}}_X=\left(1/4\right){\mathit{G}}_X$ or ${\mathit{Q}}_Y=\left(1/4\right){\mathit{G}}_Y,$ where ${\mathit{G}}_X$ and ${\mathit{G}}_Y$ are the Bragg peaks. This area covers the intensity distribution of a unidirectional CDM. The color-coded quantity is the fit residual to the best-fitting smooth 2D peaked function (SI Text, Statistical Analysis: Two-Dimensional Fitting and Fig. S4). Data from BSCCO samples at doping level p and having superconducting transition temperature $T_c$ are shown: $p=0.06,\hspace{0.17em}{T}_c=20\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_X$ in A; $p=0.06,\hspace{0.17em}{T}_c=20\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_Y$ in B; $p=0.08,\hspace{0.17em}{T}_c=45\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_X$ in C; $p=0.08,\hspace{0.17em}{T}_c=45\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_Y$ in D; $p=0.10,\hspace{0.17em}{T}_c=65\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_X$ in E; $p=0.10,\hspace{0.17em}{T}_c=65\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_Y$ in F; $p=0.14,\hspace{0.17em}{T}_c=74\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_X$ in G; $p=0.14,\hspace{0.17em}{T}_c=74\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_Y$ in H; $p=0.17,\hspace{0.17em}{T}_c=89\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_X$ in I; and $p=0.17,\hspace{0.17em}{T}_c=89\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_Y$ in J.

We, however, observe that the fit is quite poor, with values of adjusted R² of about 0.7, which corroborates visual inspection, e.g., in Fig. 3C. The culprit for the poor fit is data having multiple local (single-pixel) peaks in an asymmetric distribution. We confirm this by showing that statistically these peaks cannot be described by the simple smooth single-peaked distribution.

A standard tool for analyzing the nature of the fit is the normal quantile–quantile (QQ) plot, shown for all of the samples in Fig. S4. The fit residuals, i.e., the set of differences between a value $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|$ and its fitted value, are sorted by size and indexed by $n=1,\dots ,N.$ It is natural to expect that data points are randomly scattered around the fit function, producing a normal distribution of fit residuals. In the QQ plot, the residuals are plotted against the inverse cumulative distribution of the standardized normal distribution evaluated at $\left(n-3/8\right)/\left(N+1/4\right)$ [ref. 46 and references therein (e.g., ref. 47)]. If the fit residuals are indeed normally distributed, the QQ plot follows a straight line centered at the origin. We fitted the part of QQ plot near the origin to a straight line (black line in Fig. S4). However, for a typical dataset in Fig. S4 the significant deviations from the straight line at the high values of residuals are visually striking. They show that the largest values of $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|$ deviate from a normal scatter around a single smooth peaked-fit function.

Fig. S4.

The standard QQ plot compares the inverse cumulative normal distribution (horizontal axis) to the inverse cumulative distribution of fit residuals (vertical axis) for the dFF Fourier amplitude data, $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|,$ in Fig. S3. The residuals for each dataset are obtained from the best fitting of data in Fig. S3 to several single-peaked smooth 2D functions (SI Text, Statistical Analysis: Two-Dimensional Fitting). The color coding for different data points $\left|\stackrel{\sim }{\mathrm{\Psi }}\left(\mathit{q}\right)\right|$ is used in Fig. S3. The black line is a linear fit to the QQ plot obtained for half the data points nearest to the origin. Data from BSCCO samples at doping level p and having superconducting transition temperature $T_c$ are shown: $p=0.06,\hspace{0.17em}{T}_c=20\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_X$ in A; $p=0.06,\hspace{0.17em}{T}_c=20\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_Y$ in B; $p=0.08,\hspace{0.17em}{T}_c=45\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_X$ in C; $p=0.08,\hspace{0.17em}{T}_c=45\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_Y$ in D; $p=0.10,\hspace{0.17em}{T}_c=65\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_X$ in E; $p=0.10,\hspace{0.17em}{T}_c=65\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_Y$ in F; $p=0.14,\hspace{0.17em}{T}_c=74\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_X$ in G; $p=0.14,\hspace{0.17em}{T}_c=74\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_Y$ in H; $p=0.17,\hspace{0.17em}{T}_c=89\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_X$ in I; and $p=0.17,\hspace{0.17em}{T}_c=89\hspace{0.5em}\text{K}$ for ${\mathit{Q}}_Y$ in J.

This deviation is statistically significant, according to the comparison of the highest value of the fit residual with the probability distribution for the highest value drawn from a normal distribution, this normal distribution best describing the fit residuals. The observed highest residual value typically lies within 3–6 SDs away from the center of this probability distribution.

Demodulation in Two Dimensions, Smoothing, and Optimization Error.

To apply our analysis to 2D data, the SI-STM Z-map dFF $\psi \left(\mathit{x}\right),$ we define the demodulation directly in Fourier space as

$${\mathrm{\Psi }}_{\mathit{q}}\left(\mathit{k}\right)=\exp \left(-\hspace{0.17em}\frac{{\left(\mathit{k}-\mathit{q}\right)}^2}{2{\mathit{\Lambda }}^2}\right)\exp \left(-i\hspace{0.17em}\mathit{q}\mathrm{\cdot }\mathit{x}\right)\stackrel{\sim }{\psi }\left(\mathit{q}+\mathit{k}\right)$$ [S5]

or, in words, we shift the Fourier space origin to $\mathit{q}$ and filter by a Gaussian suppression around it, which is equivalent to a real-space Gaussian smoothing of $\exp \left(-i\hspace{0.17em}\mathit{q}\mathrm{\cdot }\mathit{x}\right)\psi \left(\mathit{x}\right)$ on length scale $1/\mathrm{\Lambda }$ (Fig. S5A) (48). The analysis of CDM requires smoothing because one needs to include only the data within a single broad intensity distribution (the one around either the x or the y axis) from the Fourier data of $\psi \left(\mathit{x}\right).$ Our choice of smoothing procedure, centered on $\mathit{q},$ instead of smoothing the whole broad distribution once, gives equal regard to every demodulation wavevector $\mathit{q}$ (Fig. S5A). Note that the demodulated pattern ${\mathrm{\Psi }}_{\mathit{q}}\left(\mathit{x}\right)$ is complex because data around $-\mathit{q}$ are not used (Fig. S5B).

Fig. S5.

(A) The dFF Fourier amplitude in a p = 0.06 sample, surrounding ${\mathit{Q}}_X$ within a square whose diagonal reaches halfway to the Bragg peak. A demodulated complex pattern in real space, ${\mathrm{\Psi }}_{\mathit{q}}\left(\mathit{x}\right),$ shown in B, is defined by wavevector $\mathit{q}$ and is smoothed by Gaussian suppression of Fourier data beyond the cutoff $\mathrm{\Lambda }$ (radius of orange circle). (B) The complex phase of ${\mathrm{\Psi }}_{\mathit{q}}\left(\mathit{x}\right)$ is color coded, whereas complex amplitude varies the brightness of color from black (zero amplitude) to full brightness (maximal amplitude in image). Points around which the complex phase winds by $\pm 2\pi$ (green/purple) are dislocations. (C) The density, along spatial direction $\alpha ,$ of demodulation residue at $\mathit{q}$ is defined as $R_{\mathit{q}}^{\alpha }\left(\mathit{x}\right)=\mathrm{Re}\left[{\psi }_{\mathit{q}}^{\ast }\left(\mathit{x}\right)\hspace{0.17em}\left(-i{\partial }_{\alpha }\right){\psi }_{\mathit{q}}\left(\mathit{x}\right)\right].$ The demodulation residue along the horizontal axis of image, $R_{\mathit{q}}^{hor},$ is the average of $R_{\mathit{q}}^{hor}\left(\mathit{x}\right),$ here multiplied by ${10}^5.$ (D) Same as C, along the vertical direction (ver) instead of horizontal.

The demodulation residue in two dimensions is a vector with components $\alpha =x,y$ (Fig. S5 C and D) and in field-of-view of finite size L is a direct generalization of the one-dimensional case:

$${\mathit{R}}_{\mathit{q}}^{\alpha }\left[\mathit{\psi }\right]\equiv {{\displaystyle \int }}^{}\frac{d^2\mathit{x}}{L^2}\mathrm{Re}\left[{\mathrm{\Psi }}_{\mathit{q}}^{\ast }\left(\mathit{x}\right)\left(-i{\partial }_{\alpha }\right){\mathrm{\Psi }}_{\mathit{q}}\left(\mathit{x}\right)\right].$$ [S6]

We find that the demodulation residue is robust to choice of value of filtering cutoff $\mathrm{\Lambda }$ in all underdoped samples. The phase-optimized wavevector chosen by demodulation residue varies negligibly as long as $\mathrm{\Lambda }$ is larger than the spacing between $\mathit{q}$ points (the $\mathit{q}$ space is discrete with spacing $2\pi /L$) and a few times smaller than the distance between X and Y intensity peaks in Fourier data of $\psi \left(\mathit{x}\right).$

As discussed in SI Text, Heterogeneity and Demodulation Residue, the phase-optimized wavevector gives a linear fit $\mathit{Q}\mathrm{\cdot }\mathit{x}+\phi$ to the phase argument function $\mathit{\Phi }\left(\mathit{x}\right)$ of the modulation $\psi \left(\mathit{x}\right)$ (see Fig. 2 B and E for 1D examples). To estimate the error of the obtained slope $\mathit{Q},$ we calculate

$${\sigma }^2\equiv \frac{{{\displaystyle \int }}^{}\frac{d^2\mathit{x}}{L^2}{\mathit{v}}^2}{{{\displaystyle \int }}^{}\frac{d^2\mathit{x}}{L^2}{\left|{\mathrm{\Psi }}_{\mathit{Q}}\left(\mathit{x}\right)\right|}^2},$$ [S7]

where ${\mathit{v}}_{\alpha }=\mathrm{Re}\left[{\mathrm{\Psi }}_{\mathit{Q}}^{\ast }\left(\mathit{x}\right)\left(-i{\partial }_{\alpha }\right){\mathrm{\Psi }}_{\mathit{Q}}\left(\mathit{x}\right)\right].$ The $\sigma$ corresponds to the SD (caused by spatial fluctuations) of the phase argument function around the linear term $\mathit{Q}\mathrm{\cdot }\mathit{x}+\phi$ (hence it is evaluated exactly at the phase-optimized wavevector $\mathit{Q}$). In the 1D examples in Fig. 2 A, ii and B, ii the $\sigma$ quantifies the SD of the phase function around the optimal fit (dotted blue line). The $\sigma$ has units of wavevector due to the use of normalized demodulation function ${\mathrm{\Psi }}_{\mathit{Q}}\left(\mathit{x}\right)$ and directly quantifies the error in measurement of phase-optimized $\mathit{Q}.$ For a discrete dataset, such as an SI-STM image, we replace the derivative by a difference at neighboring pixels, the integral by a sum over all image pixels, and the $L^2$ by the total number of pixels; and finally the wavevector error $\sigma$ is obtained (and reported in Fig. 4F) by simple conversion of units from $1/pix$ to $2\pi /a_0,$ where $pix$ is the real-space width of an image pixel.

Fig. 4.

(A, Left) Measured $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|$ for p = 0.06, within the square region of $\mathit{q}$ space (the dashed square in Fig.3B) bounded by $\left(\mathrm{0,0}\right),\hspace{0.17em}\left(\pi /2a_0,\pi /2a_0\right),\hspace{0.17em}\left(0,\pi /a_0\right),\left(-\pi /2a_0,\pi /2a_0\right).$ (A, Right) The value of demodulation residue $\left|{\mathit{R}}_{\mathit{q}}\right|=\sqrt{{\left(R_{\mathit{q}}^x\right)}^2+{\left(R_{\mathit{q}}^y\right)}^2}$ calculated from data using cutoff $\mathrm{\Lambda }=0.08\times 2\pi /a_0.$ The measured $\left|{\mathit{R}}_{\mathit{q}}\right|$ varies smoothly and drops quickly toward zero at a single discrete wavevector near the center of the image, similar to behavior in one dimension (Fig. 2). The identified discrete wavevector has length $\left|\mathit{Q}\right|=0.245\times 2\pi /a_0.$ The position of $(\mathrm{0,1}/4)\times 2\pi /a_0$ is marked by a small cross. (B–E) Same analysis as shown in A, but for a series of samples with estimated hole densities p = 0.08, 0.10, 0.14, 0.17. In each case in B–E, Left, shown is the measured $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|$ within the square region of $\mathit{q}$ space that defines the considered CDM orientation, e.g., for ${\mathit{Q}}_{\mathit{Y}}$ bounded by $\left(\mathrm{0,0}\right),\hspace{0.17em}\left(\pi /2a_0,\pi /2a_0\right),\left(0,\pi /a_0\right),\left(-\pi /2a_0,\pi /2a_0\right);$ in each case in B–E, Right, shown is the measured $\left|{\mathit{R}}_{\mathit{q}}\right|$ within the square region of $\mathit{q}$ space marked by the orange thin square on the image to its Left. The corresponding position of $\left(0,\hspace{0.17em}1/4\right)\times 2\pi /a_0$ is marked by a small cross. The pixel at which the $\left|{\mathit{R}}_{\mathit{q}}\right|$ is found to be a minimum is identified as the phase-optimized CDM wavevector for that carrier density. (F) The lengths of wavevectors ${\mathit{Q}}_X,{\mathit{Q}}_Y$ extracted from measured Bi₂Sr₂CaCu₂O₈ underdoped samples at different dopings $p$ using $\left|{\mathit{R}}_{\mathit{q}}\right|$ minimization (A–E). The error in value of the phase-optimized wavevector is obtained from spatial variation of residue (SI Text, Demodulation in Two Dimensions, Smoothing, and Optimization Error) and is comparable to the error caused by discreteness of choices for $\mathit{Q}.$ For doping $p>0.14$ the d-form factor CDMs are less pronounced and there is a larger error in value of phase-optimized $\mathit{Q}.$ Note the doping-independent trend and values consistent with commensurate value $1/4$. Shown are reported doping-dependent values of CDM wavevector length $\left|Q_A\right|$ in Bi₂Sr₂CaCu₂O_8+x and Bi₂Sr_2-xLa_xCuO_6+y (disks, colored light to dark in order for refs. 43, 44, and 33), YBa₂Cu₃O_7-x (squares, light to dark for source 132 from refs.15, 45, and 37), and La_2-xBa_xCuO₄ (diamonds, reported in ref. 45).

CDM Commensurability in Underdoped BSCCO.

When we extract the phase-optimized wavevector, representing the average wavevector of CDMs, we find values consistent with commensurate $Q_{\text{phase}-\text{opt}}=Q_0=\left(1/4\right)\left(2\pi /a_0\right)$ throughout the underdoped regime of BSCCO (SI Text, Summary of Results for All Samples). We now demonstrate the consistency of this finding and simultaneously show that the local wavevector is also $Q_0.$ The CDMs here have nonoverlapping uniaxial nature, so we focus on one direction, e.g., $\mathit{q}=q{\mathit{e}}_x$ in Fig. S6. The CDM order parameter at $q=Q_0,$ Fig. S6A (also Fig. 3F and Fig. S1H), shows multiple domains with CDM phase close to the commensurately pinned values $n\left(2\pi /4\right)$ (end of this section), indicating the local commensurate wavevector $Q_0.$ In a real-space SI-STM image (Fig. S6B), areas where CDM order parameter amplitude is high consistently show commensurate patches of a typical $4a_0$ -wide dFF pattern (Fig. 1 B and F) and reveal these patches forming commensurate domains. Every patch individually is therefore aligned with an underlying lattice, an effect observed already in ref. 49, but the patches are shifted in position with respect to each other by values a₀, 2a₀, or 3a₀.

Fig. S6.

(A) The complex phase of ${\mathrm{\Psi }}_{{\mathit{Q}}_{opt}}\left(\mathit{x}\right),$ the CDM OP at phase-optimized wavevector ${\mathit{Q}}_{opt}=\left(1/4\right)\left(2\pi /a_0\right)$ along the x axis, is color coded, whereas complex amplitude varies the brightness of color from black (zero amplitude) to full brightness (maximal amplitude in image). A phase vortex is at the center of the dashed circle. (B) Real-space SI-STM Z map for doping p = 0.08, from which image A is created. The dashed circle corresponds to the one in A. Patches of width 4a₀ (rectangles) are marked within regions of high OP amplitude identified in A, and their colors (as in A) represent the dominant value of phase across the patch, matching well multiples of $2\pi /4.$ Patches exhibit the typical dFF pattern (SI Text, CDM Commensurability in Underdoped BSCCO). The OP phase vortex obtained by smoothing in A corresponds to mutual misalignment of the 4a₀ patterns by multiples of a₀, indicating a dislocation at the meeting point of commensurate domains.

As in our earlier analysis (18) we here observe that CDM order parameter phase fluctuations are dominated by phase vortices (white circle in Fig. S6A), i.e., CDM dislocations, which seems to indicate an incommensurate CDM. In commensurate CDMs, dislocations also exist as points where multiple DCs meet and provide a total of $2\pi$ phase slip upon encircling the meeting point, and we observe this directly in real space. In the vicinity of the order parameter vortex, in real space we find a few patches of appreciable order parameter (OP) amplitude representing commensurate domains meeting at the defect. Because here the OP’s smoothing length scale (SI Text, Demodulation in Two Dimensions, Smoothing, and Optimization Error) approaches the commensurate domain size, the DC meeting point smooths into a phase vortex in Fig. S6A.

Although our data show local wavevector $Q_0$ and phase slips of various multiples of $\pi /2$ between multiple domains (Fig. 3F and Figs. S1H and S6), in principle there could be an underlying DC lattice and an incommensurate average wavevector observed over distances larger than the SI-STM field-of-view. The error in phase-optimized wavevector $Q_{\text{phase}-\text{opt}}$ induced by adding/removing into the field-of-view a single DC of such a sparse underlying DC lattice is consistent with the error bar for $Q_{\text{phase}-\text{opt}}$ obtained in SI Text, Demodulation in Two Dimensions, Smoothing, and Optimization Error. We note that by definition of the phase-optimized wavevector, an ideal DC lattice would appear as a seesaw in phase of the CDM order parameter; e.g., see deviation of phase argument from best-fit line in Fig. 2 A, ii. We do not see such features in SI-STM images.

To generate DCs in our simulations (Fig. 2 and Figs. S1 and S8), we use a Ginzburg–Landau order parameter theory for CDMs, which in cuprates has square lattice symmetry (50). The theory in ref. 50 deals with the commensurate CDM state in a region around doping 1/8, but does not deal with DCs. Compared with well-known CDW systems in layered dichalcogenides (51, 52), the energy term describing the local commensuration tendency appears as a quartic term instead of as a cubic. We focus on the CDM phase mode and find an ideal DC lattice as a possible ground state (19). The quartic term prefers $Q_0=\left(1/4\right)\left(2\pi /a\right)$ commensurate CDMs with phase locked at multiples of $2\pi /4.$ The ideal DC lattice with phase slips of $n\left(2\pi /4\right),$ n integer, and separation distance $n/\left(4\delta \right)$ describes an array of commensurate domains that accumulates CDM phase and gives average wavevector $\overline{Q}=Q_0+\delta$ when averaged over multiple domains. The Fourier satellite peaks of the DC lattice are separated by $4\delta /n$ around $\overline{Q},$ whereas $Q_0$ amplitude vanishes (Fig. 2B shows an example of $n=2$). The spatial profile of the DC lattice we obtained as a variational solution (19), using 20 harmonics $\sin \left(j\hspace{0.17em}4\hspace{0.17em}\delta \hspace{0.17em}x\right),$ $j=1\dots 20.$

Fig. S8.

DC model of a situation that may apply to La_2-x-yNd_xSr_yCuO₄. (A) Modulation is the real part of complex wave $\psi \left(x\right)=A\left(x\right)e^{i(Q_0\hspace{0.17em}x+\phi \left(x\right))}$ having commensurate domains with local wavevector $Q_0=1/4\times 2\pi /a_0$ (period $4a_0$). The amplitude $A\left(x\right)\ge 0$ varies smoothly around value 1, seen as the envelope of modulation. Phase slips are incorporated in $\phi \left(x\right)$ (B). The average $\overline{Q}=0.23\times 2\pi /a_0.$ (B) The local phase $\phi \left(x\right)$ of $\psi \left(x\right)$ in A, constructed as a DC array in the phase argument $\mathrm{\Phi }\left(x\right)=Q_0\hspace{0.17em}x+\phi \left(x\right).$ Phase slips of DCs are set to $–\pi /2.$ The distances between neighboring DCs vary randomly around average distance set by value of incommensurability $\delta =\overline{Q}-Q_0=-0.02\times 2\pi /a_0$ (SI Text, CDM Commensurability in Underdoped BSCCO). (C) Fourier amplitudes $\left|\stackrel{\sim }{\psi }\left(q\right)\right|$ (blue line) of the modulation $\psi \left(x\right)$ in A. Amplitudes are smoothed by a Gaussian of width $1.4\mathrm{\cdot }{10}^{-3}\left(2\pi /a_0\right)=2\mathrm{\cdot }{10}^{-3}\hspace{0.17em}{\AA }^{-1}.$ A narrow peak at $\overline{Q}=0.23\times 2\pi /a_0$ is captured by both a Lorentzian fit to Fourier amplitude (dashed orange line) and a phase-optimized wavevector at minimum of demodulation residue $\left|R_q\right|$ (dashed red line). Fourier satellites are strongly suppressed by the randomness in the DC array.

Summary of Results for All Samples.

Fig. S7 shows the demodulation residue for modulations along the x axis, calculated for all analyzed samples (for y-axis modulations, see Fig. 4). For underdoped samples a single phase-optimized wavevector is apparent. For higher dopings the distinction is not so clear and the optimum is within a larger area of Fourier space, in accord with the weakening of the dFF component at those dopings. The results are robust to changes of filtering cutoff $\mathrm{\Lambda }$ for the underdoped samples (SI Text, Demodulation in Two Dimensions, Smoothing, and Optimization Error).

Fig. S7.

(A–E) For a series of samples with estimated hole densities p = 0.06, 0.08, 0.10, 0.14, 0.17, Left shows measured amplitude $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|,$ within the square region of $\mathit{q}$ space (bottom dashed square in Fig. 3B) bounded by $\left(\mathrm{0,0}\right),\hspace{0.17em}\hspace{0.17em}\left(\pi /2a,\pi /2a\right),\hspace{0.17em}\hspace{0.17em}\left(\pi /a,0\right),\hspace{0.17em}\hspace{0.17em}\left(\pi /2a,-\pi /2a\right),$ which defines the CDM orientation along the x axis; Right shows the value of demodulation residue $\left|{\mathit{R}}_{\mathit{q}}\right|=\sqrt{{\left(R_{\mathit{q}}^x\right)}^2+{\left(R_{\mathit{q}}^y\right)}^2}$ calculated using cutoff $\mathrm{\Lambda }=0.08\times 2\pi /a,$ within the square region of $\mathit{q}$ space marked by the orange thin square in Left panel. The corresponding position of $\left(1/\mathrm{4,0}\right)\times 2\pi /a$ is marked by a small cross. The pixel at which the $\left|{\mathit{R}}_{\mathit{q}}\right|$ is found to be a minimum is identified as the phase-optimized CDM wavevector for that carrier density. The measured $\left|{\mathit{R}}_{\mathit{q}}\right|$ varies smoothly and drops quickly toward zero at a single discrete wavevector near the center of the image, similar to behavior in one dimension (e.g., Fig. 2). The phase-optimized wavevector $\mathit{Q}$ in continuous $\mathit{q}$ space can be at most localized within the whitest pixel area, leading to a minimal error of $0.02\times 2\pi /a.$ The error value $0.03\times 2\pi /a$ is obtained from spatial variation of residue function (SI Text, Demodulation in Two Dimensions, Smoothing, and Optimization Error).

Global CDM Lattice Commensurability.

We introduce the following hypothesis: Differences in measured CDM wave vectors over different cuprate families are due to different arrangements of DCs in the underlying $4a_0$ periodic order. In support of this, we now relate cuprate materials to our simulated DC arrangements in Fig. 2 and Figs. S1 and S8, to show consistency with various values of extracted wavevectors and consistency with varying widths of the CDM peaks. A clear signature of our proposal would be observation of satellite Fourier peaks in scattering probes when the DC lattice is present in a material; however, we have shown how the satellites might be unobservable due to disorder (e.g., compare Fig. 2 A, iii and Fig. S1C), and this effect is already recognized in layered dichalcogenides (34).

In BSCCO, our findings of $4a_0$ commensurate patches and commensurate average wavevector (SI Text, CDM Commensurability in Underdoped BSCCO) are simulated in a 1D cut by a random distribution of commensurate domains of typical size around 8a₀, with DCs whose phase slips are random multiples of $2\pi /4$ (Fig. S1 D, E, and G and Fig. 2 B, i and ii). The irregular Fourier amplitude peak with phase-optimized wavevector $\overline{Q}=\left(1/4\right)\left(2\pi /a_0\right)$ mirrors our BSCCO findings. Our simulated 1D CDMs (Fig. 2 C, ii and Fig. S1F) have similar full-width $\mathrm{\Delta }Q\mathrm{\sim }0.05\left(2\pi /a_0\right)$ of intensity peak to the BSCCO SI-STM data (Fig. 3B).

In the case of YBa₂Cu₃O_7-x, existing X-ray measurements (15) offer two restrictions: (i) The CDM intensity peak is centered on wavevector $\overline{Q}\sim 0.3\left(2\pi /a_0\right);$ and (ii) the intensity peak has full-width $\mathrm{\Delta }Q\mathrm{\sim }0.02\left(2\pi /a_0\right).$ These do not rule out a commensurate local wavevector $Q_0=1/4:$ Fig. 2 A, iii shows how in absence of disorder a DC lattice generates a sharp Fourier peak at $\overline{Q}=0.3\left(2\pi /a_0\right).$ The DC lattice also generates strong satellite peaks that are not yet reported in YBCO. Adding disorder can suppress the satellites, while giving width $\mathrm{\Delta }Q\mathrm{\sim }0.02\left(2\pi /a_0\right)$ to the intensity peak (Fig. S1C).

Neutron scattering in La_2-x-yNd_xSr_yCuO₄ (ref. 39 and references therein) shows $\overline{Q}$ near $0.23\left(2\pi /a_0\right),$ therefore a small incommensurability $\delta =-0.02,$ and sharp peaks with full-width $\mathrm{\Delta }Q\mathrm{\sim }0.01\left(2\pi /a_0\right).$ Within our hypothesis this situation reflects a DC lattice as demonstrated in Fig. S8. A total of $2\pi$ phase slip is expected on a length scale of over 20 nm. Note that the dominant satellite peak at $0.31\left(2\pi /a_0\right)$ (i.e., at distance $4\delta$ from $\overline{Q}$ because our chosen DC phase slips are $–\pi /2$) is suppressed by randomness in DC positions within their lattice. Our proposal is an alternative to the existing model of CDM patches having fixed but different widths (39).

X-ray scattering on La_2-xBa_xCuO₄ (14) finds a peak near $\overline{Q}=0.23\left(2\pi /a_0\right),$ with small incommensurability $\delta =-0.02$ compared with $Q_0=\left(1/4\right)\left(2\pi /a_0\right),$ and having a larger $\mathrm{\Delta }Q\mathrm{\sim }0.04\left(2\pi /a_0\right).$ The increased peak width compared with the above La_2-x-yNd_xSr_yCuO₄ scenario can simply be caused by any of several factors such as increased CDM amplitude variation, more randomness in positions of DCs within their lattice, or additional random DCs that average to zero phase slip and occur on top of the underlying DC lattice.

We uncovered the problem of irregular Fourier amplitude distribution within a broad CDM intensity peak and it is presently unclear whether this property can be observed in other cuprates. In the Bismuth-based cuprate family resonant X-ray scattering experiments (43, 44) have a high wavevector resolution and clearly show these broad irregular Fourier amplitude distributions. On the other hand, X-ray experiments on YBCO show smoother Fourier intensity peaks, but one must focus on the wavevector resolution: Typically (45, 53) there are only about 5 distinct amplitude values in a 1D cut of q space through the intensity peak, which washes out the possible irregularity. In comparison, SI-STM data used here have of order 50 amplitude values in a 2D profile of the intensity peak (Fig. 3D).

Phase-Resolved Imaging and Phase Demodulation Residue Analysis

We apply demodulation residue analysis to study 2D short-range ordered CDM images typical of underdoped Bi₂Sr₂CaCu₂O_8+x, e.g., Fig. 3A at p = 0.06. Fig. 3B shows the d-symmetry form factor Fourier amplitude $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|$ derived from $\psi \left(\mathit{r}\right)$ in Fig. 3A. The amplitude and phase of modulations in $\psi \left(\mathit{r}\right)$ can be decomposed into two unidirectional components along x, y. We define the demodulation residue ${\mathit{R}}_{\mathit{q}}$ for each trial $\mathit{q}$ (SI Text, Demodulation in Two Dimensions, Smoothing, and Optimization Error) over a wide range in the Fourier space inside the dashed box in Fig. 3B and use it as a phase-sensitive metric for deciding how close each $\mathit{q}$ is to the phase-averaged wavevector $\overline{\mathit{Q}}.$ In the ideal limit, a long-range ordered CDM with wave vector $\overline{\mathit{Q}}$ will have zero demodulation residue; i.e., $\left|{\mathit{R}}_{\overline{\mathit{Q}}}\right|=0$ and the Fourier amplitude will vanish for $\mathit{q}\ne \overline{\mathit{Q}}.$ However, the measured Fourier amplitude distribution is typically broad and asymmetric (Fig. 3B and figure 3.7b of ref. 20) and not well fitted by a smooth peak (Fig. 3C and SI Text, Statistical Analysis: Two-Dimensional Fitting). Hence we will seek the minimum of $\left|{\mathit{R}}_{\mathit{q}}\right|$ for which $\stackrel{\sim }{\psi }\left(\mathit{q}\right)$ retains an appreciable amplitude. For the data shown in Fig. 3 A and B we calculate the $\left|{\mathit{R}}_{\mathit{q}}\left[\psi \left(\mathit{q}\right)\right]\right|$ for every pixel identified by a colored symbol in Fig. 3E (SI Text, Demodulation in Two Dimensions, Smoothing, and Optimization Error). In Fig. 3E we plot the value of both $\left|{\mathit{R}}_{\mathit{q}}\left[\psi \left(\mathit{q}\right)\right]\right|$ and the amplitude in the d-symmetry form factor (dFF) Fourier transform $\stackrel{\sim }{\psi }\left(\mathit{q}\right)$ for each of these pixels in Fig. 3D. This shows that the procedure singles out one wavevector for the CDM in the y direction with a nearly vanishing demodulation residue, a gap between this $\left|{\mathit{R}}_{\mathit{q}}\right|$ -minimizing $\mathit{q}$ (which we denote by ${\overline{\mathit{Q}}}_Y$) and the rest of the wavevectors, and that this occurs for a wavevector within the Fourier intensity peak. The ${\overline{\mathit{Q}}}_Y$ is identified as the green pentagon within the black square box in Fig. 3D. Instructively, this demodulation residue-minimizing $\mathit{q}={\overline{\mathit{Q}}}_Y$ does not equal the wavevector at which the Fourier amplitude $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|$ is the largest. Indeed the power of the $\left|{\mathit{R}}_{\mathit{q}}\right|$ -minimization approach is that it singles out the phase-averaged ${\overline{\mathit{Q}}}_Y$ for this CDM, from a broad and asymmetric Fourier amplitude peak where ${\mathit{Q}}_A$ is unreliable (SI Text, Heterogeneity and Demodulation Residue; and SI Text, Statistical Analysis: Two-Dimensional Fitting). Most remarkably, we find that $\left|{\overline{\mathit{Q}}}_Y\right|$ has commensurate value $2\pi /4a_0$ within the error. Moreover, imaging the phase of CDMs at the commensurate ${\overline{\mathit{Q}}}_Y$ reveals in r space where the CDM phase is locked to the four expected discrete values [$n\left(2\pi /4\right);\hspace{0.17em}n=\mathrm{0,1,2,3};$ Fig. 3F], forming locally commensurate ${\mathit{Q}}_0=2\pi /4a_0$ regions of the fundamental CDM (SI Text, CDM Commensurability in Underdoped BSCCO). In this highly typical case of a Bi₂Sr₂CaCu₂O_8+x $\psi \left(\mathit{r}\right),$ the phase slips of the DCs average to zero as in Fig. 2B, confirming the fundamental ${\mathit{Q}}_0={\overline{\mathit{Q}}}_Y.$

Given this demonstrated capability of $\left|{\mathit{R}}_{\mathit{q}}\left[\psi \left(\mathit{q}\right)\right]\right|$ minimization to extract the defining Q from short-range CDM data, we next turn our attention to the doping dependence of fundamental ${\mathit{Q}}_X;{\mathit{Q}}_Y$ throughout the pseudogap regime of underdoped Bi₂Sr₂CaCu₂O_8+x (SI Text, Summary of Results for All Samples). Fig. 4A contains two side-by-side panels; Fig. 4A, Left shows measured $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|$ whereas Fig. 4A, Right is the measured $\left|{\mathit{R}}_{\mathit{q}}\left[\psi \left(\mathit{q}\right)\right]\right|$ analysis for its y-axis modulations. Fig. 4 A–E then shows a series of such pairs of measured $\left|\stackrel{\sim }{\psi }\left(\mathit{q}\right)\right|$ $\text{and}\hspace{0.17em}\left|{\mathit{R}}_{\mathit{q}}\left[\psi \left(\mathit{q}\right)\right]\right|$ for five different Bi₂Sr₂CaCu₂O_8+x hole-densities p = 0.06, 0.08, 0.10, 0.14, 0.17. In all cases, the demodulation residue-minimizing process clearly singles out the minimized values in $\left|{\mathit{R}}_{\mathit{q}}\left[\psi \left(\mathit{q}\right)\right]\right|$ for the phase-averaged CDM wavevectors. This is evident in the sharp minimum that is observed near the (0,0.25)2π/a₀ point (marked by a cross in Fig. 4 A–E, Right, $\left|{\mathit{R}}_{\mathit{q}}\left[\psi \left(\mathit{q}\right)\right]\right|$). Therefore, the most striking result as summarized in Fig. 4F is that the measured magnitudes of the average wavevectors ${\overline{\mathit{Q}}}_X;{\overline{\mathit{Q}}}_Y$ of the Bi₂Sr₂CaCu₂O_8+x CDM are all indistinguishable from the lattice commensurate values ${\mathit{Q}}_0=\left(\mathrm{0,1}/4\right)2\pi /a_0;\left(1/\mathrm{4,0}\right)2\pi /a_0,$ making the fundamental wavevectors ${\mathit{Q}}_X;{\mathit{Q}}_Y$ equal to ${\mathit{Q}}_0$ and virtually doping independent (SI Text, Summary of Results for All Samples). Moreover, the largest deviation of the conventional amplitude-derived ${\mathit{Q}}_A$ from the phase-optimized value $\overline{\mathit{Q}}={\mathit{Q}}_0$ is at lowest doping, which can be associated with the observed higher density of DCs at the same doping (SI Text, Heterogeneity and Demodulation Residue).

Ubiquity of Lattice-Commensurate CDMs

Comparison of this result with reports of a preference for a CDM periodicity of 4a₀ in YBa₂Cu₃O_7-x-based heterostructures (36), in the NMR-derived view of the lattice-commensurate CDM in YBa₂Cu₃O_7-x (37), and in the pair density wave state of Bi₂Sr₂CaCu₂O_8+x (38) points to growing evidence for a unified phenomenology of lattice-commensurate CDMs across disparate cuprate families. Of course, the wavevectors ${\mathit{Q}}_A$ of maximum intensity in X-ray diffraction patterns for YBa₂Cu₃O_7-x and La₂Sr(Ba)CuO₄ families evolve continuously with doping and appear generally incommensurate (14, 15). However, DC configurations of the type in Fig. 2A will result in such an incommensurate average wavevector $\overline{\mathit{Q}}={\mathit{Q}}_A$ even though the fundamental wavevector ${\mathit{Q}}_0$ of the CDM is commensurate, so that evolution of cuprate DC arrays (e.g., Fig. 2) can yield the incommensurate wavevector evolution detected by X-ray scattering (SI Text, Global CDM Lattice Commensurability); a related hypothesis has long been considered (39). Our application of the classic theory of CDM DC disorder (19) (Fig. 2) with CDM phase-resolved imaging and analysis reveals this as the correct picture for Bi₂Sr₂CaCu₂O_8+x. This finding motivates the hypothesis that doping dependence of ${\mathit{Q}}_A$ across all cuprate families is caused by a competition between incommensurate modulations promoted by evolving Fermi arcs and a lattice-commensurate CDM state, with this competition being resolved through DC configurations.

The $\left|{\mathit{R}}_{\mathit{q}}\right|$ -minimization technique introduced here is designed to use the additional CDM information available only from phase-resolved imaging (16, 17, 33). Remarkably, it reveals a doping-independent locking of the phase-averaged CDM wavevector to a lattice commensurate wavevector $\left|{\mathit{Q}}_0\right|=2\pi /4a_0$ oriented with the Cu-O-Cu bonds in Bi₂Sr₂CaCu₂O_8+x (Fig. 4). Moreover, we directly detect the CDM DCs between phase-locked commensurate regions that generate this situation (Fig. 3F). These observations have significant fundamental consequences for understanding the mechanism of the cuprate CDM state. They are orthogonal to a weak-coupling k-space–based picture for CDM phenomena, in which the fundamental wavevector should increase or decrease monotonically with doping or should evolve in discrete jumps even with “lattice locking.” Moreover, the commensurability is intractable as a perturbative effect of interactions in the k-space picture (40). By contrast, a lattice-commensurate CDM state has been obtained comprehensively in different types of strong-coupling r-space–based theories (2–11). For underdoped Bi₂Sr₂CaCu₂O_8+x at least, our data are far more consistent with such lattice-commensurate strong-coupling r-space theories being the intrinsic organizational principle for the cuprate CDM phenomena. Furthermore, nanoscale clusters of lattice-commensurate CDMs are the first broken-symmetry state to emerge at lightest hole doping (12, 13), multipletransport and spectroscopic measurements of cuprate quasiparticles have recently been demonstrated to be quite consistent with lattice-commensurate r-space theories (41), and YBa₂Cu₃O_7-x NMR studies (37, 42) are also most consistent with them. Explorations of universality of lattice commensurability of CDMs in other cuprate compounds can now be pursued using these phase-resolved imaging and $\left|{\mathit{R}}_{\mathit{q}}\right|$ -minimization techniques.