The asymmetric Fermi surface of Bi2201

Smit and co-authors present ARPES measurements on the cuprate superconductor Bi2201. Their data provide evidence for anisotropic electronic structure between the “nodal” directions G-X and G-Y. This shows that the electrons couple to the orthorhombic structural distortion, and should not be treated in a tetragonal scheme. This has direct consequences for evaluation of the carrier density determined by ARPES, which is discussed at length. The anisotropy is also reflected in the self-energy, and persists across a wide range of dopings and temperatures, which suggests that the anisotropy is not related to a phase transition such as nematicity or charge order.

The manuscript is remarkably well-written in a scholarly style, with highly systematic results and analysis. The data is high quality and compellingly evidences the anisotropic electronic structure. The extensive discussion carefully weighs the implications of the results, especially in relation to the well-known Presland relation. Therefore, I highly recommend this manuscript for publication, after my questions are addressed.

1. The authors should carefully describe the requirements to rule out anisotropy as an experimental artifact. For example, whether it can be derived from anisotropic stray fields in the vacuum chamber, or a distortion in the lens system or detector of the analyzer.
2. Similarly, the authors should comment on the possible role of photon momentum. The momentum of a 100eV photon is 0.05/Ang, which is the magnitude of the observed effect. Even for the lower photon energies of 28eV employed in Fig 1, the momentum of 0.01/Ang is not insignificant on the scales being discussed.
3. The authors should show the MDC fitting. In addition, they analyze the Lorentzian component of a Voigt fit. How would the results change with a different curve fit?
4. Is the carrier density fixed in the comparison of Fig 3? It shows that k_F for the tetragonal structure is smaller than that of both the GX and GY directions of the orthorhombic structure.
5. The locus of the Fermi surface is determined by MDC peaks, but this may have some error, especially in the antinodal directions where the band dispersion is flatter. Can the authors quantify this error? Perhaps to make it more compelling, the authors could overlay a tight-binding FS with carrier density taken from the Presland relation, and show that it systematically misses the FS.
6. Since the chemical substitution is non-stochiometric, it necessarily comes with some disorder. How would this manifest in the ARPES spectra and the Luttinger count?
7. Fig 2d shows a slight doping dependence to the asymmetry. Can this be accounted for simply by shifting the chemical potential in the DFT calculation? It is unclear from the tight-binding analysis since different hopping parameters are used for each doping.

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The manuscript entitled “The asymmetric Fermi surface of (Pby,Bi1−y)2Sr2−xLaxCuO6+δ” investigates the asymmetric Fermi surface of the cuprate Bi2201 using angle-resolved photoemission spectroscopy (ARPES). It reveals a significant asymmetry in Fermi momenta along orthogonal nodal directions, exceeding the orthorhombic distortion observed via X-ray crystallography. Using a tight-binding model and first-principle calculations, the authors propose a mechanism rooted in crystal-field splitting. The study presents a new, linear relationship between ARPES-derived and transport-derived doping estimates, providing insights into the electronic structure of high-temperature superconductors.
This manuscript is scientifically rigorous, novel, and well-aligned with SciPost Physics's standards for high-quality research. It makes a significant contribution to the understanding of electronic structure in high-temperature superconductors and fulfills the journal’s publication criteria. With that said, I have some points the authors should clarify:
1. Precision of Manipulator: To perform the cuts along ΓX and ΓY directions, it is necessary to rotate the sample and ensure that the cut is done precisely along the nodal direction. In the method section, it is mentioned that the experiments were conducted at I05 at the Diamond Light Source. How precise is the manipulator? It is worth mentioning this to ensure the robustness of the data.
2. MDC Analysis Presentation: In the paper, only the resultant fit of the MDCs is shown. However, the MDCs and the fitting are not displayed. The authors subtracted the full width at half maximum (FWHM) at EF from the energy-dependent Lorentzian width of the Voigt function fitted to the MDCs. As I understood, this subtraction technique was a key methodological step to ensure the anisotropy measured was intrinsic rather than influenced by extrinsic effects like experimental resolution or sample quality. Can the authors show the MDCs before and after subtraction as a function of energy and temperature, and conduct the same analysis to see if the results obtained are similar? Showing such comparative analysis will strengthen the paper and convince the readers about the robustness of their results. Such comparative analysis can be added as supplementary information.
3. Significance of Linear Relationship: The manuscript introduces a new linear relationship between ARPES-derived and transport-derived doping estimates. Could the authors elaborate on its significance and limitations more explicitly?
4. Quantification of Uncertainties: Systematic uncertainties in ARPES measurements are addressed, but an explicit quantification of experimental and theoretical errors would strengthen the conclusions.

5. Applicability to Other Cuprates: The paper focuses on Bi2201 and does not fully explore the potential applicability of the findings to other single layer cuprates. If possible, it would be interesting to put these findings in perspective with other cuprates.
6. Nematicity and Charge Order Discussion: The discussion of nematicity and charge order as alternative explanations for the asymmetry could be expanded to connect with ongoing debates in condensed matter physics. For example, resonant inelastic X-ray scattering (RIXS) studies on Bi2201 have provided significant insights into its electronic properties, particularly concerning charge order and spin excitations. A notable study by Y.Y. Peng et al. ([https://doi.org/10.1103/PhysRevB.94.184511](https://doi.org/10.1103/PhysRevB.94.184511)) utilized Cu L₃-edge RIXS to investigate charge density modulations in underdoped and optimally doped Bi2201. They observed short-range charge order with a momentum transfer of approximately 0.23 reciprocal lattice units, persisting up to optimal doping levels. This charge order was found to modulate along the Cu-O bond directions, with no evidence of modulation along the nodal (diagonal) direction. Additionally, the out-of-plane measurements indicated a lack of phase correlation, suggesting that the charge order in single-layer Bi2201 is primarily two-dimensional.
In the context of this manuscript, the authors reveal a 6-10% difference in the nodal vectors along the ΓY and ΓX directions. This asymmetry was attributed to crystal-field splitting and orthorhombic distortions in the CuO₂ plane. Connecting these findings, the RIXS-detected charge order modulating along the Cu-O bond directions aligns with the ARPES-observed Fermi surface asymmetry. Both studies highlight the significance of anisotropic electronic properties in Bi2201. The charge order observed in RIXS could influence the electronic structure probed by ARPES, potentially contributing to the detected Fermi surface asymmetries. Furthermore, the absence of charge modulation along the nodal direction, as reported in the RIXS study, complements the ARPES findings by indicating that the observed anisotropies are more pronounced along specific crystallographic directions.
7. In the sentence starting by Berben et al., a subscript is missing for R(T).

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