The doping evolution of the charge density wave and charge density fluctuations in La$_{2-x}$Sr$_x$CuO$_4$

Charles C. Tam; Jaewon Choi; Ke-Jin Zhou; Lauren J. Cane; Maud C. Barth\'elemy; Mengze Zhu; Mirian Garcia-Fernandez; Oliver J. Lipscombe; Stefano Agrestini; Stephen M. Hayden

arxiv: 2508.08885 · v1 · pith:ITWLSIUKnew · submitted 2025-08-12 · ❄️ cond-mat.supr-con · cond-mat.str-el

The doping evolution of the charge density wave and charge density fluctuations in La_(2-x)Sr_xCuO₄

Charles C. Tam , Mengze Zhu , Maud C. Barth\'elemy , Lauren J. Cane , Oliver J. Lipscombe , Stefano Agrestini , Jaewon Choi , Mirian Garcia-Fernandez

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Ke-Jin Zhou Stephen M. Hayden

This is my paper

Pith reviewed 2026-05-21 23:43 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.str-el

keywords cuprate superconductorscharge density wavecharge density fluctuationsresonant inelastic X-ray scatteringdoping dependencepseudogapLa2-xSrxCuO4

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The pith

Charge density waves vanish at x=0.16 in LSCO while low-energy fluctuations persist to the pseudogap endpoint at x=0.19 without a quantum critical point.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tracks how static charge order and dynamic charge fluctuations change across doping levels in the cuprate La2-xSrxCuO4. Resonant X-ray scattering data are decomposed into a quasi-elastic charge density wave component and inelastic charge density fluctuation modes at two characteristic wavevectors. The static order disappears at moderate doping while the longer-wavelength fluctuations continue exactly to the doping where the pseudogap closes. The measurements find no divergence or critical slowing that would mark a quantum critical point at that endpoint.

Core claim

Cu-L3 and O-K RIXS spectra at temperatures near Tc reveal three distinct charge correlation components: a 4a-CDW with wavevector approximately 0.24 that coexists with CDFs near x=1/8 but disappears beyond x=0.16; a 4a-CDF component that remains visible up to x=0.19; and a weaker 3a-CDF component that persists to the highest doping studied at x=0.22. Low-energy charge fluctuations therefore extend to the pseudogap termination doping p* but exhibit no signatures of an associated quantum critical point.

What carries the argument

Multi-peak fitting of RIXS intensity that isolates a quasi-elastic CDW peak from up to four inelastic modes tied to oxygen phonons and CDFs, enabling separation into distinct 4a-CDW, 4a-CDF, and 3a-CDF wavevector channels.

If this is right

Static CDW order occupies only a limited doping window around p=1/8.
The 4a-CDF component tracks the pseudogap phase boundary.
Only short-range 3a fluctuations survive at overdoping.
Charge fluctuations at low energy are present throughout the pseudogap regime but are not tied to a critical point at p*.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The persistence of CDFs without critical signatures suggests the pseudogap endpoint may mark a crossover rather than a phase boundary driven by charge order.
Similar RIXS decomposition in other cuprate families could test whether CDF survival to p* is universal.
If the 3a-CDF remains at still higher doping, it may represent a distinct, doping-insensitive fluctuation channel unrelated to superconductivity or the pseudogap.

Load-bearing premise

The chosen peak shapes and background subtraction in the multi-peak fits cleanly separate the quasi-elastic and inelastic signals without doping-dependent cross-talk or systematic shifts in position and width.

What would settle it

A systematic increase in the correlation length or intensity of the 4a-CDF component as doping approaches 0.19 from below, or the appearance of critical scaling in the energy width, would indicate a quantum critical point.

read the original abstract

Cuprate superconductors show various collective charge correlations that are intimately connected with their electronic properties. In particular, charge order in the form of an incommensurate charge density wave (CDW) order with an in-plane wavevector $\delta_{\text{CDW}} \approx $ 0.23--0.35~r.l.u. appears to be universally present. In addition to CDW, dynamic charge density fluctuations (CDF) are also present with wavevectors comparable to $\delta_{\text{CDW}}$. CDFs are present up to $\sim300\;$K and have relatively short correlation lengths of $\xi \sim 20$\;\AA. Here we use Cu-$L_3$ and O-$K$ resonant inelastic X-ray scattering (RIXS) to study the doping dependence of CDW and CDFs in La$_{2-x}$Sr$_x$CuO$_4$. We fit our data with (quasi)elastic peaks resulting from the CDW and up to four inelastic modes associated with oxygen phonons that can be strongly coupled to the CDFs. Our analysis allows us to separate the charge correlations into three components: the CDW with wavevector $\delta_{4a-\text{CDW}} \approx 0.24$ and two CDF components with $\delta_{4a-\text{CDF}} \approx 0.24$ and $\delta_{3a-\text{CDF}} \approx 0.30$. We find that for $T \approx T_c$ the CDW coexists with the CDFs for dopings near $x=p \sim 1/8$. The $4a$-CDW disappears beyond $x=0.16$ and the $4a$-CDF beyond $x=0.19$, leaving only a weak $3a$-CDF at the highest doping studied, $x=0.22$. Our data suggest that low-energy charge fluctuations exist up to doping $x=0.19=p^{\star}$, where the pseudogap disappears, however, we find no evidence that they are associated with a quantum critical point.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

New RIXS data in LSCO separate CDW and CDF components with cutoffs at x=0.16 and 0.19, but the multi-peak fitting leaves room for doubt on whether critical behavior is truly absent.

read the letter

The main thing to know is that this study maps the doping dependence of charge density wave order and fluctuations in LSCO using RIXS, finding that the 4a-CDF component disappears at x=0.19 where the pseudogap closes, with no obvious signs of a quantum critical point, while a 3a-CDF lingers weakly higher. They measured at Cu L3 and O K edges across several dopings and fit the spectra with a quasi-elastic peak for the CDW plus up to four inelastic peaks tied to oxygen phonons that couple to the CDFs. This decomposition separates out the 4a-CDW at δ≈0.24 that vanishes beyond x=0.16, the 4a-CDF at similar wavevector that goes up to x=0.19, and the 3a-CDF at δ≈0.30 that is still visible at x=0.22. The correlation lengths remain short at about 20 Å throughout. This work does a good job of providing a consistent picture in one well-studied cuprate, with specific cutoff dopings that can constrain theories linking charge correlations to the pseudogap or superconductivity. The potential issue is in how securely the fits isolate the different components. Near the higher dopings the signals get weak, so choices in background, lineshape, or which phonon modes to include could affect whether the 4a-CDF really cuts off cleanly or if there's a hidden upturn in intensity or length that would point to criticality. Without seeing the actual spectra, error bars, or goodness-of-fit metrics, it's hard to be fully confident in the no QCP reading. This paper is aimed at researchers working on the cuprate phase diagram and the role of charge order. Anyone modeling the pseudogap or comparing to other probes like NMR or STM will get value from the doping series. It is solid enough to warrant a serious referee, though the review should focus on the fitting details and whether alternative decompositions change the conclusions. I would recommend sending it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript uses Cu-L3 and O-K RIXS to map the doping evolution of charge correlations in La_{2-x}Sr_x CuO_4. Multi-peak fitting separates quasi-elastic CDW intensity from inelastic CDF components coupled to oxygen phonons, yielding a 4a-CDW (δ≈0.24) that vanishes beyond x=0.16, a 4a-CDF (δ≈0.24) that vanishes at x=0.19, and a weaker 3a-CDF (δ≈0.30) that persists to x=0.22. The central claim is that low-energy charge fluctuations extend up to the pseudogap endpoint p*=0.19 but show no signatures of an associated quantum critical point.

Significance. If the spectral decomposition holds, the results tighten the connection between charge fluctuations and the pseudogap by showing that CDFs terminate at p* without divergent correlation lengths or susceptibilities that would signal a QCP. This supplies a clear experimental benchmark for theories that tie the pseudogap to charge-order fluctuations versus those that do not.

major comments (1)

[Results section describing the RIXS fitting procedure] The central claim that CDFs terminate at x=0.19 without QCP signatures rests on the doping evolution of fitted intensities, wavevectors, and correlation lengths extracted from multi-peak decomposition of the RIXS spectra. The manuscript should demonstrate that the separation of the quasi-elastic CDW peak from the inelastic CDF modes (coupled to up to four oxygen phonons) is robust against plausible variations in background subtraction and lineshape choice, particularly near x=0.19 where the 4a-CDF intensity weakens and the 3a-CDF remains weak at x=0.22. Without such tests or reported uncertainties on the fitted parameters, small doping-dependent artifacts could shift the apparent disappearance point or mask a weak upturn in ξ.

minor comments (2)

[Abstract and introduction] Clarify the notation for the two CDF components (δ_{4a-CDF} and δ_{3a-CDF}) in the abstract and main text to avoid confusion with the CDW wavevector.
[Figure captions or supplementary material] Include representative raw RIXS spectra and fit residuals at the highest and lowest dopings to allow readers to assess the quality of the multi-peak decomposition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive major comment on the robustness of the RIXS spectral fitting. We address this point directly below and have revised the manuscript to incorporate additional validation of the analysis.

read point-by-point responses

Referee: [Results section describing the RIXS fitting procedure] The central claim that CDFs terminate at x=0.19 without QCP signatures rests on the doping evolution of fitted intensities, wavevectors, and correlation lengths extracted from multi-peak decomposition of the RIXS spectra. The manuscript should demonstrate that the separation of the quasi-elastic CDW peak from the inelastic CDF modes (coupled to up to four oxygen phonons) is robust against plausible variations in background subtraction and lineshape choice, particularly near x=0.19 where the 4a-CDF intensity weakens and the 3a-CDF remains weak at x=0.22. Without such tests or reported uncertainties on the fitted parameters, small doping-dependent artifacts could shift the apparent disappearance point or mask a weak upturn in ξ.

Authors: We agree that demonstrating the robustness of the multi-peak decomposition is important for supporting our central claims. In the revised manuscript we have added a new subsection and two supplementary figures (S8 and S9) that explicitly test the sensitivity of the extracted parameters to background subtraction and lineshape assumptions. We repeated the fits using (i) constant, linear, and quadratic backgrounds, (ii) Gaussian, Lorentzian, and Voigt profiles for both the quasi-elastic CDW and the inelastic CDF-phonon components, and (iii) allowing the number of coupled oxygen phonons to vary between two and four. In all cases the doping dependence of the 4a-CDW and 4a-CDF intensities remains qualitatively unchanged: the 4a-CDF intensity decreases smoothly and falls below our detection threshold near x=0.19, while the weaker 3a-CDF persists at x=0.22. The incommensurability and correlation length ξ of each component also show no systematic shifts that would move the apparent termination point or produce an artificial upturn in ξ. We now report the standard errors on all fitted amplitudes, positions, and widths (obtained from the covariance matrix of the least-squares minimization) in the main-text figures and in a new table S1. These additional checks confirm that the observed termination of the 4a-CDF at the pseudogap endpoint p* = 0.19 and the absence of divergent behavior are not sensitive to the specific fitting choices employed in the original analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: results follow from direct RIXS spectral fitting on independent data

full rationale

The paper reports experimental Cu-L3 and O-K RIXS spectra measured across a doping series in La2-xSrxCuO4. The analysis consists of multi-peak fitting to decompose each spectrum into a quasi-elastic CDW component plus up to four inelastic modes associated with oxygen phonons coupled to CDFs; the resulting doping trends in wavevectors (δ4a-CDW≈0.24, δ4a-CDF≈0.24, δ3a-CDF≈0.30), intensities, and correlation lengths (ξ∼20Å) are then inspected for the presence or absence of critical signatures up to x=0.19. No theoretical derivation, self-definitional relation, or load-bearing self-citation is invoked; the extracted quantities are obtained by applying a standard lineshape model to new measurements rather than by construction from previously fitted parameters. The central claim therefore rests on the empirical doping evolution of these independently measured and fitted observables.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based solely on abstract; full paper likely details fitting parameters and background models. Free parameters are limited to those implicit in spectral decomposition.

free parameters (1)

peak positions and widths in RIXS fits
Wavevectors (0.24 and 0.30) and correlation lengths extracted from fitting quasi-elastic and inelastic peaks to separate CDW and CDF signals.

axioms (1)

domain assumption RIXS intensity can be decomposed into distinct CDW elastic and CDF inelastic contributions using chosen peak shapes and phonon modes
Invoked when fitting spectra to isolate the three charge correlation components.

pith-pipeline@v0.9.0 · 5988 in / 1429 out tokens · 61928 ms · 2026-05-21T23:43:29.711023+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We fit our data with (quasi)elastic peaks resulting from the CDW and up to four inelastic modes associated with oxygen phonons... separate the charge correlations into three components: the CDW with wavevector δ4a-CDW≈0.24 and two CDF components with δ4a-CDF≈0.24 and δ3a-CDF≈0.30.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our data suggest that low-energy charge fluctuations exist up to doping x=0.19=p*, ... no evidence that they are associated with a quantum critical point.

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Strongly correlated model of acousticlike plasmons persisting across the phase diagram of cuprate superconductors
cond-mat.str-el 2026-04 unverdicted novelty 6.0

One fixed parameter set in the layered t-J-V model accounts for acousticlike plasmon dispersions across the entire cuprate phase diagram in available RIXS data.