Single-site diagonal quantities capture off-diagonal long-range order

I. D'Amico; I. M. Carvalho; M. Sanino; V. V. Fran\c{c}a

REVIEW 4 major objections 3 minor 60 references

Single-site diagonal quantities capture off-diagonal long-range order

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Local measurements at one site can detect a superconducting transition.

desk verdict A useful empirical observation undermined by a flawed symmetry-breaking mechanism; the paper needs major revision but deserves referee time. read the letter →

arxiv 2507.10328 v1 pith:KR7MPHDZ submitted 2025-07-14 cond-mat.str-el

M. Sanino , I. D'Amico , V. V. França , I. M. Carvalho This is my paper · ORCID

classification cond-mat.str-el

keywords single-sitediagonalobservablesoff-diagonallong-rangeorderextendedHubbardmodelparticle-holesymmetrybreakingentanglemententropyquantumphasetransitiontripletsuperconductivitydensitymatrixrenormalizationgroup

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 challenges the common assumption that detecting phases with off-diagonal long-range order, such as superconductivity, requires measuring nonlocal correlations. Working with the one-dimensional extended Hubbard model at half-filling and attractive nearest-neighbor interaction, it shows that purely local, diagonal quantities — charge and spin fluctuations, occupation probabilities, and single-site entanglement entropy — all display an extremum at the spin-density-wave to triplet-superconducting (SDW-TS) transition at $V_C=-1.1t$ for $U=1.6t$. The extremum is unchanged when the system size is doubled from $L=128$ to $L=256$, which the paper takes as evidence that it survives in the thermodynamic limit. The transition is preceded by a continuous breaking of particle-hole symmetry, and approximate identities connect charge fluctuation to spin fluctuation and to entanglement entropy, making local measurements a stand-in for nonlocal order.

What carries the argument

The load-bearing object is the single-site reduced density matrix $\rho_j$, which is diagonal in the occupation basis because the Hamiltonian conserves particle number and spin projection. Its four occupation probabilities $w_0,w_\uparrow,w_\downarrow,w_2$ determine every local observable and yield explicit formulas for charge fluctuation $\langle\delta n\rangle=w_0+w_2-(w_0-w_2)^2$, spin fluctuation $\langle\delta s^z\rangle=w_\uparrow/2$, and entanglement entropy $S=-\sum_k w_k\log_2 w_k$. The mechanism is the continuous breaking of particle-hole symmetry ($w_0\leftrightarrow w_2$) that precedes the SDW-TS transition: although the symmetry-breaking term is too small to affect the fluctuations directly, its presence allows local fluctuations to be connected to nonlocal entanglement through the approximate identities above. These identities are what turn a purely diagonal, single-site quantity into a proxy for off-diagonal long-range order.

What would settle it

Recompute the single-site descriptors at $U=1.6t$ for $L=64,128,256,512$ and, in the same runs, directly evaluate the algebraic decay of the triplet superconducting pair correlation; if the extremum shifts or disappears with $L$, or if its position falls outside the $V$ range where the off-diagonal correlation shows the expected critical behavior, the central claim fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the single-site reduced density matrix of the one-dimensional extended Hubbard model at half-filling, although strictly diagonal in the occupation basis, carries enough information to detect the emergence of off-diagonal long-range order. For $U=1.6t$, the charge fluctuation $\langle\delta n\rangle$, spin fluctuation $\langle\delta s^z\rangle$, occupation probabilities $w_0,w_2,w_\uparrow=w_\downarrow$, and single-site entanglement entropy $S$ all exhibit an extremum at $V_C=-1.1t$, the SDW-TS boundary; the feature is unchanged when $L$ goes from 128 to 256. The paper further finds that this transition is preceded by a continuous breaking of particle-hole symmetry ($w_0\neq w_2$), yet the symmetry-breaking contribution $(w_0-w_2)^2$ remains negligible in the relevant regimes, allowing the identities $\langle\delta n\rangle=1-4\langle\delta s^z\rangle$ and $S\approx\tilde S(\langle\delta n\rangle)$ to hold. The paper argues that this is why local diagonal descriptors can serve as practical probes of superconducting order.

Load-bearing premise

The central result depends on the assumption that the extremum at $V_C=-1.1t$ is a genuine thermodynamic-limit feature of the ground state and not a finite-size artifact, since the support comes from comparing only two system sizes and from trusting a previously published phase diagram rather than computing off-diagonal correlations in the same calculation.

Editorial extensions

If this is right

The SDW-TS transition at $U=1.6t$ can be located from single-site diagonal data alone, so the extremum at $V_C=-1.1t$ constitutes a valid probe of off-diagonal long-range order.
The superconducting phase domain identified locally, $-1.55t<V\le -1.1t$, matches the range obtained earlier from the decay of long-range off-diagonal triplet correlations.
The identity $\langle\delta n\rangle=1-4\langle\delta s^z\rangle$ explains the observed inverse relation between charge and spin fluctuations wherever the particle-hole symmetry-breaking term is negligible.
The approximation $\tilde S$ expresses single-site entanglement entropy in terms of charge fluctuation, so a local measurement can estimate the entanglement between a site and the rest of the system.
Because the extremum persists from $L=128$ to $L=256$, the paper concludes that the signature is a genuine thermodynamic-limit feature rather than a finite-size artifact.

Reading between the lines

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

If the mechanism is generic, single-site snapshots from quantum-gas microscopes may be enough to map superconducting regions in other one-dimensional models without measuring two-point correlators.
The derived $S(\langle\delta n\rangle)$ relation suggests a cheap experimental proxy for single-site entanglement; testing it at other fillings or interaction strengths would show how far the approximation extends.
The paper leaves open whether other transitions into off-diagonal long-range ordered phases, such as bond-order or higher-dimensional superconductors, also leave a local diagonal fingerprint; applying the same descriptors there would be a direct test.

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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

The paper reports something genuinely new: several single-site diagonal quantities (charge/spin fluctuations, occupation probabilities, entanglement) show clean extrema at the SDW-TS transition in the 1D extended Hubbard model, where earlier work said diagonal probes would be blind. The algebraic identities (4)-(8) are correct and the numerical observation at V_C=-1.1t is plausible, even if the thermodynamic-limit evidence is thin (two system sizes, no truncation details).

The soft spot is load-bearing. From their own definitions, w0 - w2 = 1 - ⟨n_j⟩. At half-filling, the global density is 1, and at the central site (away from the pinning field) the local density should be extremely close to 1. So the reported w0 ≠ w2 cannot reflect a spontaneous breaking of particle-hole symmetry; it is a local-density deviation, almost certainly a finite-size or boundary effect. The paper builds its entire narrative around this 'continuous symmetry breaking' as the mechanism, so the mechanism collapses. The relations (7) and (8) are exact at half-filling, not approximate consequences of a small (w0-w2)^2 term, and they connect only local quantities. Calling this a 'direct connection between local fluctuations and nonlocal correlations' is an overreach: single-site entanglement is not off-diagonal long-range order. The paper also never computes an ODLRO order parameter in the same runs; the identification with the SDW-TS boundary rests entirely on the phase diagram of Ref. [36].

None of this necessarily kills the empirical observation. The extrema might be robust and practically useful as experimental markers. But as written, the title and abstract claim that local diagonal quantities 'capture off-diagonal long-range order,' which is not demonstrated. The authors need to either compute a genuine ODLRO correlation in their DMRG runs or substantially soften the claim, and they should replace the symmetry-breaking mechanism with a discussion of local-density effects.

Verdict: send to peer review. The observation is new and potentially useful, and the algebra is sound, but the manuscript needs major revision. The stress-test note should be shared with the authors; it gets the core flaw right.

Referee Report

4 major / 3 minor

Summary. The paper studies the one-dimensional extended Hubbard model at half-filling with on-site repulsion U=1.6t and attractive nearest-neighbor interaction V, using DMRG on chains of length L=128 and 256. From the single-site reduced density matrix at the central site, it computes charge and spin fluctuations, occupation probabilities, and single-site entanglement entropy. The authors report that all these diagonal quantities develop an extremum at V_C=-1.1t, which they identify with the SDW-TS superconducting transition known from the phase diagram of Ref. [36]. They further claim that this transition is preceded by a continuous breaking of particle-hole symmetry (w0≠w2) and that, because the symmetry-breaking correction is negligible, the single-site quantities satisfy universal relations connecting charge fluctuations, spin fluctuations, and entanglement. The central claim is that single-site diagonal descriptors can capture off-diagonal long-range order.

Significance. If established, the result would be significant because experimental probes at the single-site level are usually restricted to diagonal quantities, and the paper offers a concrete model where such quantities display critical signatures at a superconducting transition. The algebraic derivation of Eqs. (4)-(8) is straightforward and correct, and the numerical observation of extrema is plausible. However, the strongest interpretation — that single-site diagonal quantities genuinely detect ODLRO — is not directly verified, and the proposed symmetry-breaking mechanism conflicts with the exact half-filling constraint. The paper would be more convincing with direct computation of off-diagonal correlations in the same DMRG runs and a careful scaling analysis.

major comments (4)

[Fig. 2 and the thermodynamic-limit claim] At half-filling the identities w0+w↑+w↓+w2=1 and ⟨n_j⟩=w↑+w↓+2w2=1 imply w0-w2=1-⟨n_j⟩. Therefore the reported w0≠w2 in Fig. 2(b) is exactly a local density deviation, not a spontaneous breaking of particle-hole symmetry. With open boundary conditions and an edge pinning field, ⟨n_j⟩ at the central site is not constrained to be 1, so the observed split most likely reflects boundary/pinning effects. The paper never reports ⟨n_j⟩ or how (w0-w2) scales with L, so the claim that a continuous particle-hole symmetry breaking precedes the transition is unsupported. The authors should either show that (w0-w2) extrapolates to a nonzero value in the thermodynamic limit or remove the symmetry-breaking narrative from the central argument.
[Main text, SDW-TS identification] The conclusion that the extremum at V_C=-1.1t 'persists in the thermodynamic limit' is based on only two system sizes (L=128 and L=256). No DMRG truncation parameters (maximum bond dimension, discarded weight) or error bars are reported, and the two sizes may both lie in the same finite-size regime. To support the claim, the authors should provide convergence data and at least one additional size (e.g., L=512) or a scaling analysis of the extremum position and depth.
[Eqs. (7)-(8)] The paper identifies V_C=-1.1t with the SDW-TS transition solely by comparing with the phase diagram of Ref. [36]; no off-diagonal correlator (e.g., triplet pairing-pairing correlation, Luttinger parameter Kρ) is computed in the same DMRG runs. Without a direct verification, the title's claim that single-site diagonal quantities 'capture off-diagonal long-range order' overstates what is demonstrated. A direct computation of an off-diagonal correlation for the same parameters and system sizes would substantiate the identification.
[Eqs. (7)-(8)] When w0=w2 and w↑=w↓, which holds exactly at half-filling for a particle-hole symmetric state, Eq. (8) is an exact algebraic identity for the single-site entropy, not an approximation enabled by the smallness of (w0-w2)^2. The 'connection between local fluctuations and nonlocal correlations' is therefore a local identity that does not, by itself, link local quantities to off-diagonal correlations. The authors should either provide a nontrivial derivation or temper the interpretation in the abstract and conclusion.

minor comments (3)

[Fig. 2 caption] The caption of Fig. 2 is difficult to parse (the text 'L=256 L=256 (b) (a) (c) SDW TS PS L 128 256' is garbled); please specify clearly which system sizes appear in each panel and what the vertical axes represent.
[Methods, DMRG] The pinning field applied at the edge sites is not defined. The authors should state its explicit form, strength, and how it is removed in the analysis (if at all).
[Text after Eq. (5)] In the paragraph discussing charge fluctuations, the reference to 'Eq. (5)' should be 'Eq. (4)', since the charge-fluctuation formula is Eq. (4).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims rest on new DMRG data benchmarked against an external phase diagram, and the derived relations are algebraic identities.

full rationale

The paper's central claim is empirical: DMRG calculations of single-site diagonal quantities (charge and spin fluctuations, occupation probabilities, entanglement entropy) show an extremum at V_C = -1.1t, which is identified with the SDW-TS transition using the externally established phase diagram of Ref. [36]. No parameter is fitted to the target result; the phase boundary is an external benchmark, and the comparison is a posteriori labeling rather than a derivation. The formal relations in Eqs. (7) and (8) are derived algebraically from the definitions of the reduced density matrix elements in Eq. (3) under the approximation (w0 - w2)^2 ≈ 0; this is a mathematical identity, not an input. The self-citations (e.g., Refs. [27, 32, 54-58]) support the generic fact that single-site entanglement entropy can signal phase transitions; they are not load-bearing for the specific ODLRO claim, which rests on new DMRG data and the external phase diagram of Ref. [36]. The skeptic's observation that w0 - w2 = 1 - <n_j> follows from normalization and half-filling is a mathematical identity; if w0 ≠ w2 in the numerics, then the local density at the central site deviates from 1, which would be a boundary or pinning artifact rather than spontaneous symmetry breaking. This is a correctness risk for the proposed symmetry-breaking mechanism, but it does not make the derivation circular: the extremum at V_C is not defined in terms of w0 - w2, and the claim that diagonal descriptors show critical features is not forced by any fitted parameter or self-citation. The paper is self-contained against an external benchmark, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on a previously published phase diagram, DMRG convergence, and a finite-size comparison with only two system sizes. No free parameters are fitted; the derived relations are algebraic consequences of the reduced density matrix form.

assumptions (3)

domain assumption The ground-state phase diagram of the 1D EHM from Ref. [36] is correct, including the location of the SDW-TS transition at U=1.6t around V≈-1.1t and the TS-PS transition at V≈-1.55t.
The paper identifies the extrema in local descriptors with these transitions by comparison with Ref. [36], not by computing off-diagonal correlations itself.
domain assumption DMRG calculations are converged to the ground state with sufficient accuracy for central-site observables; the authors report energy convergence to 1e-7 but give no bond dimension or truncation error.
The numeric results depend on DMRG convergence, which is stated but not fully documented.
ad hoc to paper The invariance of the extremum position from L=128 to L=256 implies thermodynamic-limit behavior.
Only two system sizes are compared, which is not a standard finite-size scaling analysis.

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Cite this review

Pith. "Pith review of Single-site diagonal quantities capture off-diagonal long-range order." pith.science (2026). https://pith.science/paper/KR7MPHDZ

@misc{pith2026250710328,
  author       = {Pith},
  title        = {Pith review of: Single-site diagonal quantities capture off-diagonal long-range order},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KR7MPHDZ}},
  note         = {Machine review of arXiv:2507.10328}
}

read the original abstract

Quantum phase transitions are typically marked by changes in quantum correlations across various spatial scales within the system. A key challenge lies in the fact that experimental probes are generally restricted to diagonal quantities at the single-site scale, which are widely believed to be insufficient for detecting phases with off-diagonal long-range order, such as superconducting states. In a striking departure from conventional expectations, we show that single-site diagonal descriptors -- charge and spin fluctuations, occupation probabilities, and entanglement -- can capture the emergence of off-diagonal long-range order in the one-dimensional extended Hubbard model at half-filling. These single-site quantities display clear critical signatures of the superconducting transition, preceded by a continuous breaking of particle-hole symmetry, consistent with a second-order phase transition. While this symmetry breaking has a negligible effect on single-site descriptors, it allows a direct connection between local fluctuations and nonlocal correlations.

Figures

Figures reproduced from arXiv: 2507.10328 by the authors.

**Figure 2.** (Color online) Diagonal descriptors at the cen [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [36]

Qu, B.-B

D.-W. Qu, B.-B. Chen, H.-C. Jiang, Y. Wang, and W. Li, Communications Physics 5, 257 (2022)

work page 2022

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