Pervasive symmetry-lowering nanoscale structural fluctuations in the cuprate La$_{2-x}$Sr$_{x}$CuO$_{4}$

D. Pelc; D. Zhai; F. Ye; I. Khayr; M. Greven; M. J. Krogstad; M. Spai\'c; N. Bielinski; R. J. Spieker; R. Osborn

REVIEW 3 major objections 5 minor 59 references

Pervasive symmetry-lowering nanoscale structural fluctuations in the cuprate La$_{2-x}$Sr$_{x}$CuO$_{4}$

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Nanoscale lattice disorder in a cuprate persists to 970 K, beyond previous limits.

desk verdict A careful, incremental extension of the prior LTO-fluctuation work, but the missing high-T Debye-Waller correction makes the headline T0 numbers insecure. read the letter →

arxiv 2502.02947 v1 pith:U5XDL5S7 submitted 2025-02-05 cond-mat.supr-con cond-mat.mtrl-scicond-mat.str-el

R. J. Spieker , M. Spaić , I. Khayr , X. He , D. Zhai , Z. W. Anderson , N. Bielinski , F. Ye

show 7 more authors

Y. Liu S. Chi S. Sarker M. J. Krogstad R. Osborn D. Pelc M. Greven

This is my paper · ORCID

classification cond-mat.supr-concond-mat.mtrl-scicond-mat.str-el PACS 74.72.-h61.05.fg

keywords La2-xSrxCuO4cupratesuperconductorsdiffuseneutronscatteringstructuralfluctuationsorthorhombic-tetragonaltransitionuniaxialstressrare-regioneffectsnanoscaleinhomogeneity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper reports that nanoscale structural fluctuations—local tilts of the CuO6 octahedra that break the crystal's tetragonal symmetry—are far more persistent than previously thought in the cuprate La2-xSrxCuO4. Using neutron and x-ray diffuse scattering, the authors show that in undoped La2CuO4 the strength of these orthorhombic fluctuations keeps following an exponential temperature dependence up to about 970 K, close to 60% of the melting point, and that the spatial correlation length remains about three lattice constants even there. They find that applying uniaxial stress along the tilt direction barely changes the exponential slope, even though the same stress enhances the ordered phase below the transition. The authors interpret the persistence and stress-insensitivity as evidence that the underlying structural inhomogeneity forms during crystal growth, is not caused by strontium substitution, and is tied to the same hidden inhomogeneity thought to shape the electronic phase diagram. They also report rod-like diffuse scattering at a nominally forbidden reflection in an overdoped sample, indicating local symmetry even lower than orthorhombic, possibly connected to extended defects.

What carries the argument

The load-bearing object is the diffuse scattering at LTO superstructure reflections such as (5/2 3/2 9), measured with time-of-flight neutron spectroscopy that separates the quasistatic response (energy transfers $\lesssim 2$ meV) from the total energy-integrated response ($\lesssim 10$ meV), and with high-energy x-rays for energy-integrated data. The argument turns on two empirical regularities: the integrated intensity follows $I \propto e^{-T/T_0}$ over a wide temperature range, and the Gaussian width of the peak follows a power law in $T - T_{\mathrm{LTO}}$. These are compared with rare-region theory, in which exponentially rare ordered puddles above a transition produce exponential scaling, and with the analogous exponential onset of superconducting fluctuations.

What would settle it

Re-measure the integrated intensity at (5/2 3/2 9) in undoped La2CuO4 from 700 K to 970 K, applying a measured Debye-Waller correction; if the corrected intensity deviates from a single exponential or yields a $T_0$ outside the reported uncertainty, the quantitative scaling claim fails.

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Extended reading notes

Core claim

The central claim is that short-range orthorhombic (LTO) fluctuations exist throughout the high-temperature tetragonal phase of La2-xSrxCuO4 and obey an exponential temperature scaling, $I \propto e^{-T/T_0}$, that persists to the highest temperatures measured rather than dying out at the structural transition. In undoped La2CuO4 the total-scattering intensity follows this law up to 970 K with $T_0 = 286 \pm 25$ K and the quasistatic channel with $T_0 = 179 \pm 26$ K, so the response becomes more dynamic as temperature rises. Correlation lengths follow a power law in $T - T_{\mathrm{LTO}}$ and at 970 K remain near three lattice constants. The exponential slope is nearly doping independent and, in an optimally doped sample, unchanged by uniaxial stress up to 500 MPa along [110], even though the same stress enhances Bragg intensity below $T_{\mathrm{LTO}}$ and shifts $T_{\mathrm{LTO}}$ upward by about 20 K. The paper reads these observations as support for a rare-region picture in which an underlying, growth-born structural inhomogeneity couples to order parameters and is insensitive to in-plane strain.

Load-bearing premise

The reported exponential decay constants rest on assuming the Debye-Waller thermal factor is negligible up to 970 K and at the high momentum transfer of the measured reflection, even though the check of that assumption only reached 700 K.

Editorial extensions

If this is right

If the fluctuations persist to 970 K in undoped La2CuO4, they cannot be blamed on strontium substitution; they must originate in the growth process or in the intrinsic perovskite structure.
Since 0.5 GPa strain barely changes the exponential slope, the mechanism setting the fluctuation scale is not the strain order parameter that drives the LTO transition, but something more hidden.
The similarity between the extracted energy scale near 100 meV and electronic scales such as magnetic superexchange and the superconducting gap implies structural and electronic inhomogeneity may share a common origin.
The rod-like scattering at (0 -3 4) in overdoped LSCO, if bulk in origin, would mean the local symmetry is lower than orthorhombic; if defect-related, it offers a way to probe dislocations or stacking faults through diffuse scattering.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension is to measure LTO diffuse scattering in crystals grown under different floating-zone conditions or with different post-growth annealing; if the exponential $T_0$ changes, that would directly confirm the growth-born inhomogeneity claim.
The quasistatic versus total scattering difference could be used to estimate the lifetime of the dynamic fluctuations; mapping that lifetime versus temperature would show whether they freeze into a glassy state on cooling or remain critical.
The apparent upward shift of $T_{\mathrm{LTO}}$ under strain, together with an unchanged $T_0$, suggests the energy scale of the hidden inhomogeneity is set by local lattice properties rather than by the bulk transition; comparing with hydrostatic pressure experiments would separate volume effects from symmetry-strain effects.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague, here's the short version. The paper is a genuine but incremental extension of Pelc et al. 2022 [15]. The new measurements—LTO diffuse scattering in undoped LCO to 970 K with energy discrimination, a triple-axis run at x=0.155, an in-situ uniaxial stress series, and a forbidden-reflection search at x=0.20—are all real additions. The authors use CORELLI properly to separate quasistatic from total scattering, and the stress-cell work is technically difficult and executed cleanly. The qualitative claims hold up: the fluctuations persist to ~60% of the melting point, become more dynamic at high T, the quasistatic correlation length is longer, and the exponential slope T0 does not change with stress even though the intensity drops 30–40%.

The soft spot is the Debye-Waller factor. The paper asserts it is negligible based on ref [15], which only went to 700 K. At Q≈15.6 Å⁻¹ and 970 K, exp(−Q²⟨u²⟩) is not obviously negligible, and its temperature dependence could easily shift the fitted T0 values by tens of Kelvin. That matters because the comparison to the prior x-ray T0 and the ~100 meV energy scale is one of the selling points. The persistence and dynamic character don't depend on this correction, but the quantitative exponential scaling claim is insecure until the authors provide a D-W estimate at the highest temperatures.

Two smaller issues. The 25 K shift in T'_LTO used to reconcile the correlation-length exponent with the mean-field 1/2 is introduced without independent justification; it reads as fitting freedom. And the abstract's "does not significantly alter" is understated: the intensity changes by 30–40% and TLTO shifts ~20 K. The authors do describe these effects in the text, so it's a framing problem, not a data problem.

Overall: this is a paper that should be reviewed. It doesn't overturn anything, but it solidifies a picture that has been building in this group for several years, and the new stress result is genuinely informative. The D-W point should be fixed with a calculation or a measurement. I'd send it to a serious referee.

Referee Report

3 major / 5 minor

Summary. This paper reports neutron and x-ray diffuse scattering measurements of La2-xSrxCuO4. The authors show that LTO structural fluctuations in the tetragonal phase of undoped La2CuO4 persist to 970 K, with integrated intensities decaying approximately exponentially with temperature (T0 = 179 +/- 26 K for the quasistatic channel and T0 = 286 +/- 25 K for the total channel) and with correlation lengths following a power law in T - TLTO. Quasistatic scattering is found to decay faster than the total scattering while exhibiting somewhat larger correlation lengths, indicating an increasingly dynamic response at high temperatures. For optimally doped samples, the authors combine neutron and x-ray data and find consistency with prior results, and they report that in situ uniaxial stress up to 500 MPa suppresses the fluctuation intensity by 30-40% but does not significantly change the exponential temperature scale. The paper also reports rod-like diffuse scattering at a nominally forbidden reflection in an overdoped sample. The authors interpret the results as evidence for an underlying nanoscale structural inhomogeneity that forms during crystal growth and that is insensitive to in-plane stress, with an energy scale of roughly 100 meV comparable to electronic scales.

Significance. If the quantitative claims hold, this is a valuable extension of prior work: the high-temperature neutron data uniquely separate quasistatic and total responses, and the in situ strain experiment addresses the robustness of the exponential scaling in a direct way. The paper is commendable for using multiple instruments, multiple doping levels, and energy-discriminated detection, and for reporting clear limitations of the large-volume x = 0.155 neutron sample. The central quantitative conclusion, however, currently rests on a Debye-Waller assertion that is not demonstrated at the highest temperatures, and the rare-region consistency check uses an adjustable transition-temperature shift. These issues are fixable with additional analysis and should be addressed before the quantitative scaling claim is accepted.

major comments (3)

[Section II (Experimental Methods), Fig. 2(c)] The exponential decay constants T0 for undoped La2CuO4 are extracted from integrated intensities at the (5/2 3/2 9) reflection, |Q| ~ 15.6 A^-1, without applying a Debye-Waller correction. The text states that this factor was 'investigated elsewhere and determined to be negligible [15]', but the cited check extended only to 700 K, whereas the exponential fit in Fig. 2(c) spans to 970 K. At this momentum transfer, exp(-Q^2 <u^2>) is strongly temperature dependent in the relevant range: even a modest increase of <u^2> from about 0.01 A^2 to 0.03 A^2 between 530 K and 970 K changes the cross section by roughly two orders of magnitude. Consequently, the reported values T0 = 179 +/- 26 K and 286 +/- 25 K, their comparison with the x-ray value T0 = 155 +/- 8 K, and the inferred ~100 meV energy scale are not secure. I ask the authors to provide a quantitative Debye-Waller estimate (for example, from high-temperature Bragg-peak intensities or a phonon model), apply it to the integrated intensities, and either quote corrected T0 values with systematic uncertainties or restrict the quantitative exponential-scaling claim to a temperature range where the correction is demonstrably negligible.
[Section IV and Fig. 4(b)] The purported consistency with the theoretically expected correlation-length exponent of 1/2 is obtained by invoking an effective transition temperature T'_LTO = TLTO - 25 K. Since this shift is not independently constrained, it functions as an adjustable parameter that can always improve agreement; as stated, the comparison is not a test of rare-region theory. Moreover, the sign of the effect appears to be described incorrectly: if the actual transition temperature lies below TLTO, plotting the correlation length against T - TLTO should make the apparent power-law exponent smaller than the true exponent (with magnitude less than 1/2), not larger. Please either fit the model with T'_LTO as a free parameter and report its fitted value and uncertainty, or present the comparison as qualitative only.
[Section III, Fig. 5] The central conclusion that uniaxial stress does not alter the exponential scaling should be supported by more explicit analysis. The strain-cell data yield T0 = 222 +/- 31 K, but there is no zero-stress measurement in the same experimental setup; the 25 MPa data are used as the baseline, even though the paper notes that the extracted correlation lengths at 25 MPa are already systematically smaller than ambient-pressure values. To substantiate the claim that the exponential slope is unchanged, please report fits of T0 and TLTO for each stress value, state the confidence intervals, and provide a formal test of slope equality across stresses. This is important because the abstract's 'insensitive to in-plane stress' claim rests on this comparison.

minor comments (5)

[Fig. 4(a) caption] The caption states that intensities were scaled to match the prior results; please specify the scaling procedure and state explicitly that the T0 fits are unaffected by the free scale factor.
[Section III] The phrase 'already at at 25 MPa stress' contains a duplicated 'at'.
[References] Reference [8] is missing its opening bracket: it reads '[8 A. N. Pasupathy' and should be '[8] A. N. Pasupathy'.
[Fig. 2(d) and Fig. 4(b)] The power-law fits (exponent 1/3) and the T'_LTO-shifted 1/2 line should be accompanied by fit ranges, uncertainties, and the number of points used; as presented, the solid and dashed lines cannot be evaluated quantitatively.
[Section IV and Fig. 6] For the rod-like scattering at the forbidden reflection, the energy windows used for the total and quasistatic channels should be given explicitly, since the comparison between the two channels is central to the claim that the response is primarily inelastic.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the T0 values and correlation lengths are direct fits to new neutron and x-ray data, and the central claims do not reduce to the paper's inputs or to the authors' prior work.

full rationale

The paper is an empirical scattering study. The central quantitative outputs—T0 = 179 ± 26 K and 286 ± 25 K for undoped La2CuO4, and T0 = 222 ± 31 K under uniaxial stress—are obtained by least-squares fits to measured integrated diffuse intensities, while correlation lengths are extracted from measured Gaussian peak widths. These quantities are not derived from a model that already contains them, and no fitted parameter is renamed as a prediction. The rare-region/inhomogeneity interpretation is qualitative and is not used to generate the quoted scales. The paper does rely on ref. [15] for prior data, the correlation-length extraction procedure, and an earlier Debye-Waller check, and several authors overlap with that reference; however, the new measurements independently extend the temperature, doping, and strain phase space, and no central claim reduces to a self-citation chain. The T'_LTO = TLTO – 25 K adjustment in Fig. 2(d) and Section IV is a post-hoc consistency check rather than a load-bearing derivation, and it does not determine the main results about persistence to 970 K, increasingly dynamic character, or stress insensitivity. The unquantified Debye-Waller contribution at high Q and 970 K is a legitimate quantitative correctness risk, but it is not a circularity because the correction is an external experimental check, not an input to the fitting model. Overall, the derivation chain is self-contained with respect to the paper's principal claims; the low score reflects only the presence of minor non-circular self-citation usage.

Assumptions & free parameters 7 free parameters · 7 assumptions · 2 invented entities

The ledger captures the fitted phenomenological constants (T0 values, power-law exponents), the ad hoc effective TLTO shift, the illustrative trapezoidal-distortion parameter, and the main domain assumptions about Debye-Waller corrections, background subtraction, energy discrimination, and the interpretive framework. The central experimental observations are new, but the quantitative interpretation rests on several assumptions that extend beyond what is directly verified in this paper.

free parameters (7)

T0 (exponential intensity decay constant) - x=0 quasistatic = 179 ± 26 K
Fit to the quasistatic neutron intensity at (2.5 1.5 9) vs. T for undoped La2CuO4 (Fig. 2c). It is a phenomenological fit parameter; the central scaling claim depends on it.
T0 (exponential intensity decay constant) - x=0 total = 286 ± 25 K
Fit to the total scattering neutron intensity for undoped La2CuO4 (Fig. 2c).
T0 (exponential intensity decay constant) - x=0.155 strained = 222 ± 31 K
Fit to the strain-cell x-ray data for optimally doped LSCO (Fig. 5a). Used to claim stress insensitivity.
Power-law exponent for correlation length = 1/3 (solid line); mean-field 1/2 assumed with adjusted TLTO
The Gaussian width vs T-TLTO is fit to a power law; the paper notes the exponent is lower than 1/2 and invokes a first-order transition with effective T'_LTO = TLTO - 25 K to reconcile (Fig. 2d, Section IV). This is a post hoc adjustment.
Effective transition temperature shift T'_LTO = TLTO - 25 K = -25 K
Ad hoc shift introduced in Fig. 2d and Section IV to make the correlation-length exponent agree with the mean-field value 1/2. Not independently determined.
Trapezoidal oxygen distortion parameter r = not fitted (r > 0)
Introduced in Appendix A to show that a trapezoidal distortion produces scattering at the forbidden (0 -3 4) reflection. No comparison to measured intensity; purely illustrative.
Intensity scaling factor for Fig. 4a = not stated
New intensities were 'scaled to match the prior results' from ref. [15] to compare slopes. A normalization choice, not a physical parameter.

assumptions (7)

domain assumption The Debye-Waller factor contribution to the temperature dependence of the diffuse intensity is negligible up to 970 K.
Invoked in Section II ('determined to be negligible [15]') and applied beyond the 700 K range of the prior x-ray check. If false, the reported T0 values are biased.
domain assumption The structural transition temperature TLTO(x) is known from prior literature (TLTO = 530 K for x=0, ~150 K for x=0.155).
Used throughout to compute relative temperature T - TLTO for scaling plots (Section II, Fig. 4). The paper explicitly says TLTO=530 K was assumed [28].
domain assumption The diffuse scattering at the LTO positions is dominated by orthorhombic octahedral-rotation fluctuations, with background adequately removed by a nearby-box subtraction.
Central to all intensity and width extractions; stated in Section II. If multi-phonon or other diffuse scattering contributes, the exponential scaling could be contaminated.
domain assumption Gaussian profiles provide a faithful measure of the diffuse peak width; the correlation length is extracted from the HWHM as in ref. [15].
Section II and IV. The paper notes Lorentzian fits give similar widths, but the Gaussian choice affects the extracted correlation lengths.
domain assumption The quasistatic channel (<= ~2 meV) and total channel (<= ~10 meV) at CORELLI reliably separate static from dynamic fluctuations.
Used to claim the response is increasingly dynamic at higher temperatures (Fig. 2c). The energy windows are set by statistical choppers; no explicit verification of the separation is provided.
domain assumption Rare-region theory (Vojta [49]) provides a valid framework for interpreting the exponential scaling.
Section IV uses it to argue the behavior indicates inhomogeneity. The paper does not derive this; it is an interpretive assumption from prior work.
domain assumption Atomic coordinates from Decroux et al. [60] are representative of LSCO for the structure-factor calculation.
Appendix A uses these coordinates to show a trapezoidal distortion yields intensity at forbidden reflections. The result is illustrative, not definitive.

invented entities (2)

Trapezoidal distortion of in-plane oxygen atoms (parameter r)
purpose: To explain weak scattering at the nominally forbidden (0 -3 4) reflection in x=0.20 LSCO.
Appendix A shows such a distortion makes the structure factor nonzero at forbidden positions, but no measured intensity is compared and no other observable is predicted. The paper itself calls it 'one possible scenario'.
Underlying nanoscale structural inhomogeneity
purpose: Proposed as the common cause of the exponential LTO fluctuation scaling and similar superconducting fluctuation behavior.
The paper adds indirect support (persistence to 970 K, stress insensitivity) but does not directly image or identify the inhomogeneity. It is an inferred entity, largely carried over from prior work [15,24].

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Cite this review

Pith. "Pith review of Pervasive symmetry-lowering nanoscale structural fluctuations in the cuprate La$_{2-x}$Sr$_{x}$CuO$_{4}$." pith.science (2026). https://pith.science/paper/U5XDL5S7

@misc{pith2026250202947,
  author       = {Pith},
  title        = {Pith review of: Pervasive symmetry-lowering nanoscale structural fluctuations in the cuprate La$_2-x$Sr$_x$CuO$_4$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U5XDL5S7}},
  note         = {Machine review of arXiv:2502.02947}
}

abstract

The cuprate superconductors are among the most widely studied quantum materials, yet there remain fundamental open questions regarding their electronic properties and the role of the structural degrees of freedom. Recent neutron and x-ray scattering measurements uncovered exponential scaling with temperature of the strength of orthorhombic fluctuations in the tetragonal phase of $La_{2-x}Sr_xCuO_4$ and $Tl_2Ba_2CuO_{6+y}$, unusual behavior that closely resembles prior results for the emergence of superconducting fluctuations, and that points to a common origin rooted in inherent correlated structural inhomogeneity. Here we extend the measurements of $La_{2-x}Sr_xCuO_4$ to higher temperatures in the parent compound (x=0) and to optimal doping (x=0.155), and we furthermore investigate the effects of in-situ in-plane uniaxial stress. Our neutron scattering result for undoped $La_2CuO_4$ complement prior x-ray data and reveal that the structural fluctuations persist to the maximum experimental temperature of nearly 1000K, i.e., to a significant fraction of the melting point. At this temperature, the spatial correlation length extracted from the momentum-space data is still about three lattice constants. The neutron scattering experiment enables quasistatic discrimination and reveals that the response is increasingly dynamic at higher temperatures. We also find that uniaxial stress up to 500 MPa along the tetragonal [110] direction, which corresponds to a strain of about 0.2%, does not significantly alter this robust behavior. Overall, these results support the notion that subtle, underlying inhomogeneity underpins the cuprate phase diagram. Finally, we uncover (for x=0.2) low-energy structural fluctuations at a nominally forbidden reflection. While the origin of these fluctuations is not clear, they might be related to the presence of extended defects such as dislocations or stacking faults.

Figures

Figures reproduced from arXiv: 2502.02947 by the authors.

**Figure 2.** Diffuse neutron scattering in the HTT phase of undoped La2CuO4 (x = 0), indicative of anisotropic short-range orthorhombic structural fluctuations. (a) Total scattering data showing the (5/2 3/2 9) peak in the HK9 plane at 630 K. The response is asymmetric along [1 1 0] and [1 1̅0], consistent with previous observations for undoped/underdoped LSCO [15]. (b) Corresponding one-dimensional momentum cuts along [1 1 0] a… view at source ↗

**Figure 3.** Asymmetric LTO fluctuations in undoped La [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Temperature dependence of (a) LTO scattering intensity and (b) correlation length for current and [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: (a) Temperature dependence of the diffuse x [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

Discussion (0). Continue with ORCID to comment.

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