Scattering interference signature of a pair density wave state in the cuprate pseudogap phase

An unidentified quantum fluid designated the pseudogap (PG) phase is produced by electron-density depletion in the CuO2 antiferromagnetic insulator. Current theories suggest that the PG phase may be a pair density wave (PDW) state characterized by a spatially modulating density of electron pairs. Such a state should exhibit a periodically modulating energy gap \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta }_{{{{{{\rm{P}}}}}}}({{{{{\boldsymbol{r}}}}}})$$\end{document}ΔP(r) in real-space, and a characteristic quasiparticle scattering interference (QPI) signature \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda }_{{{{{{\rm{P}}}}}}}({{{{{\boldsymbol{q}}}}}})$$\end{document}ΛP(q) in wavevector space. By studying strongly underdoped Bi2Sr2CaDyCu2O8 at hole-density ~0.08 in the superconductive phase, we detect the 8a0-periodic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta }_{{{{{{\rm{P}}}}}}}({{{{{\boldsymbol{r}}}}}})$$\end{document}ΔP(r) modulations signifying a PDW coexisting with superconductivity. Then, by visualizing the temperature dependence of this electronic structure from the superconducting into the pseudogap phase, we find the evolution of the scattering interference signature \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda ({{{{{\boldsymbol{q}}}}}})$$\end{document}Λ(q) that is predicted specifically for the temperature dependence of an 8a0-periodic PDW. These observations are consistent with theory for the transition from a PDW state coexisting with d-wave superconductivity to a pure PDW state in the Bi2Sr2CaDyCu2O8 pseudogap phase.

signature Λ( ) that is predicted specifically for the temperature dependence of an 8a0periodic PDW. These observations are consistent with theory for the transition from a PDW state coexisting with d-wave superconductivity to a pure PDW state in the Bi2Sr2CaDyCu2O8 pseudogap phase.
A spatially homogeneous d-wave superconductor has an electron-pair potential or order parameter Δ ( ) = Δ with macroscopic quantum phase and critical temperature Tc.
By contrast, a PDW state has an order parameter Δ ( ) that modulates spatially at wavevectors Δ ( ) = Δ( ) with a macroscopic quantum phase . In theory, such a state exhibits a particle-hole symmetric energy gap ∆ ( ) near the BZ edges, with the ∆ ( ) = 0 points connected by extended ( = 0) Fermi arcs [8][9][10] . Of necessity, such a partial gap suppresses ( ), ( ), ( ), and ( , ). Moreover, a pure PDW is defined by a pair potential modulation as in equation (1)  implying that the relict of suppressed superconductivity is a PDW. Therefore, our objective is to visualize the evolution with temperature of electronic structure, especially Δ ( ) and Λ ( ), from the superconducting into the zero-field pseudogap phase of strongly underdoped Bi2Sr2CaDyCu2O8.

Modeling the temperature dependence of the PDW state
For theoretical guidance, we use a quantitative, atomic-scale model for PDW state based upon CuO2 electronic structure and the t-J Hamiltonian, = − ∑ + ℎ. . ( , ), Here, the electron hopping rates between nearest neighbor (NN) and next-nearest neighbor (NNN) Cu orbitals are t and t', respectively, the onsite repulsive energy → ∞, thus the antiferromagnetic exchange interactions J=4t 2 /U, and the operator PG eliminates all doubly-occupied orbitals. A renormalized mean-field theory (RMFT) approximation then replaces PG with site-specific and bond-specific renormalization factors , and , based on the average number of charge and spin configurations permissible 34,35 . The resulting Hamiltonian is decoupled into a diagonalizable mean-field approximation using on-site hole density , bond field χ , and electron-pair potential Δ . This mean field t-J Hamiltonian has a uniform d-wave superconducting (DSC) state as its ground state, but PDW and DSC states are extremely close in energy, as has also been shown elsewhere [45][46][47] (Fig. 1d). However, the spectral gap defined by the position of the E>0 coherence peak reduces in the high-temperature PDW state due to the reduced Δ . We believe this discrepancy is a result of an inadequate treatment of self-energy effects in the current renormalized mean field theory, including the assumption of temperature independent Gutzwiller factors (Supplementary Note 1). Fig. 2b shows the most prominent Fourier components of the mean-fields in PDW+DSC and PDW states namely = (± 2 8 ⁄ , 0) and = 2 . All mean-fields, including the hole density and the d-wave gap order parameter 34 shown in Fig. 2c   We explore these predictions using strongly underdoped Bi2Sr2CaDyCu2O8 samples with resistive transition temperature = 37 ± 3 K and ≅ 0.08 as shown schematically by the white arrow in Fig. 1a. These samples are cleaved in cryogenic vacuum at ≈ 4. Cross correlation analysis of ( , ) at T = 0.14 and of ( , ) at T = 1.5 in this FOV versus bias voltage V, yield a normalized cross correlation coefficients around 0.9 for practically all energies (Supplementary Note 3), thus indicating that virtually no changes have occurred in spatial arrangements of electronic structure upon entering the PG phase.
The major exception is in the energy range +100 meV < < +160 meV, wherein the feature denoted coherence peak (arrow Fig. 1d) Figure 12). The implication is that the PDW state which definitely exists at lowest temperatures 35-38 , continues to exist into pseudogap phase. But in that case, since that pseudogap is often (but not always) reported to support no supercurrents, it would have to be in a strongly phase fluctuating PDW phase 32,33,50-54 .

Comparison of QPI Signature of a CDW and PDW State
Finally, we consider the widely promulgated hypothesis 15,16,17, that the pseudogap phase is a primary CDW state, whose charge density modulation breaks the translational symmetry of the cuprate pseudogap phase. First we note the very sharp distinction between these states: the mean-field order parameter of a PDW at wavevector is 〈 ↑ ↓ 〉, whereas for a CDW at wavevector it is ∑ 〈 , , 〉 . Second, while a periodically modulating energy gap is a key PDW signature (Fig. 3a), r-space energy gap modulation should be weak in a CDW state, where it is charge density which modulates. Third, the quasiparticles and their scattering interference are highly distinct for the two states. A primary CDW order by itself does not exist as a stable self-consistent solution of the RMFT t-J model at any temperatures or dopings that we have considered. However, we can study STM signatures of the CDW order non-self-consistently. Figure  To summarize: strongly underdoped Bi2Sr2CaDyCu2O8 at p~0.08 and T = 5 K exhibits the 8a0-periodic Δ ( ) modulations characteristics of a PDW coexisting with superconductivity 35,37,38 (Fig. 2d, Fig 3b). Increasing temperature from the superconducting into the pseudogap phase, seems to retain these real-space phenomena apparently thermally broadened but otherwise unchanged ( Fig. 3c-f). More obviously, the measured scattering interference signature 10 Λ( ) evolves from correspondence with Λ ( ) predicted for an 8a0periodic PDW coexisting with superconductivity 35 into that predicted for a pure 8a0-periodic PDW above the superconductive Tc in the pseudogap phase (Fig. 4). Furthermore, this signature is highly distinct from Λ( ) predicted for a 4a0-periodic CDW (Fig. 5). The clear inference from all these observations is that the Bi2Sr2CaDyCu2O8 pseudogap phase contains a PDW state, whose quantum phase is fluctuation dominated.

Methods
Single crystals of Bi2Sr2CaDyCu2O8 with hole doping level of p ≈ 8% and Tc = 37±3 K were Competing Interests: The authors declare no competing interests.

Data availability
All data are available in the main text, in the Supplementary Information and on Zenodo 55 .
Additional information is available from the corresponding author upon reasonable request.

Code availability
The data analysis codes used in this study are available from the corresponding author upon reasonable request.

Renormalized mean-field theory of the extended t-J model
The extended t-J model on a square lattice is given by where creates an electron at the lattice site with spin . The hopping amplitude is taken to be and ' when , are the nearest-neighbor (NN) and next-nearest-neighbor (NNN) sites, respectively. < , > and ( , ) denotes only NN, and both NN and NNN sites, respectively. The Gutzwiller projector projects out all configurations with doubly occupied sites from Hilbert space. Finally, represents the spin operator at site , and is the superexchange coupling between spins residing at NN sites. The no-doubleoccupancy constraint can be implemented by employing the Gutzwiller approximation, in which the projection operator is replaced by site-dependent Gutzwiller renormalization factors and for hoppings and superexchange coupling, respectively. The resulting renormalized Hamiltonian now reads, Further progress can be made by mean-field decoupling of the renormalized Hamiltonian in density and pairing channels with ensuing mean-fields hole density , bond-field , magnetic moment , and pair potential ∆ defined as =< Ψ Ψ >, where, |Ψ > is the unprojected ground state wavefunction. A direct diagonalization of the resulting mean-field Hamiltonian will not yield the lowest energy state, however, as the Gutzwiller factor themselves depend on the local mean-fields. Instead, the ground state energy =< Ψ | |Ψ > has to be minimized with respect to |Ψ > under constraints that the total electron density is fixed and |Ψ > is normalized 1 . This leads to following renormalized mean-field Hamiltonian for paramagnetic states ( = 0). = ∑ + ℎ. .
( , ), where, = − , ∆ Here, δij,<ij> = 1 for NN sites and 0 otherwise. In this work, we have focused only on paramagnetic states since we are interested in charge ordering without any long-range spin ordering as very few experiments suggested the presence of any long-range magnetic order coexisting with charge order in Bi2Sr2CaCu2O8+δ. In this scenario, the Gutzwiller renormalization factors are simply given by the following expressions 1  7) can be diagonalized by using a spin-generalized Bogoliubov transformation, yielding the following Bogoliubov-de Gennes (BdG) equation The BdG equation has to be solved self-consistently as the matrix elements depend on the mean-fields, which, in turn, depend on the eigenvalues ( , ) and eigenvectors . The paramagnetic ground state of the t-J model treated within aforementioned renormalized mean-field theory (RMFT) is a uniform d-wave superconductor (DSC). However, we are interested in pair density wave (PDW) solutions which have been shown to be very close in energy to the DSC state within RMFT 1-3 as well as in more rigorous numerical schemes like variational Monte-Carlo 4,5 and tensor networks 6 Here, the modulation wavevector is chosen to be = (±1/8, 0) based on experimental evidences 7,8,9 . A bond-centered PDW state with coexisting d-wave superconductivity (PDW+DSC) can be obtained as a self-consistent solution using a finite ∆ < ∆ , whereas a pure PDW state can be obtained by setting ∆ = 0 in the initial seed. We note that a computationally more efficient scheme to study unidirectional modulating states (in absence of disorder) is obtained by exploiting translational invariance in direction orthogonal to modulations. Here, BdG equations on 2D lattice are Fourier transformed in the orthogonal direction to yield quasi-1D BdG equations. Details of this scheme can be found in Ref. [3]. Fig. 2(a), (c)-(f) in the main-text have been obtained using this scheme.
Results presented in the main-text were obtained using the parameter set = 400 meV, = −0.3 , = 0.3 . Further, we chose hole-doping = 0.125, which is larger than the doping = 0.08 at which experiments discussed in the main-text were performed because of the following reasons. First, it has long been known that the t-J model overestimates the doping scale of DSC dome by almost a factor of two. Experiments find the DSC dome to be in hole doping range ~ 0.05-0.3 whereas in the RMFT t-J model (with the aforementioned parameter set) it turns out to be in the range 0.01-0.45 (at = 0) 3 . If we account for this scale difference, then = 0.125 will be closer to the experimental doping of = 0.08. Second, it's hard to get converged PDW solutions at very low dopings as the derivatives of Gutzwiller factors, entering in the on-site potentials [Supplementary Eq. (10)], fluctuate strongly even with a small change in local doping 10 . This is more severe when solving the impurity problem. Finally, our conclusions mainly depend on just one premise: the low-temperature state is PDW+DSC and the hightemperature state is pure PDW, which does not depend on the actual doping level as long as it remains below a critical level ( ~ 0.18 at = 0) to realize these states.
For the aforementioned parameter set, a self-consistent pure PDW state is obtained in temperature range 0 < < 0.11 whereas the PDW+DSC state is found as a stable solution for 0 < < 0.085 . Both PDW and PDW+DSC states have almost equal energy per site, which is a few meV larger than the uniform DSC state 1,2,3,11 . This tiny energy difference between PDW+DSC and DSC state can be overcome by a variety of means, such as disorder which is not accounted for in the calculation. We have effectively assumed such effects to be present, leading to the PDW+DSC state at low-temperatures (in the range 0 < < 0.085 ) and the pure PDW state at higher temperatures (in the range 0.085 < < 0.11 ). In the PDW+DSC state, increasing temperature leads to a sharp decrease in the uniform DSC component (∆( = )) as shown in the main-text Figure 2e. For 0.05 < < 0.085 , ∆( = ) is very small but finite. This 'fragile PDW+DSC' state is a stable solution of the RMFT equations and not a computational artefact. This result is verified by the observation that lowering the self-consistency tolerance by an order of magnitude yields the same state.
In the main-text Fig. 2c and 2d, we showed spatial variation of hole density and d-wave gap order parameter, respectively. To complete the discussion of mean-fields, Supplementary Figure 2a-c show the spatial variation of NN bond mean-field in PDW+DSC (at = 0.01 , 0.04 ) and pure PDW state (at = 0.09 ). We find that the modulations in are typically of the size ~0.1t in PDW+DSC at low temperatures and ~0.05t in pure PDW states at higher temperatures. The bare bond fields are not physical observables, however. The physical expectation value of the bond operator ( ) in the Gutzwiller projected state is the bond mean-field scaled by the Gutzwiller hopping factor: = 12 . We can define the NN bond order at a given lattice site i as = ( , + , + , + , )/4, where ± ( ) represent NN sites to i along x(y)-direction.
As evident from Supplementary Figure 2d, the size of modulations in the bond order turns out to be an order of magnitude smaller than the bare mean-field. Similar to the case of hole density, the reduction in the modulation amplitude of bond variables for higher temperatures is a consequence of the reduction in the PDW gap order parameter (Fig. 2e in the main-text). Finally, we note that the bond order in both PDW and PDW+DSC states has a dominant d-form factor 11 .
In order to compute local density of states (LDOS), we first obtain lattice Green's functions ( ) using the eigenvalues and eigenvectors of the BdG matrix Here, 0 is a small artificial broadening set to be 0.01t, and the sum runs over all the eigenvalues. The diagonal lattice Green's function yields total LDOS at a site: where, represents imaginary part and the factor 2 accounts for spin degeneracy. Differential conductance measured in an STM experiment is, however, proportional to the sample's LDOS evaluated at the STM tip position 13 . Thus, we must compute the continuum LDOS few angstroms above the exposed BiO layer in Bi2Sr2CaCu2O8+δ for a meaningful comparison with the experimental data. Accordingly, we obtain continuum Green's function ( , ; ) via a basis transformation 14 from lattice to continuum space where the matrix elements of the transformation are given by the Wannier function ( ) centered at lattice site i.
The imaginary part of the diagonal continuum Green's function yields LDOS at a continuum point .
We have obtained the continuum LDOS at a height ~4Å above the BiO layer in Bi2Sr2CaCu2O8+δ employing a first-principles Cu-3dx2-y2 Wannier function obtained using the Wannier90 package, identical to that used in Ref. [3,15] and very similar to that in Ref. [16].
Supplementary Fig. 1a shows the continuum LDOS map at = ∆ in the pure PDW state at = 0.09 . The LDOS shows a periodicity of 4 . Supplementary Fig. 1b shows spectra at eight Cu positions marked in the panel 1a. Sharp features present at higher energies are expected to be broadened by inelastic scattering, which has been shown in Ref. [17] to be essential to account for the spectral lineshapes in underdoped cuprates. In that work, it was shown that the effects of inelastic scattering can be simply incorporated by adding a linear-in-energy term = | | to the constant artificial broadening 0 used in calculation of lattice Green's function [Supplementary Eq. (16)]. Using the experimental fits presented in Ref. [17], we set = 0.25. Supplementary Fig. 1c shows the continuum LDOS incorporating the linear inelastic scattering. All LDOS, ( , ), and Λ ( , ∆ ) results presented in the main-text, and gap map results presented in Supplementary Figure 7 have been obtained after accounting for the inelastic scattering.
We note that a finite value of artificial broadening 0 used in our calculations is responsible for non-zero LDOS at zero bias in the PDW+DSC state, as seen in Fig. 2a. Indeed, with decreasing artificial broadening, the zero-bias LDOS in PDW+DSC state approaches 0 due to presence of nodes in the quasiparticle spectrum 18 , as evident from Supplementary Figure 3. On the contrary, the zero-bias LDOS saturates at a finite value in pure PDW state due to the presence of Bogoliubov-Fermi surface 1,18 .

Supplementary Note 2 PG gap ∆ ( ) modulation detection
We determine the gap map ∆ ( ) by measuring the energy of the coherence peak in each / spectrum at > 0. Supplementary Figure 5b shows the magnitude of the powerspectral-density Fourier transform ∆ ( ) of the gap map ∆ ( ) in Figure 5a. There is strong disorder in ∆ ( ) surrounding = 0. The feature at a length of 1/5 in the (0, 0)-(1, 1) direction is related to the BiO supermodulation. The feature at about 20 degrees off the (0, 0)-(1, 0) direction at a length about 1/6 is possibly related to the electronic disorder. In Supplementary Figure 5, we show ∆ ( ) intensities before and after the exponential background has been subtracted. After the background is subtracted, the maxima at ≈ (0, ±1/8)2 / 0 and ≈ (±1/8, 0)2 / 0 become clearly visible. This analysis provides one type of experimental evidence of the 8 0 modulations in ∆ ( ).
We apply a computationally two-dimensional lock-in technique to obtain the amplitude ( ) of the gap modulation Δ ( ) at . Δ ( ) is multiplied by • and integrated over a Gaussian filter to obtain the complex-values lock-in signal 9,19 Where q denotes the wavevector of interest and σ the average length-scale in r-space. This technique is implemented in q-space where = 1/ is the cut-off length in q-space. is specified to capture only the relevant image distortions.

Supplementary Note 3 Atomic precision image registration
In the temperature dependence experiments, ( , 5 K) and ( , 55 K) are measured in the same field of view with sub-unit-cell resolution. The data are processed by performing the Lawler-Fujita procedure 20 that maps the data onto a perfectly periodic lattice without lattice distortions. The data are subsequently corrected using shear transformation to maintain the C4 symmetry of the CuO2 crystal lattice. After the topographs are corrected, ( , 5 K) and ( , 55 K) are registered to the exact same FOV with atom-by-atom precision as shown in Supplementary Figure 6a and b. Subtraction of ( , 5 K) from ( , 55 K) gives rise to ( ) in Supplementary Figure 6c. The differences between ( , 5 K) and ( , 55 K) are noise and distortions in individual unit cells. They are not relevant to the demonstration from ( ) that the FOVs of 5 K and 55 K are identical.
The differential conductance map ( , ) is simultaneously acquired with ( ). Applying the same image processing procedures of correcting ( ) to ( , ) gives rise to the temperature induced electronic structure changes. The electronic structures are measured in a wide energy range from -800 mV to 800 mV which includes the PG energy range. The cross-correlation coefficient between ( , ) at 5 K and 55 K are around 0.9 in the large energy range (Supplementary Figure 6d). This method provides meaningful subtraction of high ( > ) and low ( < ) temperature data to detect temperature induced differences of the electronic structures at atomic scale.

Supplementary Note 4 Predicted temperature-evolution of gap map ∆1( )
We calculated the temperature evolution of the gap map Δ ( ). Δ ( ) is defined as the energy of the coherence peak at > 0, i.e., the same definition as the experimental measurement in main-text Figure 3. The gap modulation in the PDW+DSC state has a periodicity of 8 0 ( Supplementary Figure 7a and b). The amplitude of the y-averaged gap modulation is ~0.14 at = 0 and ~0.13 at = 0.04 . The gap modulation in the pure PDW state has a periodicity of 4 0 (Supplementary Figure 7c). The amplitude of the yaveraged gap modulation is ~0.05 at = 0.09 , which is much smaller compared to the PDW+DSC state. This is a consequence of the reduction in the PDW gap order parameter with increasing temperature (Fig. 2d). In this prediction the modulation periodicity of Δ ( ) changes from 8 0 to 4 0 in the transition from the PDW+DSC state to the pure PDW state. In experiments we have observed that Δ ( ) modulates at 8 0 (inset in main-text Figure 3a) at ≪ . However, the modulation periodicity of Δ ( ) could not be determined at = 55K = 1.5 due to the presence of large regions with indeterminate coherence peaks (see white regions in Figure 3f). Therefore, the predicted temperatureevolution of the gap map Δ ( ) could not be tested.
We note that the gap modulation is also possible in a state with coexisting charge density wave (CDW) and uniform DSC. In particular, a d-form factor (dFF) bond density wave

Supplementary Note 5 Bogoliubov quasiparticle scattering interference calculations
Bogoliubov quasiparticle scattering interference (BQPI) is a consequence of impurity scattering. To study the BQPI characteristics of the PDW+DSC and PDW states, we consider a point-like potential scatterer with impurity potential located at the lattice site * in the middle of an × square lattice. The resulting system is described by the following Hamiltonian where, is given by Supplementary Eq. (7), and the impurity Hamiltonian can be expressed as We set = 56 and = 3 . The Hamiltonian can be diagonalized following the same procedure used for diagonalizing . The resulting BdG equations have the same form as The upper cut-off of the energy sum is set to ∆ = 0.05 = 20 meV ( = 400 meV) to match with the experiment.
The as obtained ( , ) maps exhibit largest intensity at PDW driven charge order Bragg peaks = ± , = 0, 1, 2, … , 7 in PDW+DSC state and = ± (2 ), = 0, 1, 2, 3, in the pure PDW state (the fundamental charge order harmonic occurs at = in PDW+DSC state and at = 2 in pure PDW state as explained in the main-text), see Supplementary Fig. 9a, d. Accounting for the discommensurate short-range nature of the charge order in Bi2Sr2CaCu2O8+δ 24 will smear the Bragg peaks and reduce their intensity. The exact amount of suppression is not clear, though. In order to emphasize the QPI wavevectors emerging from impurity scattering, we have chosen to suppress the Bragg peaks by a factor = 100 ( Supplementary Fig. 9b, e). Finally, the resulting ( , ) maps are symmetrized by adding their 90°-rotated versions to account for the orthogonal domains of unidirectional charge modulations seen in the experiments 25 ( Supplementary  Fig. 9c, f). To further illustrate the effects of suppression of charge order Bragg peaks we show Λ ( , ∆ )-maps with suppression factors = 1, 10, 50, 100, 1000, in Supplementary Fig. 10. Clearly, if shown to scale ( = 1), the Bragg peaks will obscure all wavevectors emerging from impurity scattering. We found that a better match with the experimental result can be obtained by using = 100, although the qualitative features do not change significantly with once the Bragg peaks are suppressed somewhat, around = 50.
Energy integrated BQPI map Λ ( , ∆ ) in CDW state is constructed non-self-consistently via two independent methods. The first method is setting gap order parameter in selfconsistent PDW state (at = 0.09 ) to zero while keeping bond order and on-site potential modulations intact ( Supplementary Fig. 11a and main-text Fig. 5b). The other method is taking the normal state Hamiltonian from the uniform DSC state solution (at = 0.09 ) and adding a term producing a d-form factor bond ordered charge density wave (dFF-BDW) with wavevector = (1/4, 0) ( Supplementary Fig. 11b). The amplitude of the charge density wave is set to be the same as the uniform DSC state gap field. This state becomes equivalent to that in Supplementary Fig. 8 if the coexisting DSC state in the later is removed. To calculate Λ ( , ∆ ), an impurity Hamiltonian is added and subsequently the corresponding total Hamiltonians are diagonalized in the real space. This procedure is equivalent to a T-matrix calculation. Λ ( , ∆ ) in the pure CDW state obtained from both methods exhibit features very different from the Λ( , ∆ ) observed in experiments. We have presented Λ ( , ∆ ) map from Supplementary Figure  11a in the main-text Figure 5b. Moreover, we measure the arc-like feature in the experimental Λ( , ∆ ) and theoretical Λ ( , ∆ ). The extension of the arc is quantified by the angle subtended by the arc. We fit each arc of a circle about (±1, ±1) 2π/ point using least square fit (see Supplementary Figure 13). This procedure is carried out for six temperatures in both the theory and the experiment. The measured arc extension increases as a function of temperature from superconducting to pseudogap phase (Supplementary Figure 14a). This measurement agrees with the predicted arc extension in Λ ( , ∆ ) from PDW+DSC state to pure PDW state (Supplementary Figure 14b).

Energy-evolution of QPI signatures of the pseudogap phase and the PDW state
To avoid the 'setup' effect in the experiments, we calculate the ratio of the total density of states We take the power spectral density Fourier transform ( , V) of ( , V). The ( , V) are summed up to ∆0, the energy that the Bogoliubov quasiparticles cease to exist 26 . ∆0 is around 20 meV in the 8% hole-doped Bi2Sr2CaDyCu2O8 sample studied in this paper. The energy evolution of the experimental ( , V) maps from 8 meV to 20 meV and the corresponding calculated ( , V) maps are presented in Supplementary Figure 15. The energy evolution of the wavevectors are visualized in a supplementary movie of ( , , 55 K) from 2 mV to 20 mV. The wavevectors evolve dispersively with energy only by a small amount.

Legends of Additional Supplementary Files
Supplementary Movie 1. Determination of ∆ from a movie of ( , ) at = 4.2 K. ∆ is defined as the energy that the Bogoliubov quasiparticles cease to exist. Supplementary Movie 2. Energy evolution of quasiparticles in the pseudogap phase shown in a movie of of ( , ) at = 55 K.  Fig. 4 of the main-text. Largest intensity occurs at non-dispersing charge order Bragg peaks = ± , = 0, 1, 2, … , 7. b. Same as in (a) with charge order Bragg-peaks suppressed for a better visualization of QPI wavevectors emerging from impurity scattering and to account for short-range discommensurate nature of charge order seen in the experiments. c. Symmetrized map obtained by adding the map in (b) and its 90° rotated version. d. BQPI ( , )-map at energy = 0.03 in pure PDW state at temperature = 0.09 . Largest intensity occurs at non-dispersing charge order Bragg peaks = ± (2 ), = 0, 1, 2, 3. e. Same as in (d) with charge order Bragg-peaks suppressed. f. Symmetrized map obtained by adding the map in (e) and its 90° rotated version. Figure 11. Λ ( , 20 meV) for CDW state constructed non-selfconsistently at temperature = 0.09 . a. Λ ( , 20 meV) for a CDW state constructed by setting the pair field to zero in the pure PDW state that is obtained self-consistently at = 0.09 . b. Λ ( , 20 meV) for a CDW state constructed by taking the normal state Hamiltonian from the uniform DSC state solution at = 0.09 and, subsequently, adding a d-form factor charge density wave term.