Half–Bogoliubons as the intermediate states for phase coherence in underdoped cuprates

Domain: physics
Published: 2025-12-03
Editorial outcome: minor_revision

Abstract
Superconductivity is achieved by the pairing of electrons and phase coherence between the Cooper pairs. According to the Bardeen-Cooper-Schrieffer theory, quasiparticles with Bogoliubov dispersion exist in the superconducting state, leading to the particle-hole symmetric coherence peaks on the single particle tunneling spectrum. Here we report the observation of two kinds of tunneling spectra showing only one side of the coherence peak in the underdoped cuprate superconductor Bi2Sr2-xLaxCuO6+δ, and we name the electronic states as half-Bogoliubons. Based on the model that two doped holes bind into a local pair within the 4a0 × 4a0 plaquette, we attribute the half-Bogoliubons to one-hole or three-hole intermediate states for phase coherence. An entanglement of distinct half-Bogoliubons would lead to the dynamic hopping of charge freedom, resulting in the phase coherence between the local pairs. Our results unravel a unique process for establishing the phase coherence through exchanging a charge between regions with local pairs.

Introduction
Superconductivity occurs when electrons pair up to form Cooper pairs and then condense into the superconducting condensate through phase coherence. The superconducting state is protected by the superconducting gap Δ, and the superfluid density governs the phase stiffness. According to the Bardeen-Cooper-Schrieffer (BCS) theory1, the elementary excitations of the superconducting ground state are Bogoliubov quasiparticles, which are the quantum-mechanical particle-hole mixture2. The annihilation operator of excited Bogoliubov quasiparticle (Bogoliubon) is a linear combination of electron and hole operators with the amplitudes of uk and vk via γk = ukck + vkc†-k; uk and vk obey the constraint for the coherence factors |uk|2 + |vk|2 = 1 for any momentum k (normalization)2–4. In BCS formalism, the Bogoliubov dispersion forms two branches separately at positive (+) and negative (–) energies Ek ± = ± √ξk2 + |Δk|2, where ξk is the quasiparticle energy counted from the Fermi energy (EF)5. The Bogoliubov band back-bending results in two symmetric coherence peaks at the gap energies ±Δ, and the BCS coherence factors are closely related to the spectral weight (|vk|2 below EF and |uk|2 above)6. Bogoliubov quasiparticles and superconducting gapped features are the natural consequence of the superconducting state7,8.

In cuprates, superconductivity originates from hole doping to the Mott insulators9–11. One peculiar point when compared to BCS superconductors is that the electron pairing and phase coherence in cuprates do not happen simultaneously12–14. However, what is sure is that the charge carriers in the superconducting state of cuprates are still condensed bosons that carry twice of the electron charge15, just like the Cooper pairs of BCS superconductors. Meanwhile, spectroscopy evidence has demonstrated the BCS-like particle-hole symmetric spectrum in cuprates2,6,16 besides a background due to the strong correlation effect17, which is especially true in samples at the optimal doping. Thus, it has been confirmed that the elementary excitations in the superconducting state are still Bogoliubov quasiparticles, just like in BCS superconductors8,18,19. However, because of the existence of pseudogaps10 at high energies and the correlation effect, this symmetric feature may be strongly distorted17. In the extremely underdoped region, although superconducting coherence peaks can still be detected by resistivity or magnetization measurements, the coherence peaks are very weak or invisible, showing a very weak coherence weight of quasiparticles.

Another important issue in cuprates is the existence of the 4a0 × 4a0 plaquette with internal stripes in the pseudogap phase20,21, and this order can also emerge in the region of the vortex cores22,23. Recent experiments provide evidence that the pseudogap phase contains an 8a0-period pair density wave (PDW) state24 and the 4a0-period plaquette is attributed to the secondary charge density modulation of such PDW23,25. These spectroscopic evidences strongly indicate the preformed incoherent Cooper pairs in the pseudogap phase14,26,27. Moreover, the spatial occupation of the 4a0 × 4a0 plaquettes systematically evolves with increasing hole doping, and statistics tell that each 4a0 × 4a0 plaquette contains approximately two holes on average, which is believed to be closely related to the preformed local Cooper pairing28,29. Consistently, recent investigation suggests a pairing glue originating from the quantum phase strings created by the hopping of holes in the spin background30. An electronic state with a 4a0 × 4a0 plaquette structure was discovered with approximately two holes within each plaquette, and it was proposed that local pairing may be formed through these two-hole bound states28,29. This two-hole ground state ansatz30 may capture the significant ingredients of the pairing force in cuprate superconductors. The key mystery now is to understand how the phase coherence is established between the preformed pairs within neighboring 4a0 × 4a0 plaquettes in order to have the superconducting state with the macroscopic phase coherence.

In this work, we report the observation of two kinds of asymmetric tunneling spectra showing only one coherence peak either at positive or negative energies in some regions of the underdoped Bi2Sr2-xLaxCuO6+δ (x=0.75, La-Bi2201) by using scanning tunneling microscopy or spectroscopy (STM/STS). Interestingly, the spectrum obtained by merging or averaging these two kinds of spectra mimics that of a complete Bogoliubov dispersion, and we thus name these one-side-peaked spectra as a result corresponding to half-Bogoliubons. It is found that the so-called half-Bogoliubon peak exhibits a strong temperature dependence, but is not suppressed by a magnetic field up to 9 Tesla applied perpendicular to the ab-plane. Based on the picture concerning the ground state of a two-hole bound state in one 4a0 × 4a0 plaquette, we attribute the half-Bogoliubons as the intermediate states for the phase coherence between the local paired states.

Results
Characterization of La-Bi2201 sample
Figure 1a shows the schematic phase diagram of the La-Bi2201 system which holds the antiferromagnetic (AF) order until the nominal hole density per Cu reaches p ≈ 0.10. Superconductivity emerges with further doping, and the maximum critical temperature Tc is about 35 K at the hole doping level of p ≈ 0.1631. The La-Bi2201 sample with a nominal La doping level of x = 0.75 studied in this work corresponds to the hole doping level of p ≈ 0.11432, which is higher than the threshold for the insulator-superconductor transition (p ≈ 0.10). Figure 1b, c display the magnetic and transport characterizations of the La-Bi2201 samples, respectively. Both curves show the bulk superconducting transition at about 15 K. From the broad magnetic transition, we conclude that the superconductivity is still inhomogeneous and maybe fractional. Figure 1d shows a typical atomically resolved topography of the BiO cleavage surface, and the supermodulation can be clearly observed on the surface. Figure 1e shows a set of tunneling spectra in a wide bias-voltage range recorded at points along the arrowed line in Fig. 1d. Although the spectra have a globally similar shape, they vary in positions, especially at high energies. Meanwhile, one can see that there is a broad hump around 0.8–0.9 eV and the hump energy is slightly spatially dependent. These details reveal that the electronic structure in space is inhomogeneous to a certain extent as expected for underdoped cuprates9. In addition, Fig. 1e shows a gap heterogeneity near ±100 meV, which is similar to the previously reported heterogeneous gap in Bi-based cuprate superconductors33,34. In the extremely underdoped sample with p ≈ 0.08, a charge transfer gap was observed with the gap edge at about −0.2 and +1.5 eV28. The gap is filled with partial density of states (DOS) with further doping, see for example in the sample with p ≈ 0.1028. The spectra measured in the present sample follow the trace of the filled charge transfer gap and the asymmetric high-energy differential conductance with respect to EF as observed in the sample with p ≈ 0.1028. The asymmetry of high-energy conductance is consistent with a previous report on lightly hole-doped Ca2-xNaxCuO2Cl2 (Na-CCOC)20, and it reveals a larger probability for electron extraction than for injection, as anticipated for a doped Mott insulator17,35. In the low-bias range, an energy gap can be observed at energies within ±100 meV in Fig. 1e. This gap is also observed in La-Bi2201 samples with lower doping levels28, and it is regarded as a large pseudogap or the so-called AF pseudogap when the charge-transfer gap is completely filled36. It should be noted that the coherence peaks associated with the superconducting gap are not visible here because of the very small superconducting gap energy, very diluted superfluid density and the very wide bias-voltage range with 10 mV ac modulation for the lock-in technique.

Tunneling spectra with half-Bogoliubons
In the La-Bi2201 sample with p ≈ 0.10, accompanied by the filling of the charge transfer gap, we observed the electronic structure with 4a0 × 4a0 basic plaquettes, and meanwhile, there are three unidirectional bars in each plaquette28. It is interesting to see whether such a peculiar electronic structure exists in the present sample with a higher doping level of p ≈ 0.114. Figure 2a shows the differential conductance mapping measured at +15 mV in a region with corresponding atomically resolved topography shown in Supplementary Fig. 1. One can clearly see the 4a0 × 4a0 plaquettes with three unidirectional bars in one plaquette, following the similar electronic structure observed in the sample with p ≈ 0.1028. The plaquettes and the unidirectional bars can be distinguished through the auto-correlation analysis (Supplementary Fig. 2). The density of the plaquettes corresponds well to the hole doping level if we assume two holes occupying one plaquette as we did in previous work28; the density increases obviously in the present sample with a higher doping level and the plaquettes almost cover the whole field of view. The calculated value from the density of the plaquettes is also close to the nominal doping level. The electronic structure is similar to the checkerboard order observed in Na-CCOC19,20. The differential conductance mapping at −15 mV is shown in Fig. 2b, and one can also see the plaquettes and the unidirectional bars. However, the outlines of these 4a0 × 4a0 plaquettes at −15 mV become more blurred than those at +15 mV on average. In both cases, one can see that some plaquettes become overlapped showing some regions with extended stripy bars or more than three stripy bars exist on one patch.

In the differential conductance mapping, besides the 4a0 × 4a0 plaquettes, we also find some short-range stripe patterns. As an example, two of them are shown within the white dashed square in Fig. 2a, b. The stripe patterns consist of several unidirectional bars with an interval of about 4a0/3, and the lengths of the patterns are a bit larger than 4a0. For example, the bright stripes near spot No. 1 show up at negative biases, while those near spot No. 2 show up at positive biases. However, these stripes show particle-hole asymmetry. One can see that the brightness of the stripes near spot No. 1 at negative biases becomes darker at positive biases, while the ones near spot No. 2 at positive biases are darker at negative biases. The particle-hole asymmetry also appears on the tunneling spectra measured at these points, and the examples are shown in Fig. 2c. One can see a clear peak at about −11 mV on the spectrum measured at spot No. 1 or at about +11 mV on the spectrum measured at spot No. 2, but the peak is absent at about +11 mV on the spectrum measured at spot No. 1, or at about −11 mV on the spectrum measured at spot No. 2. These spectra show only one side of the coherence peak, and the coherence peaks are absent at ±15 mV. In regions with moderate brightness, for example at spot No. 3 in Fig. 2a, b, one can see only some kinked features on the spectra at about ±11 mV, showing asymmetric feature. These regions may correspond to the ones with very fragile superconductivity, and thus the coherence peaks are very shallow. We will show data later to support this conclusion. Interestingly, the energies corresponding to the kinks in the spectrum measured at spot No. 3 are roughly consistent with the negative-bias side coherence peak of the spectrum measured at spot No. 1 and the positive-bias side peak of the spectrum measured at spot No. 2. In Fig. 2d, the blue and red curves illustrate the merging result of the negative-bias side of the spectrum measured at spot No. 1 and the positive-bias side of the spectrum measured at spot No. 2. Meanwhile, the green curve in Fig. 2d shows the averaged result of the two spectra measured at spots No. 1 and No. 2. Both methods yield the typical spectrum of a complete Bogoliubov dispersion for a perfect superconductor, i.e., there are two sharp coherence peaks at symmetric energies of positive and negative sides37. This is the key finding of our present work.

Based on the BCS theory, when the superconducting gap opens in a band, the Bogoliubov dispersion forms two branches at positive and negative energies Ek ± = ± √ξk2 + |Δk|2 (Fig. 2e). The Bogoliubov band structure results in the coherence peaks in the energy-dependent DOS shown in Fig. 2a, b. In regions with moderate brightness, for example at the gap energies, as illustrated by the simulated result for a d-wave superconducting gap based on extended Dynes formula38 (see Fig. 2f). The amplitudes of the coherence peaks qualitatively reflect the phase coherence stiffness of condensed Cooper pairs. In the BCS framework, the DOS of coherence peaks at positive energy can be simply quantified by |uk|2, while that at negative energies is evaluated by |vk|2. In BCS superconductors, the low energy electronic structure behaves DOS(+E) = DOS(−E) because of particle-hole symmetric Bogoliubov excitation |uk|2(+E) = |vk|2(−E). Therefore, the observed extremely particle-hole asymmetric spectra shown in Fig. 2c do break the common understanding of superconducting excitation in the BCS theory.

Local particle-hole asymmetry of the unidirectional charge stripes
In order to clearly visualize the detailed structures of unidirectional stripes, we enlarge the region within the dashed square in Fig. 2a, b and show them in Fig. 3a, b. The structural and spatial layout of positive and negative stripes is visually demonstrated, and they are along the directions of the Cu-O-Cu bonds. By doing autocorrelation analysis (see Supplementary Fig. 2), we verify that the spatial spacing of modulation for these two kinds of stripes is nearly 4a0/3. This spacing is consistent with the stripe distance of the nematic electronic structure within a 4a0 × 4a0 plaquette in the sample with p ≈ 0.1028. Although most of the plaquettes have the size of 4a0 × 4a0 in the present sample, the local stripes shown in Fig. 3a, b exceed the length of 4a0. By taking a comparison of the dI/dV mappings measured at the positive and negative biases, the stripes near the arrowed line C1 are very bright in the mapping measured at the positive bias but are very weak in the mapping at negative energies. Consequently, some spectra measured along C1 in Fig. 3c show clear coherence peaks at about +11 mV, however, some spectra do not show the coherence peak on the positive side but rather two roughly symmetric shoulders as shown in Fig. 3e, probably due to the inhomogeneity. Moreover, the coherence peaks disappear on the negative bias side, leaving only the kinked feature at about −11 mV. In contrast, the stripes along the arrowed line C2 are very bright in the dI/dV mappings measured at negative bias but vague at positive bias. Meanwhile, the coherence peaks appear only at about −11 mV but unclear at about +11 mV on the spectra measured along C2 as shown in Fig. 3d. At the positions where the mapping intensity is moderate at both positive and negative bias, for example at green dots near the stripes shown in Fig. 3a, b, the spectra do not show coherence peaks but only kinked feature at about ±10 mV. It should be noted that the superconducting gapped feature always exists even when the coherence peaks are absent on the spectra, and the gapped feature is clearer in the corresponding −d3I/dV3 curves shown in Supplementary Fig. 3. Except for the most salient features, spectra in Fig. 3c-e also show some humps at higher energies beyond the superconducting gap, and they may correspond to the pseudogap at relatively lower energies ranging from 20 to 50 meV33,39.

These extremely particle-hole asymmetric spectra are observed in underdoped superconducting samples across distinct doping levels, with the energy positions of the half-side peaks on both the positive and negative energy exhibiting a notable degree of inhomogeneity. Figure 4a, b displays typical half-side-peaked spectra recorded on samples with a slightly lower doping level of p = 0.108, featuring peaks at negative and positive bias, respectively. The peak positions are centered around ±15 mV of the sample bias. Similarly, Fig. 4c, d illustrates the inhomogeneity of half-side-peaked spectra measured on samples with p = 0.114. For both doping levels, the half-side peaks predominantly localize at approximately ±15 mV, with the absolute values of the peak positions spanning from ~10 to 18 mV. The inhomogeneity in the energy range of the half-side peaks is consistent with the heterogeneous superconducting gap in Bi220133,34. In Supplementary Fig. 4, we show the spatial map corresponding to the half-side peaks and shoulder-like spectral features, and provide a comparison across different doping levels. We also estimate the degree of asymmetry-related inhomogeneity in Supplementary Fig. 5, and compare it with typical doping inhomogeneity observed in similar systems28. The degree of asymmetry-related inhomogeneity decreases with increasing doping and the result is consistent with the spatial occupation of particle-hole asymmetric electronic states evaluated in Supplementary Fig. 4. These statistical results qualitatively reveal that the spatial occupancy of the half-side-peaked spectra decreases, while the population with the particle-hole symmetric feature increases with higher hole concentration. This trend directly reflects the enhancement of phase coherence with increasing hole doping.

For a well-defined superconducting tunneling spectrum, the gap magnitude Δ characterizes the pairing strength, while the sharpness of the coherence peaks indicates the phase coherence stiffness. The coherent spectral weight (weight of the coherent peak) can be suppressed by increasing temperature14,40,41 or external magnetic field42. To examine the thermal effect and magnetic field influence on the asymmetric coherence peaks observed in this work, we provide the temperature and magnetic field dependence of the spectra with features of half-Bogoliubons. In Fig. 5a-c, we show the temperature evolution for three kinds of typical spectra as displayed in Figs. 2 and 3. One can see that the peaks of the half-Bogoliubons at about +10 and −10 mV are both significantly suppressed by increasing the temperature from 0.3 to 20 K. Moreover, the particle-hole symmetric gap which shows a pair of moderate kinks on the spectra at about ±10 mV is filled up by DOS with increasing temperature. Therefore, the particle-hole symmetric gap with moderate kinks reasonably reflect the superconducting pairing energy scale. Then, we measured the spectra by applying magnetic fields up to 9 T with the direction perpendicular to the CuO2 plane. This field roughly equals to the upper critical field (Hc2) as previously reported for La-Bi2201 samples at hole concentration p = 0.11432. Figure 5d, e shows the tunneling spectra measured under μ0H = 0 and 9 T, and neither positive nor negative peaks are suppressed by the magnetic field. In Supplementary Fig. 6, we show another two sets of data which further strengthen the field-insensitive nature. In Figs. 4b and 5d, a broad peak emerges at approximately 50 mV. Additionally, a similar broad feature is observed between −50 and −10 mV in Fig. 5e. These energy scales have been reported to correlate with the antinodal region of the momentum space, probably sharing the same origin as the pseudogap33,39. Upon a close scrutiny, it is also evident that a small peak has emerged at +11 mV in Fig. 5d and it shows no marked response to the magnetic field. Therefore, we interpret this spectrum as a state which has not reached the status of superconducting phase coherence yet. It could be the intermediate state when the half-side-peaked spectra of the same kind in Fig. 5a is evolving to a particle-hole symmetric one with perfect superconductivity by accumulating DOS on the positive energy side (see Supplementary Fig. 7). To sum up, the magnetic field insensitive nature distinguishes the sharp particle-hole asymmetric peaks from the superconducting coherence peaks, the latter should be sharp and possess a sensitive magnetic field dependence. Then we can naturally raise the questions: what is the intrinsic nature of these states, and what is their relationship with superconductivity?

Half-Bogoliubons and the relationship with superconductivity
To answer these questions, we must consider what happens to the electronic states in the underdoped cuprates. In the Mott insulator, the on-site Coulomb repulsion U together with p-d orbital hybridization results in a charge transfer Mott insulator, and the inter-copper superexchange interaction J stabilizes a robust spin-half AF order9,11,43. When holes with modest density are doped into the CuO2 planes, the spectral weight transfers from high energy scale to fill up the charge transfer gap, forming a prominent pseudogap at about 200 meV10,28,44,45. The competition between the kinetic energy, spin interaction and Coulomb repulsion makes the pseudogap phase break the C4 symmetry within CuO2 unit cells leading to a locally stripy electronic structure46–49. The onset doping level for the emergence of the pseudogap is coincident with the appearance of the 4a0-period plaquettes which persist from the region with AF phase until global superconductivity is established22,28,29. In Supplementary Fig. 8, we estimate the densities of the 4a0 × 4a0 plaquettes at different hole doping levels through compairing the real space conductance modulation and the inverse Fourier transform of the 4a0-period wave vector. If we assume that each 4a0 × 4a0 plaquette contains approximately two holes, the hole density per Cu in the present sample is roughly about 0.112, which is quite close to the nominal doping level. Meanwhile, the spatial occupancy of the 4a0 × 4a0 plaquettes increases with enhanced hole concentration. After the spatial occupation of plaquettes exceeds a threshold corresponding to the transition from insulator to superconductor, the particle-hole symmetric spectrum shape gradually emerges, similar to that of a d-wave superconductor. According to the single particle tunneling mechanism, the positive (negative) bias side of the spectrum is dictated by injecting (extracting) electrons into (from) the system. Thus, if there is a coherent unoccupied hole DOS at a location, it would be able to observe a DOS peak on the positive bias voltage. On the contrary, if locally we have a coherent occupied DOS of electrons, it would allow us to see a peak at the negative bias when the electrons are taken away. This is exactly what we have witnessed here in the present experiment. In view that the merged and the averaged spectra shown in Fig. 2d mimics that of the complete Bogoliubov dispersion for a d-wave superconductor, it is reasonable to attribute the spectrum with one-sided coherence peak in Fig. 2c at about −11 or +11 meV to the half branches of the Bogoliubov dispersion. The half branch of the Bogoliubov dispersions Ek ± = ± √ξk2 + |Δk|2 (Fig. 2c, d) can be generally understood as two exclusive excited states of the ground state with approximately two holes in each 4a0 × 4a0 plaquette. Since the plaquette of carriers may reflect electron correlation nature and play a crucial role in local pairing50, it is reasonable that the extremely particle-hole asymmetric electronic states detected on the background of 4a0-period plaquettes may provide clues for the superconducting phase coherence.

Methods
Sample preparation and characterization
High-quality single crystals of Bi2Sr2-xLaxCuO6+δ (x=0.75) were grown by using the traveling-solvent floating-zone technique65. Temperature-dependent resistivity measurements were carried out with a physical property measurements system (PPMS-9T, Quantum Design). The DC magnetization measurements were performed with a SQUID-VSM-7T (Quantum Design).

STM/STS measurements
The STM/STS measurements are performed with a cryogenic ultrahigh vacuum STM (USM-1300, Unisoku Co., Ltd.). Single crystals were cleaved at room temperature in ultrahigh-vacuum conditions up to 1.2 × 10−10 torr. Sharp electrochemically etched tungsten tips were used in this study, and the calibration was carried out on the single-crystalline Au(111) surface. The tunneling spectra were recorded by using a standard lock-in amplifier technique with an ac oscillation with the magnitude of 1% Vset and at the frequency of 973 Hz. All the tunneling data in the main text were taken at a temperature of about 1.5 K except for those specially noted.

Data availability
The experiment data that support the findings of this study are available from the corresponding author upon request. The source data underlying Figs. 1–5 have been deposited in the figshare database under accession code https://doi.org/10.6084/m9.figshare.30269581. Source data are provided with this paper.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Authors have performed STM/S measurements on the underdoped Bi2201 and observed spectra that show asymmetric ‘coherence peak’, for which an intensity of the coherence peak at one side of the bias voltage is high while the one for the other side is absent in other location. The authors claim that the combination of these half-peaked spectra resemble that of a conventional particle-hole symmetric density of states of a d-wave BCS superconductor. The authors call such a spectrum of the half-peaked coherence peak ‘half-Bogoliubons’. The authors suggest that the ‘half-Bogoliubons’ is the intermediate states for the phase coherence, corresponding to the three-hole and one-hole states as the excited ones from the local pairing state of two holes.\nI am concerned about a systematic error in the measurement such that so-called ‘checkerboard’ states or ‘4a0x4a0 plaquette with unidirectional charge stripe’ in the author’s language observed at relatively low energies are strongly setpoint condition dependent. Figure C1 below is taken from the figure 1.7 in A. R. Schmidt Ph. D thesis, Cornell University (2009) from J. C. Davis group, demonstrating that such states appear as a systematic error in the measurements. For example, when dI/dV(r,-25mV) is measured on the underdoped Bi2212 at V0=150mV bias setpoint, a ‘checkerboard’ pattern is obvious. However, it’s virtually absent in dI/dV(r,-25mV) at V0=50mV, suggesting that the ‘checkerboard’ modulation is originated from high energy modulations via a denominator in the formula,\ndI/dV(r,E)∝N(r,E)/∫_0^(eV_0)N(r,V)dV.\nThis indicates that, at least, the dI/dV is not a good quantity to represent electronic structures of the underdoped Bi2212. Although the authors studied Bi2201, there is a strong correspondence in the electronic structure to those in underdoped Bi2212. Thus, I suspect that the ‘4a0x4a0 plaquette with unidirectional charge stripe’ in the manuscript are strongly setpoint dependent. A setpoint effect mainly comes from the integrated local density of states in the denominator in the equation above, and ‘normalization’ on the local density of states N(r,E) is enforced by the junction formation. This means that the coherence peak height can be extrinsically enhanced or decreased as well as the modulating structure, depending on a choice of V0. Therefore, I don’t think that the authors can unambiguously reach to the conclusion made in the manuscript based on erroneous measures. The manuscript is not suitable for publication in Nature Communications. My point-by-point response is as follows.\nFigure C1. Setpoint dependence in topography, dI/dV(r,-25mV) and Z(r,-25mV). This is the figure1.7 of A. R. Schmidt Ph. D thesis, Cornell University (2009) from J. C. Davis group.\n1. There might be redundancy to my comments above, but it has been demonstrated that the density wave observed in the pseudogap states is unidirectional with d-symmetry form factor such that the in-plane Ox and Oy degree of freedom is modulated with   out of phase (K. Fujita et al., PNAS 111, E3026 (2014)), and this is also true for Bi2201 (T. A. Webb et al., Phys. Rev. X 9, 021021 (2019)). These unidirectional d-form factor density waves were identified in quantities,\nZ(r,E)=dI/dV(r,+E)/dI/dV(r,-E) and R(r,E)=I(r,+E)/I(r,-E) that are setpoint independent, and their intensities are maximized at the pseudogap energy (Y. Kohsaka et al., Nature 454, 1072 (2008), M. H. Hamidian et al., Nature Phys. 12, 150 (2016)). Examining the paper by T. A. Webb et al., Phys. Rev. X 9, 021021 (2019), I suspect that the features the authors observed as the 4a0x4a0 plaquettes and the unidirectional charge stripes are originated from the unidirectional d-form factor density wave due to the setpoint effect. As it is evident from the Figure 1.7 above that the 4a0x4a0 plaquettes and the unidirectional charge stripes is strongly setpoint dependent, the authors need to reconsider where these signals come from.\n2. In p.5, l. 9, ‘The spectra look generally uniform, indicating that the electronic structure is almost uniform in a large energy range’. However, I notice that there is a hump around 0.8-0.9V, and its position is spatially different. Similarly, dI/dV varies in positions at the negative bias voltages, especially at high energies. In addition, I also see a gap heterogeneity near +/-100mV, as in the papers by W. D. Wise et al., Nature. Phys. 5, 213 (2009) and T. A. Webb et al., Phys. Rev. X 9, 021021 (2019), which reported the heterogeneous gap in Bi2201. I don’t think this statement is correct.\n3. In p. 5, l.11, ‘In the extremely undoped sample” → “In the extremely underdoped sample”?\n4. In p. 8, l. 16, ‘Cu chains’ → ‘Cu bonds’\n5. In p.10, l.7, ‘This magnetic field insensitive nature distinguishes the sharp particle-hole asymmetric peaks from the superconducting coherence peaks.’. This observation is based on a few spectra, so I don’t think this claim is statistically valid as the electronic structures are heterogeneous (see e.g. W. D. Wise et al., Nature. Phys. 5, 213 (2009)). So, I am not sure if this property is a representative intrinsic electronic structure in this compound. The authors need to validate statistically.\n6. p. 13, l. 9, ‘states with for phase coherence’ → ‘states for phase coherence’?\n7. p. 26, l.10, ‘demonstrates the breaken of a ‘two-hole’…’ →’ demonstrates the breakdown of a ‘two-hole’…’?\n8. p. 28, The topography shown in Fig. S1a looks to me that the STM tip is not single. The supermodulation structure is so weak, the missing atoms are not clear, and the ‘$’ signs are absent. Is this topo simultaneously taken with the maps in Fig2?\n9. p.29, Fig. S2c,d,e,f show the autocorrelation maps, and their insets show line profiles from the origin. However, values of the autocorrelation, as indicated in the inset, is too large. By definition, a value of the autocorrelation at the origin is 1, which is supposed to be the maximum overall. A normalization is not properly performed in these figures. The authors need to check the algorithm.\n10. p. 29, l. 6, ‘Autocorrelation map of a’ → ’Autocorrelation map of b’?\n11. p. 29, l. 7, ‘…full map of a.’ → ’…full map of b.’?", "reviewer_2": "Li et al. present scanning tunneling spectroscopic studies on underdoped cuprate superconductors, reporting spatially dependent particle-hole asymmetry that they attribute to so-called \"half-Bogoliubons\" Two distinct categories of dI/dV spectra, exhibiting pronounced coherence peaks either at the occupied or unoccupied states, are observed and interpreted as signatures of these fractionalized Bogoliubov quasiparticles. Based on these findings, the authors attempt to elucidate the emergence of phase coherence in the doping process of cuprates. While the experimental data are carefully collected and the topic is of potential interest to the community, I find that the core claims of the manuscript are not convincingly supported by the presented evidence. In its current form, I cannot recommend publication. My main concerns are as follows:\nIn summary, while the topic is of potential importance, the current manuscript does not provide sufficient evidence to substantiate its key claims. I recommend that the authors perform more comprehensive spatially resolved measurements and re-evaluate their interpretation in light of the concerns above.", "reviewer_3": "Han et al. report an STM observation of two distinct superconducting gaps in underdoped cuprates. The two types of spectra exhibit coherence peaks at either the positive or negative bias, but never both, a phenomenon that has not been previously clarified. The authors attribute these novel spectral features to “half-Bogoliubons,” as fully developed Bogoliubov quasiparticles are expected to produce symmetric coherence peaks according to BCS theory. They further propose that these half-Bogoliubons originate from intermediate states in the development of phase coherence.\nOverall, I find the reported phenomenon intriguing and believe it advances our understanding of phase coherence in d-wave superconductors including cuprates. However, before I can recommend publication, several issues need to be addressed:\n1. Since the key finding of this work is the discovery of two types of half-peaked spectra, more quantitative analysis is needed. I would like to see statistical data on these features in the doped cuprates—for example, how many locations exhibit a positive coherence peak, how many exhibit a negative peak, and how many display only shoulder-like features.\nAdditionally, I suggest the authors provide a comparison of these spectra across different doping levels.\n2. In Fig. 2, the authors present intensity differences at +11 mV and -11 mV but show dI/dV maps at +15 mV and -15 mV. While these biases are close, it would be clearer to present dI/dV maps directly at +11 mV and -11 mV to better illustrate the spatial distribution of the two spectral types.\n3. The claim that “the spectra measured along C1 in Fig. 3c show clear coherence peaks at about +11 mV” is not entirely convincing. By visual inspection, the fifth and sixth spectra from the top appear to exhibit only shoulders, similar to the case in Fig. 3e. The authors should provide a more precise description of whether the coherence peaks are indeed well-defined.\n4. The magnetic-field-dependent data in Fig. 4d and 4e require further clarification. In Fig. 4d, a broad peak appears at 50 mV, which is absent in Fig. 4a. Additionally, a peak is visible at +11 mV. Given this, how can it be claimed that there is no coherence peak at positive energy in this case? Similarly, in Fig. 4e, there is a broad feature between -50 mV and -10 mV. Is this a broad coherence peak or does it have another origin? The authors should provide a comment on these observations.\n5. On Page 5, the authors attempt to explain why the coherence peaks are absent in the large-bias-scale dI/dV spectra of Fig. 1e. To enhance clarity, I suggest they provide the measurement parameters, such as the modulation bias used in the lock-in detection, throughout the manuscript. This would help readers better understand the potential reasons for the absence of coherence peaks.\n6. The merging method used for the STM spectra is somewhat unconventional, as it simply combines half of two spectra. Would it be possible for the authors to instead take an average of the two spectra? This might yield a similar or even clearer result."}, "author_response": ""}, {"round": 2, "reviewer_comments": {"reviewer_1": "I would like to thank the authors for providing the answers to my comments. My major concern was whether the so-called ‘checkerboard’ states seen in g(r,E) at low energies is a true entity or not in the present manuscript. In the data the authors additionally supplied in the reply (Fig. R1), it was demonstrated that ‘unidirectional charge stripe’ structures within 4a0x4a0 plaquette, which is a signature of the ‘checkerboard’, disappeared in Z, L and R maps, as shown in Fig. R1(i-l) (please see comments below and the attachment). This concludes that the main findings in the manuscript are derived from erroneous measurements without proper prescriptions. Therefore, it’s not suitable for a publication in the Nature Communications. My new comments are in orange.\nOriginal Comments of Reviewer #1:\nI am concerned about a systematic error in the measurement such that so-called ‘checkerboard’ states or ‘4a×4a plaquette with unidirectional charge stripe’ in the author’s language observed at relatively low energies are strongly setpoint condition dependent. Figure C1 below is taken from the figure 1.7 in A. R. Schmidt Ph. D thesis, Cornell University (2009) from J. C. Davis group, demonstrating that such states appear as a systematic error in the measurements. For example, when dI/dV(r,-25mV) is measured on the underdoped Bi2212 at V0=150mV bias setpoint, a ‘checkerboard’ pattern is obvious. However, it’s virtually absent in dI/dV(r,-25mV) at V0=50mV, suggesting that the ‘checkerboard’ modulation is originated from high energy modulations via a denominator in the formula,\nFigure C1. Setpoint dependence in topography, dI/dV(r,-25mV) and Z(r,-25mV). This is the figure1.7 of A. R. Schmidt Ph. D thesis, Cornell University (2009) from J. C. Davis group.\nThis indicates that, at least, the dI/dV is not a good quantity to represent electronic structures of the underdoped Bi2212. Although the authors studied Bi2201, there is a strong correspondence in the electronic structure to those in underdoped Bi2212. Thus, I suspect that the ‘4a × 4a plaquette with unidirectional charge stripe’ in the manuscript are strongly setpoint dependent. A setpoint effect mainly comes from the integrated local density of states in the denominator in the equation above, and ‘normalization’ on the local density of states N(r,E) is enforced by the junction formation. This means that the coherence peak height can be extrinsically enhanced or decreased as well as the modulating structure, depending on a choice of V0. Therefore, I don’t think that the authors can unambiguously reach to the conclusion made in the manuscript based on erroneous measures. The manuscript is not suitable for publication in Nature Communications.", "reviewer_2": "The authors have addressed most of my comments in detail and have provided the necessary and sufficient additional data analysis and figures. These improvements significantly enhance the clarity and rigor of the experimental interpretations. Therefore, I recommend the manuscript for publication in principle, provided the following minor issues are adequately addressed:\n1) In response to my previous Comment 2, the authors emphasized that the observed particle-hole asymmetric spectra are not indicative of the true superconducting state, as they persist above Ts and diminish gradually with increasing temperature. However, it remains unclear what fundamentally determines the intensity of the asymmetric peak. The authors should elaborate on the microscopic origin or mechanism underlying the intensity of this feature. Furthermore, it would be helpful to clarify whether and how this intensity varies systematically with doping level.\n2) The authors attribute the spatially irregular particle-hole asymmetry to doping inhomogeneity, as often seen in underdoped cuprates. I suggest that the authors estimate the degree of asymmetry-related inhomogeneity, e.g., in Fig. S4, and compare it with typical doping inhomogeneity observed in similar systems. This quantitative comparison would strengthen the argument that the asymmetric spectra arise from coherent excitations of the two-hole state.", "reviewer_3": "The resubmitted manuscript by Professor Wen and colleagues provides a comprehensive and thoughtful response to the reviewers’comments. I appreciate the authors’efforts in addressing my previous concerns regarding the STM measurements and STS analysis, and I agree with their explanations. I would like to recommend it to be accepted."}, "author_response": "We thank the reviewer for his/her careful review. The major concern of the reviewer is about the observation of ‘unidirectional charge stripe structures within 4a0×4a0 plaquette’, which has been published already in our previous work [npj Quan. Mater. 8, 18 (2023)] and a similar work in another paper [Nat. Phys. 19, 1301–1307 (2023)] by another group. Our present manuscript shows a further work on the intermediate states of the local electronic pattern, which we name as the ‘half-Bogoliubons’ based on previous conclusions. We understand that the setpoint effect is very important for the scanning tunneling spectroscopy measurements. However, we disagree with this referee about the questioning of the fundamental observations here based on a set of unpublished data in a Ph.D thesis in a totally different system (both samples and doping levels are very different). We also note that the reviewer seems not to have read our last reply carefully, and he/she does not pay attention to what we comment on the Z-map and even miss the characteristic spots of the L-map. We should emphasize again that none of the methods can get the exact value of the local density of states (LDOS), but this should not be a reason to question all the data for revealing the basic electronic structure through the measurements and analysis of the mapping of dI/dV. If following the viewpoint of this reviewer, all the STM/STS works should be questioned because no one can get the real LDOS, which is certainly unjustified.\nI am not sure if the authors understand the setup effect. The modulating structures can be an issue that is on top of the uniform local density of states. A characteristic feature in energy in dI/dV spectrum, such as a peak position is not affected by the setup effect in general. Again, let me bring up symtoms of the setup effect in a data. One should suspect a presence of the setup effect, firstly if a modulating structure in a dI/dV map depends on the setup voltage V0, secondly if a modulating structure in a dI/dV map doesn’t depend on the bias voltage, and lastly if a contrast of the modulation in a dI/dV map flips with the bias voltage. If a dI/dV(r,E) map doesn’t show any of signatures above, then the setup effect is neither present nor negligible in dI/dV(r,E). Otherwise, one needs extra cares for electronic structure evaluations. dI/dV maps for the underdoped cuprates obviously fall into some (all) of these criteria including the authors’ data, as I have pointed out. g(r,E) is not a good observable for representing a non-dispersive modulation structure in the underdoped cuprates since local density of states are scrambled with spurious modulations due to the setup effect. The authors haven’t provided any systematic evaluations of the setup effect for the data.\nLet’s further justify our observations here and also in previous publications. The method of Z operation may remove the influence of the setpoint effect. However, the dimensionless quantity Z = g(r, +E)/g(r, –E) provides the information of N(r, +E)/N(r, –E). It does not reflect the real LDOS if it is particle-hole asymmetric at positive and negative bias, even if the denominator is normalized away. We do not think it is a better quantity than the differential conductance when showing the LDOS. In the reply below, we provide additional evidence that the ‘plaquettes with unidirectional charge stripes’ are the intrinsic electronic structures instead of originating from the setpoint effect. Although it may be difficult for the reviewer to change his/her inherent thought, we suggest the reviewer carefully read our reply and think it over.\nI noticed that Fig. 4 might be a good example of the setup effect although a, b and c, d are at the different doping levels. The authors described in the caption that these spectra are taken at different setpoint biases (-300mV, -150mV, -100mV and -100mV for a, b, c, and d, respectively). There are many curves and it’s not easy to trace each one of curves, but roughly speaking, in Fig. 4a, features at -100mV and those at -15mV are correlated. On the other hand, corresponding features are anti-correlated in Fig. 4c. Since the denominator (integrated density of states) in the dI/dV formula is equal to the setup current, the integrated density states are conserved everywhere in the FOV, forcing to have a crossing point in the spectra in Fig. 4c as indicated by the vertical red line. These effects may cause spurious modulations if electronic modulations are present at higher energies. Different setup biases can also result in a phase flip between -100mV and -15mV. This would be one of the examples of the setup effect.\nCopied from Fig4.\nThe texts in the light gray box are the comments and replies in the first round of review.\nOriginal Comments of Reviewer #1 of the First Review:\nI am concerned about a systematic error in the measurement such that so-called ‘checkerboard’ states or ‘4a0×4a0 plaquette with unidirectional charge stripe’ in the author’s language observed at relatively low energies are strongly setpoint condition dependent. Figure C1 below is taken from the figure 1.7 in A. R. Schmidt Ph. D thesis, Cornell University (2009) from J. C. Davis group, demonstrating that such states appear as a systematic error in the measurements. For example, when dI/dV(r,-25mV) is measured on the underdoped Bi2212 at V0=150mV bias setpoint, a ‘checkerboard’ pattern is obvious. However, it’s virtually absent in dI/dV(r,-25mV) at V0=50mV, suggesting that the ‘checkerboard’ modulation is originated from high energy modulations via a denominator in the formula,\nⅆI/ⅆV (r,E)=N(r,E)/(∫_0^(eV_0)▒N(r,V)ⅆV)\nFigure C1. Setpoint dependence in topography, dI/dV(r,-25mV) and Z(r,-25mV). This is the figure1.7 of A. R. Schmidt Ph. D thesis, Cornell University (2009) from J. C. Davis group.\nThis indicates that, at least, the dI/dV is not a good quantity to represent electronic structures of the underdoped Bi2212. Although the authors studied Bi2201, there is a strong correspondence in the electronic structure to those in underdoped Bi2212. Thus, I suspect that the ‘4a0×4a0 plaquette with unidirectional charge stripe’ in the manuscript are strongly setpoint dependent. A setpoint effect mainly comes from the integrated local density of states in the denominator in the equation above, and ‘normalization’ on the local density of states N(r,E) is enforced by the junction formation. This means that the coherence peak height can be extrinsically enhanced or decreased as well as the modulating structure, depending on a choice of V0. Therefore, I don’t think that the authors can unambiguously reach to the conclusion made in the manuscript based on erroneous measures. The manuscript is not suitable for publication in Nature Communications.\nResponse: We thank the reviewer for the careful review and thoughtful comments based on the principle of the STM measurements. The reviewer is totally right that the measured dI/dV is related to but not simply equal to the local density of states (LDOS) because of the setpoint effect. Thus, we should take care of the measured dI/dV data in the analysis. However, we do not think the Z mapping is a valid method to get the spatial evolution of LDOS, and it is the ratio between the LDOS at positive and negative energies. The division operation in the Z mapping will remove or significantly reduce the in-phase LDOS signal with similar amplitudes at positive and negative energies, for example, this was reported in the quasiparticle interference due to the impurities on the surface (e.g., Nature 363, 524 (1993)). Due to the presence of the setpoint effect which is inevitable in all STM experiments, most of the published papers of the dI/dV mappings by STM measurements were just showing the dI/dV mapping, which indeed should be carefully checked.\nBelow in Fig. R1, we reanalyze our data and show that the setpoint is not a big issue for demonstrating the ‘4a0×4a0 plaquette with unidirectional charge stripe’ observed in our experiment. As shown in Fig. R1(i), the ‘checkerboard’ patterns do become a bit blurred in the Z mapping, but it does not mean that these patterns do not exist in the LDOS mappings. In contrast, we should emphasize that the patterns ‘4a0×4a0 plaquettes with unidirectional charge stripe’ are clearly visualized and should be an intrinsic feature. Especially the spectra with one-side peak should be an intrinsic effect and determined by the local electronic properties. We argue in this way because of the following reasons.\nThe dI/dV mapping measured at +15 and –15 mV shown in Fig. 2 and Fig. 3 in our manuscript were conducted with the same setpoint conditions, i.e., Vset = –250 mV and Iset = 100 pA. According to the equation given by the referee above, the same setpoint condition would yield the same denominator or its spatial distribution. If the spatially resolved inhomogeneity of the dI/dV mapping were coming from the setpoint effect, they would exhibit the same feature in the field of view, namely, the mapping of dI/dV should show the same distribution of the bright and dark regions.\nBut this is not the case in the data.\nOriginal Comments of Reviewer #1 of the Second Review:\nI disagree with the authors. N(r,E) itself is highly heterogeneous in Bi2201 and it's not necessary for dI/dV map to show the same distribution of the bright and dark regions between opposite biases.\nResponse: The referee misunderstood what we tried to present in the previous response. Actually, we tried to explain that the observed ‘4a0×4a0 plaquettes with unidirectional charge stripes’ are real entities, and the related electronic structure cannot be induced by the setpoint effect. We agree with the reviewer that the local density of states N(r,E) is highly heterogeneous in Bi2201. Since the denominator of dI/dV at a certain location is the same for both positive and negative sample biases when measuring the conductance mappings, if the observed patterns were induced primarily by the setpoint effect, the mappings measured at positive and negative biases would show the same distribution.\nThis argument is directly against the reviewer’s statement of “the ‘4a0×4a0 plaquette with unidirectional charge stripe’ in the manuscript are strongly setpoint dependent”, because we have observed different patterns at opposite biases but with the same setpoint.\nThe authors have demonstrated in Fig. R1 that the q=3/4 peaks are gone in Z and R. This indicates that features of the unidirectional charge stripes within the plaquette are suppressed. Indeed, Z(r,25mV) and R(r,25mV) don’t show those features. It should also be noted that the L map is less effective in mitigating the setup effect.\nWe should emphasize that the division operation weakens the ‘checkerboard’ patterns in the Z mappings shown in Fig. R1(i), because of the reduction of in-phase oscillations in the real space at positive and negative energies. However, the ‘checkboard’ pattern is far from completely diminished as the corresponding Fourier transform Fig. R1(m) is showing the wave vector of the ‘checkboard’ pattern. A similar conclusion was obtained in a previous work (Science 343, 393 (2014)), and the authors showed that the Z mapping does not eliminate the charge orders at low energies with different setpoint conditions but changes the energy-dependent behavior of them.\nFig. R1 | 4a0×4a0 plaquettes with unidirectional charge stripes for the sample La-Bi2201 (p = 0.114). a,b, The conductance g(r,E) map at energy E = +25 meV and −25 meV in the same 30 30 nm2 field of view. c,d, Tunnel-current magnitude I(r,E) obtained concurrently with a and b, respectively. Setpoint condition for a-d: Vset = −250 mV, Iset = 100 pA. e-h, Fourier-transfrom of a-d shows predominant peaks around 1/4 (red arrows) and 3/4 (green arrows) in Cu-Cu lattice units 2π/a0. i, The conductance-ratio map Z(r,25 meV) = g(r,+25 meV)/g(r, −25 meV). j,k, The setpoint independent L(r,E) map L(r,±25 meV) = g(r,±25 meV)/I(r,±25 meV). l, Tunnel-current ratio map R(r,25 meV) = I(r,+25 meV)/I(r, −25 meV). m-p, Fourier-transfrom of i-l, respectively. The colour scale of d, k, l is reversed when compared to others because of the negative tunneling current I(r, −25 meV) at a negtive sample bias.\nOriginal Comments of Reviewer #1 of the Second Review:\nI don't understand the logic that the in-phase oscillation at positive and negative energies is an intrinsic property of the checkerboard and canceled out in Z or R maps. I carefully checked real space electronic structures of the ‘checkerboard’ patterns in the Z map shown in Fig. R1(i). In fact, the ‘unidirectional charge stripe’ structure that is a ‘characteristic’ signature of the ‘checkerboard’ is virtually gone in Z map (please see the separate file and flip back and forth).\nResponse: The reviewer may be an expert in scanning tunneling microscopy, but the reviewer believes that it is a basic operation to remove the setpoint effect by Z mapping. However, we should emphasize again that none of the methods can get the real spatial distribution of the density of states. Based on the formula adopted by the reviewer\ng(r,E)=N(r,E)/(∫_0^(eV_0)▒N(r,V)ⅆV)\nthe dimensionless quantity Z = g(r, +E)/g(r, –E) provides the information of N(r, +E)/N(r, –E). Suppose we have the same spatial modulation of LDOS for both negative and positive energies, we call this the in-phase case here, then the Z- or R-treatment would certainly weaken the modulation. Thus, we do not think that the Z(r,E)-map is a better quantity than the differential conductance g(r,E). Based on the definition of Z = g(+E)/g(–E), the spectrum of a uniform superconductor is a constant value versus energy, and the Z-map would be a featureless one at each energy. The same case occurs for the R-map.\nThe denominator in g(r,E) formula is common at both positive and negative bias voltages in the first place. Thus, even if N(r,E) is not modulated at E<< PG, the denominator can be modulated since N(r,E~ PG) is strongly modulated, leading to g(r,E) modulations at E<< PG. The authors didn’t provide systematic evaluations of their data.\nReturning to the observation of ‘plaquette with unidirectional charge stripes’, our group and another group have both reported similar observations on similar Bi-2201 samples in published papers [Li, H. et al. npj Quantum Mater. 8, 18 (2023); Ye, S. et al. Nat. Phys. 19, 1301 (2023).], and we do not think it is coming from the setpoint effect although the setpoint voltages are different in the experiments by the two groups.\nI checked the references (Li, H. et al. npj Quantum Mater. 8, 18 (2023); Ye, S. et al. Nat. Phys. 19, 1301 (2023)) and have to point out that none of them provided a systematic evaluation of the setup effect.\nAs mentioned in the previous reply, the division operation in the Z mapping will remove or significantly reduce the in-phase LDOS signal with similar amplitudes at positive and negative energies, for example, the quasiparticle interference due to the impurities on the surface.\nSignatures of the quasiparticle interference have indeed been visualized in Z-map very well. In Na-CCOC in which the checkerboard states are obvious in dI/dV maps, T. Hanaguri et al., Nat. Phys. 3, 865 (2007) demonstrated that the Bogoliubov QPI are visualized in Z-map very nicely. For Bi2201, T. A. Webb et al., Phys. Rev. X 9, 021021 (2019) and Ye, S. et al. Nat. Phys. 19, 1301 (2023) have demonstrated that the Bogoliubov QPI is also visualized in Z-map.\nThe situation is the same in the present case, and we address this issue in the newly added Supplementary Fig. 10. Based on the inverse Fourier transform operation to a pair of scattering spots corresponding to the unidirectional real-space period of 4a and 4a/3, the line cuts show that the spatial evolutions of the differential conductance mainly show the in-phase oscillations. There are only tiny differences in the local oscillation amplitude or the local phases. Therefore, the real-space oscillation and the spot intensity in the Fourier transform image are certainly weakened. Due to the limited area of the measurement in real space, the scattering spots with larger q usually have a weaker intensity than those with smaller q in the Fourier transform result. Such as the situation in our case, in Supplementary Fig. 10, one can see that the oscillation amplitude corresponding to the real space period of 4a/3 is only about 1/3 of that corresponding to the real space period of 4a. The division operation will significantly reduce the in-phase oscillations, and the intensities both decrease for the spots corresponding to 4a/3 and 4a real-space period. Since the initial intensity of the 4a/3 spot is already very weak, it is understandable that they are weakened in the Z- and R-maps.\nThese statements don’t rule out the possibility that the checkerboard modulations are originated from the denominator of the integrated density of states in the dI/dV formula.\nTo prove that the setpoint effect does not play an important role in determining the appearance of ‘the plaquettes with the unidirectional charge stripes’, we adopted a method to manually tune the setpoint. Thanks to the relatively low doping levels of the sample, we can find that the plaquettes exist in different places of the differential conductance mappings measured at positive and negative energies. The results are shown in Supplementary Fig. 11(a and b). The different places of the plaquettes induce the strengthened or weakened shapes of the plaquettes in the Z-map, see Supplementary Fig. 11(c). Meanwhile, we manually modify the setpoint conditions (by integrating the measured dI/dV curves to a certain bias voltage, for example -50 and -20 mV, as the denominator for each location along a line scan), and then calculate the LDOS at different energies, one can see that the feature of the plaquette with the unidirectional charge stripes still exists even when the setpoint condition is manually set to –20 mV.\nI don't think one can numerically change/modify the setpoint conditions. Integrated density of states up to the setpoint bias is always equal to the setpoint current which is guaranteed by the feedback prior to do a spectroscopy. However, a ‘setpoint current’ obtained by the numerical integral won’t be uniform in space due to the electronic heterogeneity. Therefore, a spectrum is normalized by the inhomogeneous current that is different from position to position in this case. Therefore, the authors cannot manually change/modify the setpoint bias.\nBased on the additional data and analysis, we argue that ‘the plaquette with the unidirectional charge stripes’ is an intrinsic electronic state in our present samples and is not due to the setpoint effect. The weakened feature of Z- and R-maps in real and q space is due to the cancelation of the in-phase oscillations of the dI/dV intensities of the related structures at positive and negative energies.\nWe gently remind that one cannot object to our observation by simply referring to a set of unpublished data in a PhD thesis. Especially, the samples and doping levels are very different in that study and ours, which also makes no sense to compare for these two cases.\nOne can cite a PhD thesis in a journal if data is unpublished. The thesis is open to public and is available from Davis group website. Phenomenology has been proven to be highly similar for Bi2212 and Bi2201. See M. H. Hamidian et al., Nature Phys. 12, 150 (2016) and T. A. Webb et al., Phys. Rev. X 9, 021021 (2019).\nIn addition, the 4a0×4a0 charge order is ubiquitous in cuprates and can exist in a large doping range across the phase diagram (Nature 430, 1001 (2004), Nat. Phys. 19, 1301 (2023)). These STM observations are consistent with direct observation of 4a0-period charge order in La-Bi2201 by resonant inelastic x-ray scattering (REXS) (Science 343, 390 (2014), Phys. Rev. B 94, 184511 (2016)). The REXS results rule out the possibility that the 4a0-period charge order is induced by a setpoint effect in the STM measurement, and reveal the consistency of 4a0×4a0 charge order that appears in surface and bulk. Altogether, we believe the 4a0×4a0 charge orders appearing on dI/dV maps is the reflection of intrinsic real space electronic structure caused by hole doping.\nOriginal Comments of Reviewer #1 of the Second Review:\nFirst, I have to point out that none of these papers pay attention or systematically evaluated the setpoint effect. Science 343, 393 (2014) mentioned the setup effect for their STM results in the supplementary information, but they only consider situations for |V0|>ΔPG. Second, unidirectional d-form factor Density Wave identified at the pseudogap energies also shows 4a0 periodic modulation that is visualized in Z and/or R-maps, which are setup independent.\nResponse: We respectfully disagree with the reviewer about this comment. The mentioned papers report systematic evolution of the low-energy charge order states in Bi2212 samples with different doping levels, and the modulation wave vector in underdoped samples, as reported, is in close similarity to our findings, although the origins could be different. We believe these publications have the same base for justification as ours and the authors did pay attention and systematically evaluated the setpoint effect. More interestingly, the reviewer mentioned the density wave in the pseudogap region or the d-form factor density wave in Bi-2212. As far as we know, these density waves have an antiphase feature for the opposite biases [M. Hamidian, et al. Nat. Phys. 12, 150-156 (2016)], which certainly gives a clear pattern in the Z- or R-map.\nScience 343, 393 (2014) reported results for the Bi2212 that is the same materials as those in Fig. C1. A phenomenology should be the same.\nOriginal Comments of Reviewer #1 of the Second Review:\nThird, I recall Fig. C1 that the ‘checkerboard’ patterns are significantly diminished for V0=50mV where V0<<ΔPG~80mV.\nResponse: We thank the reviewer for mentioning this set of data again and its seemingly weakened feature of the ‘checkerboard’ patterns when the setpoint bias is smaller. This is actually very easy to understand. Thanks to the high quality of the data in Fig. C1, we are able to extract the data from the figure and conduct a comparative analysis, and the results are shown in Fig. RR1 below. We admit that the ‘checkerboard’ patterns are weakened for V0 = 50 mV, but they do exist because the scattering spots are still clearly seen in the Fourier transform results.\nData shown in Fig. RR1 are the dI/dV maps at the same bias voltages but taken with different setup bias voltages. In particular, the pseudogaps are spatially heterogeneous, and it’s natural for a signature of the pseudogap states to be locally observed at E<50mV, such that checkerboard structures can still be seen even with V0=50mV. The fact is that the amplitudes for the checkerboard are systematically diminished with lowering the setup bias voltages as also evident from Fig. RR1(a-d) although the authors adjusted color intensities in the Fourier transform in Fig. RR1(e-h) to give impressions that they are not diminished (High and Low give wrong impressions). These results indicate that a characteristic energy scale of the charge order or a source of the checkerboard states come from relatively higher energy scales. The authors also confirmed that the checkerboards are setup dependent as in Fig. RR1. If this is a true entity, it must be observed in a setup independent fashion. The authors need to perform systematic measurements at different setup bias voltages and demonstrate that observed modulating structures are setup bias independent.\nWe have also noticed that the data presented in Fig. C1 are collected from a Bi2212 sample with Tc = 45 K, and this sample is reported with prominent Bogoliubov quasiparticle interference [Y. Kohsaka, et al. Nature 454, 1072-1078 (2008)]. Moreover, the superconducting gaps exhibit strong spatial inhomogeneous in the Bi2212 system and the gap value Δ_SC is reported to range from 20 meV to 70 meV [Q. Gu, et al. Nat. Commun. 10, 1603 (2019)]. Since the V0 = 50 mV in Fig. C1 is close to the gap energy Δ_SC≈eV0, the differential conductance will inevitably be influenced by the intensive Bogoliubov quasiparticle interference, this could be another reason why the ‘checkerboard’ modulations become weaker at lower setpoint bias.\nI don’t understand the logic here. Z(r,-25mV) shows Bogoliubov QPI modulations that is setup bias voltage independent (see Z(r,-25mV) in Fig. C1). Also, Y. Kohsaka, et al. Nature 454, 1072-1078 (2008) reported that the Bogoliubov QPI is terminated around 32mV. This suggests that average Δ_SC is about 32mV, but not 50mV. Moreover, T. Hanaguri et al., Nat. Phys. 3, 865 (2007), T. A. Webb et al., Phys. Rev. X 9, 021021 (2019) and Ye, S. et al. Nat. Phys. 19, 1301 (2023) have reported that the Bogoliubov QPI are visualized in Z-map very nicely in Na-CCOC and Bi2201, respectively. As I mentioned, there are strong modulations near Δ_PG~80mV. If one avoids contributions from the density wave modulations in the pseudogap states by choosing a lower setup bias voltage, one can mitigate contributions in the denominator as shown in Fig. RR1.\nFig. RR1 | Fourier analysis on differential conductance mappings in Fig. C1. a-d, The differential conductance mapping at E = −25 meV and measured under different setpoint bias V0. e-h, Fourier transform results carried on a-d, respectively.\nOriginal Comments of Reviewer #1 of the Second Review:\nFourth, it is the fact, R. Comin et al., Science 343, 390 (2014) has revealed that the symmetry of the charge order is d-symmetry in YBCO. Thus, results of the scattering measurements in Science 343, 390 (2014) and Phys. Rev. B 94, 184511 (2016) are consistent with the d-form factor density wave.\nResponse: We kindly point out that the paper Science 343, 390 (2014) reports the charge order driven by Fermi-Arc instability in Bi2201, and it’s not about YBCO. Moreover, the paper Phys. Rev. B 94, 184511 (2016) reports the observation of charge order in Bi2201 with wide doping levels by resonant inelastic x-ray scattering, which is not inconsistent with our results. Furthermore, in the whole paper we cannot see the conclusions about the “d-form factor” density wave.\nI apologize that I provided a wrong reference. This is the correct one, R. Comin et al., Nature Mat. 14, 795 (2015), which has revealed that the symmetry of the charge order is predominantly d-wave in YBCO as well as in Bi2201.\nAs a supplement to the discussion on the setpoint effect, we conduct analysis on L(r,E) ≡ g(r,E)/I(r,E) based on the differential conductance g(r,E) = dI/dV(r,E) and concurrently measured tunnel-current magnitude I(r,E). Here, L(r,E)∝N(r,E)⁄(∫_0^E▒N(r,E)dε), which is a way to weaken the setpoint effect (Surf. Sci. 181, 295-306 (1987)). From the L mappings shown in Fig. R1(j) and Fig. R1(k), one can clearly see the ‘4a0×4a0 plaquettes with unidirectional charge stripes’, and its corresponding Fourier transform Fig. R1(n) and Fig. R1(o) demonstrated the validity. This is also a proof that the observed plaquettes are an intrinsic property of the sample.\nOriginal Comments of Reviewer #1 of the Second Review:\nI disagree with the authors. The authors demonstrated that the real space patterns in both amplitudes and phases in Fig. R1(j,k) are highly different from those shown in Fig. R1(b,c). I carefully checked real space electronic structure of the ‘checkerboard’ pattern in the L maps shown in Fig. R1(j,k). In fact, the ‘unidirectional charge stripe’ structure that is a ‘characteristic’ signature of the ‘checkerboard’ is virtually gone in L maps (please see the separate attachment and flip back and forth).\nResponse: We thank the reviewer for the attention to our data and related analysis. However, cautious must be taken when having these analyses. The characteristic scattering spots do exist in the Fourier transform results for the L-maps. Here, we present it again in Fig. RR2 below and indicate the scattering spots by circles in Fig. RR2b. Meanwhile, the plaquette with the unidirectional charge stripes can also be observed in real space in Fig. RR2a, and we highlight some of them with blue rectangles. We hope the reviewer agrees with us on this point.\nIn the L map, the signatures of the checkerboard have been greatly reduced compared to those in the raw dI/dV. While remnant/residual features can be captured in the Fourier transform, their amplitudes are quite small as evidenced in the real space image highlighted in blue. Basically, no stripe features are visible. Spatial structures are almost gone.\nFig. RR2 | L mapping and related analyses. a, L(r,E) mapping repeatedly present here, which is the same as Fig. R1j. Local modulations are marked by blue rectangles. b, The Fourier transform results of a, which is the same as Fig. R1n. The green and red circles mark the positions of characteristic spots.\nTo clarify this issue more clearly, we also present below another set of new data in Fig. RR3 with a higher resolution. One can see clear plaquettes with the unidirectional charge stripes in the L-map (Fig. RR3c). We hope the reviewer will be satisfied with this set of data.\nBoth plaquette and charge stripes intensities are highly diminished in the L map as shown in Fig. RR3, and their amplitudes are close to the background noise level. While the authors claim in the Fig. RR3 caption that ‘L mapping result showing clear plaquettes with unidirectional charge stripes’, I don’t think it’s true. There are plaquettes and charge stripes much more than those the authors highlighted in this FOV, but majority of these features are effectively suppressed in L map.\nFig. RR3 | L mapping result showing clear plaquettes with unidirectional charge stripes. a, Differential conductance mappings g(r,+10 mV). b, Tunnel-current magnitude I(r,+10mV) obtained concurrently with a. Setpoint condition for a and b: Vset = −150 mV, Iset = 100 pA. c, L mapping defined as L(r,+10 mV) = g(r,+10 mV)/I(r,+10 mV). The green arrows indicates the unidirectional charge stripes within plaquettes.\nIn order to make the point clear to readers, we have added Fig. S8 to the Supplementary Materials to show the data in Fig. R1. Meanwhile, we have added some related discussions to Supplementary Note 8. We hope the reviewer will be satisfied with the revised version.\nOriginal Comments of Reviewer #1 of the First Review:\n1) There might be redundancy to my comments above, but it has been demonstrated that the density wave observed in the pseudogap states is unidirectional with d-symmetry form factor such that the in-plane Ox and Oy degree of freedom is modulated with π out of phase (K. Fujita et al., PNAS 111, E3026 (2014)), and this is also true for Bi2201 (T. A. Webb et al., Phys. Rev. X 9, 021021 (2019)). These unidirectional d-form factor density waves were identified in quantities,\nZ(r,E)=dI/dV(r,+E)/dI/dV(r,-E) and R(r,E)=I(r,+E)/I(r,-E) that are setpoint independent, and their intensities are maximized at the pseudogap energy (Y. Kohsaka et al., Nature 454, 1072 (2008), M. H. Hamidian et al., Nature Phys. 12, 150 (2016)). Examining the paper by T. A. Webb et al., Phys. Rev. X 9, 021021 (2019), I suspect that the features the authors observed as the 4a0×4a0 plaquettes and the unidirectional charge stripes are originated from the unidirectional d-form factor density wave due to the setpoint effect. As it is evident from the Figure 1.7 above that the 4a0×4a0 plaquettes and the unidirectional charge stripes is strongly setpoint dependent, the authors need to reconsider where these signals come from.\nResponse: We thank the reviewer for this critical issue. We have noticed the related references, and we thank the reviewer for recalling our attention to the related analysis. The reviewer is right that the analyses in these related works were carried out at high energies, and the data are the asymmetric part of dI/dV or I at positive and negative energies. Here we should emphasize again that the Z mapping and the R mapping do not reflect the information of the LDOS but the asymmetric part of the differential conductance and the tunneling current, respectively. We also calculated R(r,25mV) = I(r,+25meV)/I(r,–25meV) and show the result in Fig. R1(l). It can be seen that there is an absence of ladder-like d-form factor density waves in R(r,25meV) map, and its Fourier-transformed result in Fig. R1(p) shows bright peaks at wave vectors (cid:4)⃗~ (1/4∙2π/a0,0) and (0,1/4∙2π/a0), and the signal at (cid:4)⃗~ (3/4∙2π/a0,0) and (0,3/4∙2π/a0) are weaker. We note that the result is different from the related work (Science 315,1380-1385(2007)), possibly due to the relatively symmetric signal of the unidirectional bars with the real-space period of 4a0/3 and asymmetric signal of 4a0×4a0 at positive and negative energies, which is opposite to the situation at high energies near the pseudogap. Therefore, the sublattice-phase-resolved analysis used in Nature Phys. 12, 150 (2016) is not required for the data analysis in the present work. In our point of view, deriving the signal of some density waves from the asymmetric part at positive and negative energies is inappropriate.\nIn order to make the difference clearer to readers, the R(r,25mV) mapping has been plotted in the Supplementary Materials Fig. S8(l). Meanwhile, we have added some related discussions to Supplementary Note 8: “Moreover, there is an absence of ladderlike d-form factor density waves in R(r,25meV) map as shown is Fig. S8l. The Fourier transform of R(r,25meV) map shows bright peaks at wave vectors (cid:4)⃗~ (1/4∙2π/a0,0) and (0,1/4∙2π/a0), and the signal at (cid:4)⃗~ (3/4∙2π/a0,0) and (0,3/4∙2π/a0) are weaker. These evidences demonstrate that the 4a0×4a0 plaquettes is different from d-form factor density waves as previous reported (Science 315,1380-1385(2007))”\nCombining with the responses to the questions/concerns above, we hope the reviewer will be satisfied with the revised version.\nI don’t understand the point that the authors tried to claim. The ‘checkerboard’ structures appear to be absent in R(r,25mV) as shown in Fig. R1(l) (please see the separate attachment). As previously described, the ‘checkerboard’ patterns appearing in dI/dV at low energies depends on V0 and are virtually absent in Z, L, and R maps. This indicates that the ‘checkerboard’ is originated from spatial modulations in the denominator of eq. (1), which is the integrated local density of states up to V0. For V0 >  PG, the density wave modulations near  PG contributes to the integrated local density of states, inducing an erroneous dI/dV modulations as the checkerboard at low energies. For V0 <  PG, a checkerboard is virtually absent in dI/dV at low energies (Fig. C1) since the denominator does not include contributions from the density wave modulations near  PG. These observations suggest that electronic modulations near  PG, which is the d-form factor Density Wave, is the origin of a ‘checkerboard’ states at low energies.\nThe authors demonstrated that ‘unidirectional charge stripe’ structures within 4a0x4a0 plaquette, which is a signature of the ‘checkerboard’, disappeared in Z and L maps as shown in Fig. R1(i-l) (see also the attachment). These results reconfirm that g(r,E) is not a good observable for studying electronic modulations in the underdoped regime of cuprates.\nResponse: We thank the reviewer for the careful review. However, as we elaborated above, we cannot agree with this statement. As mentioned in the previous reply, the division operation in the Z-, R-maps will remove or significantly reduce the in-phase LDOS signal with similar amplitudes at positive and negative energies. For example, the quasiparticle interference patterns are due to the impurities on the surface.\nI don’t understand this. As mentioned above, signatures of the quasiparticle interference have indeed been visualized in Z-map very well. For Na-CCOC in which the checkerboard states are obvious in dI/dV maps, T. Hanaguri et al., Nat. Phys. 3, 865 (2007) demonstrated that the Bogoliubov QPI are visualized in Z-map very nicely. For Bi2201, T. A. Webb et al., Phys. Rev. X 9, 021021 (2019) and Ye, S. et al. Nat. Phys. 19, 1301 (2023) have demonstrated that the Bogoliubov QPI is also visualized in Z-map. Similarly, for Bi2212 Y. Kohsaka, et al. Nature 454, 1072-1078 (2008) in Z-map.\nHere, the situation is the same in the present case, and we address this issue in the newly added Supplementary Fig. 10. Based on the inverse Fourier transform operation to a pair of scattering spots corresponding to the unidirectional real-space period of 4a0 and 4a0/3, the line cuts show that the spatial evolutions of the differential conductance mainly show the in-phase oscillations. There are only tiny differences in the local oscillation amplitude or the local phases. Therefore, the real-space oscillation and the spot intensity in the Fourier transform image are certainly weakened. Due to the limited area of the measurement in real space, the scattering spots with larger q usually have a weaker intensity than those of smaller q in the Fourier transform result, such as our present case shown in Supplementary Fig. 10. One can see that the oscillation amplitude corresponding to the real space period of 4a0/3 is only about 1/3 of that corresponding to the real space period of 4a0. The division operation in Z- and R-map will significantly reduce the in-phase oscillations, and the intensities both decrease for the spots corresponding to 4a0/3 and 4a0 real-space period. Since the initial intensity of the 4a0/3 spot is already very weak, it is comprehensible that they are not clear in the Z- and R-maps.\nHowever, we should emphasize that the plaquettes with unidirectional charge stripes do exist in the Z-map, even with a manual setpoint voltage of −20 mV, which corresponds to an energy much smaller than the pseudogap. Therefore, we believe that plaquettes with unidirectional charge stripes are the nature electronic property in very underdoped cuprates, instead of originating from the setpoint effect.\nI disagree with the authors. I don’t think the authors can numerically change the setpoint effect. Before spectroscopy, a z-piezo voltage for the tip is ramped up/down until the setpoint current is obtained for a junction formation, and then feedback is turned off for a spectroscopy. This is the basic operation for the STS. Thus, a current (setup current) or an integrated density of states up to the setup bias is uniform everywhere in the map. The authors are manually changing the ‘setpoint voltage’ by numerically calculating the integrated g(r,E) from 0 to -20mV and normalized it pixel by pixel. However, a ‘current’ or an integrated density of states obtained by numerical calculation in this case is not uniform anymore because of the electronic heterogeneity, while it was uniform as guaranteed by the feedback. Therefore, the algorithm the authors introduced doesn’t change/modify setpoint effect at all since g(r,E) is normalized by unknown ‘current’ values at each pixel.\nOriginal Comments of Reviewer #1 of the First Review:\n2) In p. 10, l.7, ‘This magnetic field insensitive nature distinguishes the sharp particle-hole asymmetric peaks from the superconducting coherence peaks.’. This observation is based on a few spectra, so I don’t think this claim is statistically valid as the electronic structures are heterogeneous (see e.g. W. D. Wise et al., Nature. Phys. 5, 213 (2009)). So, I am not sure if this property is a representative intrinsic electronic structure in this compound. The authors need to validate statistically.\nResponse: We thank the reviewer for raising these issues. We agree with the reviewer that the electronic structures are heterogeneous. Actually, the half-side-peaked spectra can be observed in underdoped samples with different doping levels. In fact, the statement of the “magnetic field insensitive nature” of the half-side-peaked spectra is important since the spectrum is unique and the phenomenon has been repeatedly observed.\nIn Fig. R2 below, we have shown another set of data which further strengthen the field-insensitive nature, and we updated the spectra as Supplementary Fig. S5 in the revised manuscript. Here, we should emphasize that these half-side-peaked spectrum we observed do not obey the Bogoliubov dispersion in which the coherence peaks should be symmetric and the intensity of the coherence peaks is strongly related to the phase coherence. Here, the half-side-peaked spectrum is interpreted as an intermediate state, which has not reached the status of superconducting phase coherence yet. It may be related to how strong the binding of a hole (electron) to the two-hole bound state, thus it is hard to break this state by a magnetic field of a few tesla. We have addressed this issue in the revised version. In the revised version, we have added the related description to page 11, line 6 in the main text as: “In Supplementary Fig. 5, we have shown another set of data which further strengthen the field-insensitive nature”. We also added discussions for this concern in page 15, line 14 as: “The magnetic field insensitive nature indicates that the half-side-peaked spectrum has not yet reached the status of superconducting phase coherence”.\nFig. R2 | Repetitive magnetic field independence experiment of the “half-Bogoliubons”. a,b Tunnelling spectra measured at H = 0 T and 9 T magnetic field, and it shows that neither positive nor negative energy “half-Bogoliubon” peaks are suppressed by a 9 T magnetic field. Setpoint condition: Vset = −250 mV, Iset = 100 pA.\nOriginal Comments of Reviewer #1 of the Second Review:\nThe authors only showed another set of spectra, which I don’t think it’s statistically enough. I am not convinced by the datasets the authors provided.\nResponse: We thank the reviewer for the rigorous scientific scrutiny. Moreover, what we would like to explain is that the magnetic field independent nature of the half-side peaks is reproducible. To demonstrate the reliability, we present the additional data in Supplementary Fig. 6 showing magnetic-field independent behavior of the half-side peaks. At either positive or negative energies, the “half-Bogoliubon” peaks are hardly suppressed by a magnetic field.\nIt might be reproducible for a spectrum, but a fundamental question is if this is a representative property of this compound. I agree with the authors that there are many spectra with highly particle-hole asymmetric peaks, which are indeed presented in the manuscript. However, the field-dependent measurements are presented only for a few spectra. While the authors claim that the absence of the field-dependence is reproducible, data presented in the manuscript must be statistically valid.\nReviewer #3\n(Remarks to the Author)\nI think that the resubmitted manuscript by Professor Wen and colleagues provides a thoughtful response to the other reviewers’ comments again. I agree with their explanations. I would like to recommend it to be accepted."}]

Decision letter