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Quantum Spin Glass in the Two-Dimensional Disordered Heisenberg Model via Foundation Neural-Network Quantum States

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Pith's one-line read The two-dimensional disordered Heisenberg model hosts a quantum spin-glass phase for intermediate bond disorder.

desk verdict Plausible first large-scale evidence for a 2D quantum spin glass, but the central overlap value is small and lacks an independent check in the regime where it matters. read the letter →

arxiv 2507.05073 v3 pith:DLYOR4VQ submitted 2025-07-07 cond-mat.dis-nn cond-mat.str-elquant-ph

Luciano Loris Viteritti , Riccardo Rende , Giacomo Bracci-Testasecca , Jacopo Niedda , Roderich Moessner , Giuseppe Carleo , Antonello Scardicchio This is my paper · ORCID
classification cond-mat.dis-nncond-mat.str-elquant-ph PACS 75.10.Nr75.10.Jm75.50.Lk
keywords quantumspinglassdisorderedHeisenbergmodelneural-networkstatesEdwards-AndersonorderparameterbonddisorderfrustratedmagnetismvariationalMonteCarloHolstein-Primakoffexpansion
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 tries to establish that the ground state of the square-lattice spin-1/2 Heisenberg model with random ferromagnetic and antiferromagnetic nearest-neighbor bonds is a quantum spin glass for a broad intermediate range of disorder. Using a neural-network variational wave function optimized once over hundreds of disorder realizations, the authors find that both ferromagnetic and Néel order extrapolate to zero in the thermodynamic limit for 0.2 ≲ p ≲ 0.8, while the Edwards-Anderson self-overlap Q stays finite, with Q = 0.049(3) at p = 0.5. A semiclassical 1/S spin-wave expansion around classical spin-glass minima gives a compatible value at S = 1/2, indicating that quantum fluctuations do not destroy the glassy order. If correct, this would resolve a longstanding question in favor of a stable quantum spin-glass phase in two dimensions.

What carries the argument

The engine of the argument is the Edwards-Anderson self-overlap $Q^{2}$ = E_J[(1/$L^{4}$) ∑_{r,r'} ⟨S_r · S_{r'}⟩^2], which is finite in a glass, zero for a spin liquid, and equals the square of the magnetization for ordered magnets. To obtain reliable disorder-averaged values, the authors use a Foundation Neural-Network Quantum State, a transformer-based variational ansatz whose inputs include the bond couplings J on equal footing with the spin configurations, so a single optimization minimizes the disorder-averaged energy over roughly 600 realizations at once. The supporting semiclassical machinery is a Holstein-Primakoff expansion around the classical ground state: at leading order in 1/S the spin-wave Hamiltonian is quadratic and must be diagonalized by a Bogoliubov transformation, and from it the quantum correction to the overlap is computed and evaluated at S = 1/2.

What would settle it

Perform an exact or non-variational ground-state computation on comparable disorder ensembles at p = 0.5 for L = 8 and L = 10, then check whether the 1/L-extrapolated Edwards-Anderson overlap lands at approximately 0.049(3) or goes to zero; an extrapolation consistent with zero would falsify the quantum spin-glass claim.

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Extended reading notes

Core claim

The central claim is a phase diagram with three ground states as the probability p of antiferromagnetic bonds is tuned: ferromagnetic order for p ≲ 0.2, Néel order for p ≳ 0.8, and, in between, a quantum spin glass characterized by zero spontaneous magnetization but a finite Edwards-Anderson overlap Q. The overlap is measured through the squared spin-spin correlations averaged over disorder, $Q^{2}$ = E_J[(1/$L^{4}$) ∑_{r,r'} ⟨S_r · S_{r'}⟩^2_J]. At p = 0.5, finite-size extrapolations over L = 4 to 14 and 600 disorder realizations give Q = 0.049(3), and the same analysis across 0.2 ≲ p ≲ 0.8 leaves Q finite. The numerical result is consistent with the leading 1/S spin-wave correction to the classical spin-glass overlap, establishing agreement between the fully quantum variational calculation and the semiclassical expansion. The finite Q in this region rules out quantum spin liquids and valence-bond solids, both of which would have vanishing overlap.

Load-bearing premise

The whole spin-glass conclusion rests on the assumption that a single variational wave function, optimized to minimize the disorder-averaged energy, yields unbiased estimates of the small overlap Q in the region 0.2 < p < 0.8, where the method is not checked against any non-variational calculation.

Editorial extensions

If this is right

  • The ground state of the two-dimensional disordered Heisenberg model is not paramagnetic, spin-liquid, or valence-bond ordered in the intermediate region; it is a distinct quantum spin-glass phase with no spontaneous magnetic moment.
  • Spin-glass order survives quantum fluctuations down to S = 1/2, in contrast to the classical two-dimensional Heisenberg spin glass, where order exists only at T = 0.
  • The transition from ferromagnetic to spin-glass order occurs near p ≈ 0.2 and from spin-glass to Néel order near p ≈ 0.8, with the paper leaving open the possibility of a narrow spin-liquid window near the Néel boundary.
  • Disorder-averaged ground-state observables can be computed with a single variational optimization across many disorder realizations, with only modest loss in accuracy compared to training on each realization separately.
  • The overlap order parameter provides a clean numerical diagnostic for distinguishing glassy from liquid-like nonmagnetic phases in frustrated disordered magnets.

Reading between the lines

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

  • Editorial extension: if the finite-Q region is a true glass, the model should exhibit slow, non-thermalizing dynamics and aging in real-time evolution; the authors do not test this, but it is a direct consequence of glass order.
  • Editorial extension: the semiclassical agreement suggests the 1/S spin-wave treatment can predict other properties, such as dynamical structure factors, for the same S = 1/2 disordered ensemble, not just the static overlap.
  • Editorial extension: because the Foundation Neural-Network Quantum State takes the bond couplings as input, the same trained architecture could be evaluated at disorder strengths p that were not used in training, providing a fast transferability test and a way to refine the phase boundaries.
  • Editorial extension: in doped cuprate models where oxygen vacancies create effective ferromagnetic bonds, a finite-Q region would predict spin-glass-like freezing of copper moments at low temperature, a signature that could be searched for in experiment.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies the S=1/2 square-lattice Heisenberg model with random nearest-neighbor ±1 bonds, Eq. (1)-(2). Using a Foundation Neural-Network Quantum States (FNQS) ansatz optimized simultaneously over many disorder realizations, the authors compute disorder-averaged ferromagnetic, Néel, and Edwards-Anderson overlap order parameters for system sizes L=4 through L=14. They report a phase diagram in which both ferromagnetic and Néel order vanish in the thermodynamic limit for 0.2≲p≲0.8, while the overlap Q remains finite, with the central value Q=0.049(3) at p=0.5, and they interpret this as a quantum spin-glass phase. Independent support is provided by a semiclassical large-spin expansion around classical ground states, which yields a compatible value of the overlap at S=1/2. The numerical method is benchmarked against exact diagonalization for one 6×6 realization at p=0.7 and against quantum Monte Carlo in the clean antiferromagnetic limit p=1.0. The supplementary material additionally shows that a spin-liquid state would give a vanishing overlap in the thermodynamic limit.

Significance. If the central claim holds, this would be the first large-scale numerical evidence for a stable quantum spin-glass phase in the two-dimensional disordered Heisenberg model, settling a question that previous exact-diagonalization studies could only address on 4×4 clusters. The paper also demonstrates a potentially powerful methodological advance: a single FNQS optimization that averages over hundreds of disorder realizations. Credit is due for the clean-limit QMC benchmark, the ED comparison at 6×6, the explicit self-averaging analysis, the spin-liquid control calculation in the SI, and the fact that the semiclassical computation derives its coefficients from the Hamiltonian rather than fitting to the FNQS result. The main weakness is that the central order parameter Q is extrapolated from a variational method in a regime where Q is small, and the only unbiased validation in the disordered regime does not actually test Q.

major comments (3)
  1. [Methods V A and Numerical Results III A] The central claim—a finite Q in 0.2≲p≲0.8—rests entirely on FNQS estimates. Equation (3) minimizes the disorder-averaged energy, so the variational principle bounds E, not Q, and Q is a squared-correlation observable with no variational bound. The only unbiased check inside the disordered region is a single 6×6 realization at p=0.7 (Fig. 5), and it compares spin-spin correlations and the structure factor, not Q. Since the extrapolated value at p=0.5 is Q=0.049(3), a variational bias of 0.01–0.02 in Q—smaller than the energy-resolution benchmarks quoted in §V A—would change the conclusion. I request a direct benchmark of Q and its disorder distribution against exact diagonalization over many realizations at L=6 for, e.g., p=0.5 and p=0.7, and, if feasible, an independent non-variational check (e.g., DMRG/PEPS on cylinders) in the claimed spin-glass window.
  2. [Numerical Results III A, Eq. (4), Fig. 3] The finite-size data for Q are not corrected for the known diagonal self-correlation: for r=r′, ⟨S_r·S_r⟩²=9/16, which contributes 3/(4L) to Q. At L=14 this contribution is 0.054, comparable to the extrapolated value 0.049(3), and it is larger at smaller L. The fits in Fig. 3 are performed directly on Q without subtracting or explicitly modeling this term. Please repeat the extrapolation with the diagonal term removed from Q² or with a fixed 9/(16L²) term in the fit function, and show that the positive intercept survives; otherwise the finite-Q conclusion may be controlled by the fitting form absorbing the self-correlation term.
  3. [Semiclassical analysis, §III B and SI §§III-V] The semiclassical corroboration is less quantitative than the main text suggests. Equation (28) evaluates a 1/S expansion at S=1/2, a regime where the expansion parameter is not small, and the SI explicitly states that a magnon-scattering contribution of the same order as the retained (c⁽¹⁾)² term is omitted. The agreement in Fig. 4 is therefore suggestive but not an independent validation of the FNQS value of Q. The manuscript should state this limitation in the main text and soften the 'great agreement' wording in the Fig. 2 caption.
minor comments (6)
  1. [Methods V B] The text cites Quantum Monte Carlo data as [52], while Fig. 6 and the main text cite [51]; please make the reference consistent.
  2. [SI Fig. 1] The caption uses J1/J2=0.5 while the text and Eq. (1) define the frustrated point as J2/J1=0.5; please make the notation consistent.
  3. [SI Eq. (23)] In the expression for c⁽¹⁾_{ij}, the second correlator should involve site j rather than site i, in agreement with Eq. (21a); as written the expression is inconsistent with the definition of a two-point correction.
  4. [Fig. 4] The axis label 'Spin Wave OverlapFNQS Q=0.049(3)' is missing a space, and the definition of the plotted quantity Q_SW/(4S²) should be given in the caption rather than only in the text.
  5. [Data Availability] The statement 'Hugging Face NQS models' would be more useful with a direct URL or repository identifier.
  6. [Discussion and Fig. 1] The Discussion raises the possibility of a spin-liquid sliver between the antiferromagnetic and spin-glass phases near p≈0.8, but Fig. 1 draws a single boundary at p_AFM≈0.8; the phase diagram should indicate the uncertainty in this boundary.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the FNQS estimate of Q is an energy-optimized variational observable, the spin-wave prediction is derived from the Hamiltonian without tuning to Q, and the cited FNQS framework is independently benchmarked against ED and QMC.

full rationale

No load-bearing circular step could be exhibited. The central claim is that Q remains finite in 0.2≲p≲0.8; Q is computed as an expectation value on the variational state optimized for the disorder-averaged energy (Eq. 3), not fitted to the target observable. The finite-size extrapolations in Fig. 3 and Methods VB are standard scaling analyses of an independently computed observable. The FNQS method is cited to the authors' Ref. [47], but the present paper contains external checks: exact Lanczos data on a 6x6 realization at p=0.7 and QMC benchmarks at p=1.0 (Methods VA and VB), so the method does not rest solely on the self-citation. The semiclassical calculation is also not circular: c1 and c2 entering Eq. (25) are computed from the spin-wave Hamiltonian (SI Eqs. 8-10, 26) directly from the disordered Heisenberg couplings, and the FNQS value Q=0.049(3) is compared with, not imposed on, the spin-wave result. A legitimate concern is that the variational wave function has no variational bound on Q and that no unbiased check of Q is performed inside the claimed spin-glass region, where Q is small; however, that is a validation or correctness risk, not a reduction of the prediction to its input by construction. Under the hard rules, circularity requires quoting a specific equation or fitted parameter that makes the output equal to the input, and no such reduction exists here.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free physical parameters are introduced; the model has fixed coupling distribution and no fitted constants. The central claim rests on four domain assumptions: the classical SG ground state as a starting point, the truncated spin-wave expansion, the unbiasedness of the FNQS ansatz, and the validity of one of the finite-size scaling forms. None of these is an ad hoc invented entity.

assumptions (4)
  • domain assumption The classical ground state of the binary-bond Heisenberg model in 2D has spin-glass order at T=0
    The semiclassical expansion expands around a classical spin-glass minimum; the paper relies on the established result that classical Heisenberg SG order exists only at T=0 in 2D (Refs. 48,49), but does not prove the ground state is a spin glass for all p.
  • domain assumption A Holstein-Primakoff expansion truncated at quadratic order, with the 1/S^2 term taken as the square of the leading correlation, is sufficient to estimate Q at S=1/2
    The SI explicitly states that a magnon-scattering term also contributes at 1/S^2 and is omitted; retaining only the square of the linearized correlation is a modeling choice, not a systematic expansion.
  • domain assumption The FNQS variational ansatz accurately represents the true ground state for all disorder realizations and system sizes up to L=14
    The method is validated only at 6x6 against ED for one realization and at p=1.0 against QMC; in the claimed spin-glass regime there is no independent non-variational check. This is the load-bearing assumption of the numerical result.
  • domain assumption One of the finite-size scaling forms used (quadratic, power-law, or linear on the last three points) correctly describes the L-dependence of Q
    The thermodynamic-limit value depends on the assumed scaling form. The quoted Q=0.049(3) is an estimate from these fits, and the systematic error from the choice of form is not included in the quoted error.

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Cite this review

Pith. "Pith review of Quantum Spin Glass in the Two-Dimensional Disordered Heisenberg Model via Foundation Neural-Network Quantum States." pith.science (2026). https://pith.science/paper/DLYOR4VQ

@misc{pith2026250705073,
  author       = {Pith},
  title        = {Pith review of: Quantum Spin Glass in the Two-Dimensional Disordered Heisenberg Model via Foundation Neural-Network Quantum States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DLYOR4VQ}},
  note         = {Machine review of arXiv:2507.05073}
}
read the original abstract

We investigate the two-dimensional frustrated quantum Heisenberg model with bond disorder on nearest-neighbor couplings using the recently introduced Foundation Neural-Network Quantum States framework, which enables accurate and efficient computation of disorder-averaged observables with a single variational optimization. Simulations on large lattices reveal an extended region of the phase diagram where conventional magnetic long-range order vanishes in the thermodynamic limit, while the Edwards-Anderson order parameter remains finite, signaling the emergence of a quantum spin-glass phase. These findings, supported by a semiclassical analysis based on a large-spin expansion, provide compelling evidence that the spin glass-order is stable against quantum fluctuations, unlike the classical case where it disappears at any finite temperature.

Figures

Figures reproduced from arXiv: 2507.05073 by the authors.

Figure 1
Figure 1. FIG. 1. Ground-state phase diagram of the disordered [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Finite-size scaling of the overlap order parameter [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Spin-spin correlations for a disorder realization [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Finite-size extrapolations of the order parameters as a function of 1 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Distributions of the order parameters [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 1
Figure 1. Figure 1: FIG. 1. Finite-size scaling of the overlap order parameter [PITH_FULL_IMAGE:figures/full_fig_p012_1.png]
Figure 2
Figure 2. Figure 2: FIG. 2. Finite size extrapolations of the corrections to the overlap, for [PITH_FULL_IMAGE:figures/full_fig_p018_2.png]

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Forward citations

Cited by 2 Pith papers

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