Emergent Pair Density Wave Order Across a Lifshitz Transition

1. The reported "four-Fermi-point" phenomenology in the PDW regime is intriguing and potentially important.
2. The study combines DMRG and tDMRG and explores multiple $J_K$, aiming at a coherent real-space and spectral picture.

1. Key phase-identification claims ("dominant PDW"/"quasi-long-ranged uniform SC") are not yet fully supported by standard 1D diagnostics.
2. Numerical-method details and boundary/resolution issues (e.g., possible Friedel oscillations under OBC, momentum transforms, convergence/robustness checks) are insufficient for reproducibility and verification.
3. Presentation issues hinder validation, including ambiguous/misleading gap terminology and inconsistent/unclear SM (Appendix) figure captions/references.

The doped one-dimensional Kondo-Heisenberg (KH) chain is studied using DMRG/tDMRG, and PDW/uniform SC tendencies and their signatures in single-particle spectra are discussed. The topic is interesting, and the observation that the PDW regime can be characterized by four Fermi points in the single-particle spectrum is particularly intriguing. However, several central claims are not sufficiently supported by standard 1D diagnostics, and some aspects of the presentation are not well polished (and parts of the discussion are insufficient), which hinders verification. Therefore, I do not think that the current version meets the high standard of SciPost Physics. Before further consideration, I strongly encourage the authors to address the following points to improve the manuscript.

1. The authors state that "At $1/8$ hole doping and $J_K=3$, $J_H=1$, the PDW is the dominant order and is accompanied by a subsidiary charge density wave (CDW); when $J_K=4$, $J_H=1$, the KH model exhibits quasi-long-ranged uniform superconducting order with no oscillatory component." However, since the pair-pair correlation is not compared with other competing correlations (e.g., density-density correlations), it remains unclear whether the PDW or uniform SC correlation is actually dominant at long distances; moreover, because the data are shown only on a linear scale, it is also difficult to judge whether the behavior is truly quasi-long-ranged (power-law) or not. In addition, I am concerned that the oscillations of the correlator seen in Fig. 1(a) might simply reflect spatial modulations of the local density due to boundary-induced Friedel oscillations under open boundary conditions. If possible, please provide a comparison with competing correlations; if that is difficult, the authors should at least add further discussion and verification checks that facilitate assessing the robustness of these claims.

2. It is noted that the pairing correlations change from sign-alternating (oscillatory) to being of one sign upon tuning parameters, but the physical meaning of this change is not explained in a transparent way. Since the pairing operator used here is a nearest-neighbor spin-singlet, this behavior should not be interpreted as a change of internal pairing symmetry (e.g., $p$-wave $\to$ $s$-wave) without additional evidence. A more natural possibility is a change in the center-of-mass momentum of the dominant pairing correlations (e.g., $K=\pi$ for PDW versus $K=0$ for uniform SC). Please clarify this point explicitly and discuss the physical origin of the momentum change, if possible.

3. Figure captions and the main text refer to a "charge gap" (white line) and to the opening of a "Mott gap" upon increasing $J_K$. First, it is unclear what is meant by a "Mott gap" in the present context: at the doped filling considered, a genuine Mott (charge) gap associated with commensurability/Umklapp is not generally expected. Moreover, if the charge gap is defined via $\pm 1$ particle addition/removal energies, it is more appropriately interpreted as a single-particle (pair-breaking) gap in a Luther-Emery-like situation rather than an insulating charge gap. Please clearly define which gap is plotted in the spectral figures and what is meant by the term "Mott gap" in the discussion.

4. The main text discusses the emergence of in-gap excitations and states "see SM for photoemission results at $J_K=1$," but I could not locate where this is actually discussed in the Supplemental Material (the relevant Appendix/section/figure is not clearly indicated). Please explicitly specify the exact SM location (section and figure number) where the $J_K=1$ photoemission spectrum is presented and discussed.

5. In general, when a spin gap opens one expects the spin-spin correlation to decay exponentially, and a larger spin gap typically implies a shorter correlation length. In the present manuscript, however, it is not clear whether such a trend is actually supported by Fig. 6 or whether it is merely invoked as a qualitative expectation. Please clarify more explicitly what can be concluded from Fig. 6.

6. The captions of Fig. 7 and Fig. 8 do not appear to match the actual content/parameter labels shown in the figures. Please check and correct the captions (and any related in-text references) so that the stated parameter values unambiguously match the plotted data.

7. It is not clear to me what the main message of Appendix C is. While the spectra indeed change significantly with $J_K$ (which is evident from simply inspecting the figures), the current text largely describes what is already visible in the plots without providing a sufficiently sharp physical interpretation or a clear criterion for why these differences matter for the manuscript's central claims. Please revise Appendix C to state explicitly what each panel/parameter set is intended to demonstrate and how it supports the main conclusions.

8. In Fig. 9 the dispersion indeed shows two minima. However, such a two-minima structure can also appear in other 1D models (e.g., even in the standard half-filled Hubbard chain), and by itself it does not uniquely imply that the doped system has four Fermi points. In its present form, it is therefore unclear what the authors intend to demonstrate with Fig. 9 and how it quantitatively supports the central claim of a Lifshitz-type change (two $\to$ four Fermi points) in the PDW regime. Please clarify the intended message of Fig. 9 and provide a more direct connection to four Fermi points.

Minor points:

• Some important information about the numerical calculations is missing. In particular, please specify the boundary conditions used for each observable, how distances are defined in the finite chains for real-space correlation functions, and how resolution/finite-time effects in the tDMRG spectra are controlled.

• In the $t_1$-$t_2$-$J$ model Hamiltonian, double occupancy is forbidden; therefore it would be more appropriate to use projected fermion operators (e.g., $\tilde{c}$) in the Hamiltonian and in the definition of Green's functions, rather than the unprojected $c$ operators.

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