Unraveling square and kagome lattice frustrated magnetism by scalable tensor networks

Jutho Haegeman; Xing-yu Zhang; Yuchi He

REVIEW

Unraveling square and kagome lattice frustrated magnetism by scalable tensor networks

Xing-Yu Zhang, Jutho Haegeman, Yuchi He

Strongly Correlated Electrons · Submitted 6 Nov 2026 · 77 references

Pith's verdict: A strong numerical methods paper with an excellent QMC benchmark, but the kagome spin-liquid stability claim rests on a scaling proxy the authors acknowledge but do not independently test.

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arXiv ID2611.09881 (version 1)

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CitationJutho Haegeman, Xing-yu Zhang, Yuchi He. "Unraveling square and kagome lattice frustrated magnetism by scalable tensor networks." arXiv:2611.09881 (2026). https://pith.science/paper/XSNKLEIE

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Editor's note

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Reading this one, the thing to know is that the numerical work is strong and the kagome spin-liquid claim is more conditional than the abstract suggests. The J2=0 benchmark against QMC is excellent: energy to eight digits, magnetization 0.30723(12) versus 0.307447(2), with data and scripts posted. The parallelization (1DSlice/2DSlice) and the randomized loop contraction for the J2 coupling look like real contributions, and the direct scaling-dimension estimate for the square lattice (2Δm = 1.020(16)) is a useful data point for the intermediate-regime debate.

Where I'd push back is the kagome stability conclusion. The claim that 2Δm = 3.95(11) > 3 makes the spin coupling irrelevant depends on identifying the fitted slope of log m² vs log(1/ξ) with the infrared scaling dimension of the spin operator. The authors explicitly call this a proxy, and the stress-test concern is right that the alternative reading — a finite-D variational crossover toward 120° order — is not excluded. The fit itself is four points (D=10,12,14,16), the error bar is purely least-squares, and the supporting 2×2 VBS susceptibility is three points with Δb = 1.75(25). The statistical separation from the threshold 3 is large, but the systematic risk sits in the functional form, not the noise.

Credit where due: the assumptions are stated, not hidden, and the methods section is unusually detailed. The randomized contraction is benchmarked against direct contraction at D=8, and the data and scripts are on GitHub. The physical conclusions are consistent with prior proposals in the literature; the genuinely new content is the scalable implementation and the direct finite-correlation-length estimates.

Who this is for: tensor-network practitioners and the frustrated-magnetism crowd, especially anyone tracking the kagome spin-liquid question. It deserves a serious referee — the methods and benchmark justify it. The referee should press on the scaling-proxy identification and ask for an independent cross-check, or for language that matches the evidence. My recommendation: send it to peer review.

Referee report

Significance

If the scaling-dimension inference is accepted, the kagome result would be a substantial step toward settling the ground-state nature of the kagome Heisenberg model, and the square-lattice phase-diagram results would sharpen the location of the VBS transition. The technical contributions are strong and well supported: the J2=0 benchmark against high-precision QMC (Fig. 5) is excellent; the randomized loop contraction is benchmarked against direct contraction at D=8 and against an oversampling extrapolation at D=16; and the parallel scaling benchmarks are nearly ideal. The paper also states that code and data are publicly available, which is a concrete reproducibility strength. However, the spin-liquid conclusion is conditional on a scaling proxy that the authors explicitly label as a proxy and do not independently validate, so the significance of the central claim is proportionate to that assumption rather than to the numerical quality of the underlying calculations.

Major objections (4)

[Eq. (2) and the paragraph following Fig. 4] The load-bearing step is the sentence 'Using 2Δm as a proxy for the infrared scaling dimension of the spin-spin coupling term.' The fitted quantity m is the average magnitude of the local spin expectation value in a 1x1 iPEPS, which at finite D can spontaneously break the 120-degree spin symmetry; it is not a two-point spin correlation function. For the ordered square Heisenberg model the paper uses the ordered form m^2 = m∞^2 + a/ξ (Fig. 5b), whereas Eq. (2) assumes the spin-liquid form m^2 ∝ ξ^{-2Δm}. The authors state the assumption with 'Assuming gapless spin liquids', but they do not test the proxy against any independent quantity. A concrete test would be to extract Δm from the finite-ξ decay of the spin-spin correlation function, or to fit m^2 = m∞^2 + a ξ^{-2Δm} and require m∞^2 to be consistent with zero. Without such a test, the value 2Δm=3.95(11) is a least-squares slope in a four-point window, not an established infrared scaling dimension.
[Fig. 4 and Methods, kagome data set] The kagome fit in Fig. 4 uses only D=10,12,14,16 with one χ per D; the Methods state that for the kagome data 'Only one χ is used per D in Fig. 3.' Four points across a narrow ξ range cannot constrain corrections to Eq. (2), so the reported error 0.11 is purely statistical and does not include the uncertainty from the assumed leading-power form. The authors should either show χ-varied points at fixed D, extend to larger D, or include a subleading term in the fit. As it stands, the stability conclusion rests on a single power law being exact over the available ξ window.
[Supplementary Note 3 and Supplementary Fig. 3] The kagome VBS-susceptibility fit uses only three points (D=8,9,10) and gives Δb=1.75(25), whose lower 1σ edge is essentially at the stability threshold Δb=1.5. The main text's claim of oscillatory valence-bond susceptibilities coexisting with Néel susceptibilities relies on this fit. The authors acknowledge that data at larger D would sharpen the comparison, but this is a load-bearing caveat: a three-point fit with a free exponent cannot distinguish a susceptibility from a small incipient ordered moment. The consistency of Δb with Δm=1.97(6) noted in the Supplementary Note is useful, but it uses the same finite-correlation-length scaling assumption and therefore is not an independent validation.
[Field-theory discussion after Eq. (2)] A stable U(1) Dirac spin liquid also requires the 4π-flux monopole operator to be irrelevant, yet Eq. (2) measures only the spin-operator sector. The manuscript infers from 2Δm close to 4 that monopole effects are subdominant or have a small prefactor, while the two estimates it cites bracket the threshold (2ΔM≈2.5 from large-Nf and 2ΔM≈3.7(3) from noncompact QED3 QMC). Given that the abstract states 'our data indicate a stable gapless spin liquid on the kagome lattice', the authors should either restrict the claim more explicitly to stability against the spin and VBS operators, or provide evidence bearing directly on the monopole sector.

Minor objections (4)

[Abstract and conclusion] The abstract says the data 'indicate a stable gapless spin liquid', while the conclusion says 'possibly stable gapless spin liquid'; these should be aligned so that the abstract does not overstate the hedged conclusion.
[Fig. 4 caption] The caption calls the fitted slopes α and describes them as 'an estimate for 2Δm'; the main text should define α=2Δm explicitly and use one symbol consistently.
[Fig. 1 caption] The Fig. 1 caption is cluttered and mixes the panel descriptions of several panels into one paragraph; splitting the caption into individual panel-specific entries would improve readability.
[Methods, one-shot extrapolation] The distinction between optimizing at χ and evaluating at a larger χ is explained carefully in Supplementary Note 4, but the main text would benefit from a one-sentence summary of why the one-shot scheme is used instead of the two-step evaluation scheme.

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The paper

The paper claims

Tensor-network fits give 2Δm = 3.95(11) for the kagome Heisenberg model — above the 3 threshold, so the gapless spin liquid is stable — and 2Δm = 1.020(16) for the square lattice, below it.

The reading

This paper claims that the correlation length imposed on infinite tensor-network states by finite bond dimensions can stand in for system size, and that fitting the decay of the squared local magnetization against that length yields the scaling dimension of the spin operator, a quantitative probe of whether a gapless spin liquid is stable. The fitted exponents are $2\Delta m = 3.95(11)$ for the kagome Heisenberg model and $2\Delta m = 1.020(16)$ for the square-lattice $J_1$-$J_2$ model at $J_2/J_1 = 0.5$. Since the space-time dimension $d+1 = 3$ separates an irrelevant from a relevant spin-spin coupling, the kagome value indicates a stable gapless spin liquid, while the square-lattice value says the coupling is relevant and leaves competing scenarios open for the intermediate regime. The claim matters because both models are long-standing testbeds where the existence and stability of spin liquids have resisted exact methods; the paper reaches $10^{-8}$ energy precision in the unfrustrated benchmark, identifies isotropic plaquette order in the VBS region, and attributes the new reach to parallelization and randomized low-rank contraction techniques.

What carries the argument

The load-bearing mechanism is finite-correlation-length scaling applied to infinite projected entangled-pair states (iPEPS). At finite bond dimension $D$ and environment bond dimension $\chi$, the transfer matrix of the infinite tensor network has a finite leading-eigenvalue gap, so correlations decay over a length $\xi(D,\chi)$ that acts as a soft infrared cutoff; for gapless phases this cutoff plays the role that system size plays in finite-size scaling. The paper uses two scaling laws, $E(D,\chi) = E_\infty + e'/\xi^3$ for the energy density and $m^2(D,\chi) \propto \xi^{-2\Delta m}$ for the squared staggered magnetization, and the stability test is $2\Delta m > 3$ (coupling irrelevant) versus $2\Delta m < 3$ (relevant). The numerical machinery carrying this out is a $\pi$-mirror-symmetric $2\times2$ ansatz built from a single independent tensor, VUMPS environment determination, a randomized QB loop contraction that lowers the next-nearest-neighbor coupling cost from $O(D^6\chi^3)$ to $O(D^4\chi^3)$, and a two-level parallelization whose strong scaling is near-ideal, reaching bond dimensions $D = 8$–$16$ with environments up to $\chi = 2048$.

Extended reading notes

Core claim

The paper's central quantitative claim is that finite-correlation-length scaling of the squared local magnetization, $m^2(D,\chi) \propto [\xi(D,\chi)]^{-2\Delta m}$, provides a proxy for the infrared scaling dimension of the spin-spin coupling, and that the fitted exponents split the two models: $2\Delta m = 3.95(11)$ for kagome, $2\Delta m = 1.020(16)$ for the square lattice at $J_2/J_1 = 0.5$. A coupling whose scaling dimension exceeds the space-time dimension $d+1 = 3$ is irrelevant, so the kagome fit supports a stable gapless U(1) Dirac spin liquid, whereas the square-lattice fit makes the spin-spin coupling relevant and the gapless state infrared-unstable; the authors state their conclusion as a 'possibly stable' gapless spin liquid on kagome and 'competing scenarios' for the square-lattice intermediate regime. In support, the paper reports oscillatory valence-bond susceptibilities coexisting with antiferromagnetic ones on kagome, isotropic plaquette order in the square-lattice VBS region for $J_2/J_1 \ge 0.52$, and an energy extrapolation linear in $1/\xi^3$ at $J_2/J_1 = 0.5$, the signature of linear dispersion, with the unfrustrated benchmark agreeing with quantum Monte Carlo to eight significant digits.

Load-bearing premise

The load-bearing premise is that the slope of $\log m^2$ versus $\log(1/\xi)$ is twice the infrared scaling dimension of the spin operator, so that crossing the space-time dimension 3 decides whether the spin-spin coupling is irrelevant and the gapless phase stable; the authors themselves label this identification a proxy. If that proxy or the finite-correlation-length scaling form fails, the kagome stability conclusion does not follow, and the kagome valence-bond susceptibility additionally rests on only three bond dimensions, $D = 8, 9, 10$.

Editorial extensions

If this is right

If $2\Delta m = 3.95(11)$ is correct, the kagome Heisenberg ground state is a gapless U(1) spin liquid whose spin correlations decay almost as fast as free Dirac-fermion bilinears ($2\Delta m$ near 4), with the $4\pi$-flux monopole either irrelevant or suppressed by a small prefactor.
The square-lattice $J_1$-$J_2$ model at $J_2/J_1 = 0.5$ is not stable under the same criterion ($2\Delta m = 1.020(16) < 3$), so the observed linear dispersion is compatible with proximity to the Néel–spin-liquid critical point or a direct transition, which is why the intermediate regime stays open.
The valence-bond-solid region of the square-lattice model is isotropic plaquette order for $J_2/J_1 \ge 0.52$, narrowing the symmetric region on the VBS side relative to structure-factor-based estimates.
The randomized loop contraction reveals an approximate low-rank structure in iPEPS correlation contractions that the paper suggests may be general, making longer-range couplings affordable at essentially the same cost as nearest-neighbor ones.
Because temperature and measurement resolution act as infrared cutoffs in experiments, iPEPS data at matched cutoff values can guide spin-liquid identification in scattering and transport measurements.

Reading between the lines

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

Editorial extension: sweeping the same $2\Delta m$ versus 3 test across $J_2/J_1$ in the square-lattice model would turn the proxy into a phase-diagram diagnostic — a crossing above 3 would pinpoint a stable gapless regime, while its absence would favor a direct (deconfined) transition.
Editorial extension: the kagome exponent's proximity to 4 suggests the infrared spin response is almost free-fermion-like; if so, neutron-scattering continua in kagome candidate materials should show the same algebraic decay, giving an experimental target for the $\xi^{-2\Delta m}$ scaling seen numerically.
Editorial extension: the valence-bond susceptibility margin is thin ($\Delta_b = 1.75(25)$ against the $\Delta_b > 1.5$ criterion, within one error bar), and the paper itself calls for larger $D$; the next bond-dimension increment may decide whether the oscillatory valence-bond correlations are a true coexisting susceptibility or a precursor of weak order.

Original abstract

We extend the scalability of variational infinite two-dimensional tensor network methods with application to investigate frustrated magnetism of the square-lattice $J_1$-$J_2$ model and kagome-lattice Heisenberg model. In the sign-problem-free limit of the square-lattice model, we achieve high ground-state energy precision comparable to quantum Monte Carlo methods. With no obstruction beyond this limit, our method obtains direct estimates of order parameters. For the square-lattice $J_1$-$J_2$ model, we determine isotropic plaquette order in the valence-bond solid region. For the model near $J_2/J_1=0.5$, and the kagome lattice Heisenberg model, our data indicate linear-dispersion gapless states. To probe the candidate spin-liquid nature of the highly frustrated cases, we go beyond inferring the absence of symmetry-breaking orders and analyze susceptibilities that characterize gapless spin liquids. By estimating scaling dimensions, our data indicate a stable gapless spin liquid on the kagome lattice, and competing scenarios for the intermediate regime of the square-lattice model. Our developments include parallelization and randomized low-rank methods and largely expand the frontier of high-precision variational numerics for quantum phases of matter.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-09: 1 request, 6,362 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.69 · human participants 0.01 · controlled comparison 0.15 · randomized assignment 0.02 · states quantitative result 0.31 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We extend the scalability of variational infinite two-dimensional tensor network methods with application to investigate frustrated magnetism of the square-lattice $J_1$-$J_2$ model and kagome-lattice Heisenberg model. In the sign-problem-free limit of the square-lattice model, we achieve high ground-state energy precision comparable to quantum Monte Carlo methods. With no obstruction beyond this limit, our method obtains direct estimates of order parameters. For the square-lattice $J_1$-$J_2$ model, we determine isotropic plaquette order in the valence-bond solid region. For the model near $J_2/J_1=0.5$, and the kagome lattice Heisenberg model, our data indicate linear-dispersion gapless states. To probe the candidate spin-liquid nature of the highly frustrated cases, we go beyond inferring the absence of symmetry-breaking orders and analyze susceptibilities that characterize gapless spin liquids. By estimating scaling dimensions, our data indicate a stable gapless spin liquid on the kagome lattice, and competing scenarios for the intermediate regime of the square-lattice model. Our developments include parallelization and randomized low-rank methods and largely expand the frontier of high-precision variational numerics for quantum phases of matter.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We extend the scalability of variational infinite two-dimensional tensor network methods with application to investigate frustrated magnetism of the square-lattice $J_1$-$J_2$ model and kagome-lattice Heisenberg model.	contribution 0.97	0.93	0.09	asserted 0.99
s1	In the sign-problem-free limit of the square-lattice model, we achieve high ground-state energy precision comparable to quantum Monte Carlo methods.	result 0.96	0.95	0.09	asserted 1.00
s2	With no obstruction beyond this limit, our method obtains direct estimates of order parameters.	method 0.39	0.85	0.06	asserted 0.99
s3	For the square-lattice $J_1$-$J_2$ model, we determine isotropic plaquette order in the valence-bond solid region.	result 1.00	0.96	0.10	asserted 1.00
s4	For the model near $J_2/J_1=0.5$, and the kagome lattice Heisenberg model, our data indicate linear-dispersion gapless states.	result 1.00	0.94	0.18	hedged possibility 0.99
s5	To probe the candidate spin-liquid nature of the highly frustrated cases, we go beyond inferring the absence of symmetry-breaking orders and analyze susceptibilities that characterize gapless spin liquids.	contribution 0.58	0.94	0.08	asserted 0.77
s6	By estimating scaling dimensions, our data indicate a stable gapless spin liquid on the kagome lattice, and competing scenarios for the intermediate regime of the square-lattice model.	result 0.69	0.95	0.41	hedged possibility 0.98
s7	Our developments include parallelization and randomized low-rank methods and largely expand the frontier of high-precision variational numerics for quantum phases of matter.	contribution 0.91	0.93	0.12	asserted 1.00

What kind of paper the classifier reports this is: simulation study 0.63 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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A. Menczer and Ö. Legeza, Massively Parallel Tensor Network State Algorithms on Hybrid CPU-GPU Based Architectures, Journal of Chemical Theory and Compu- tation21, 1572 (2025)

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TeneT.jl contributors, TeneT.jl,https://github.com/ XingyuZhang2018/TeneT.jl(n.d.), open-source Julia package

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J. Demmel, L. Grigori, M. Hoemmen, and J. Langou, Communication-optimal parallel and sequential QR and LUfactorizations,SIAMJournalonScientificComputing 34, A206 (2012). 12 Supplementary Information Supplementary Note 1: Plaquette-order convergence atJ2/J1 = 0.55for theJ 1-J2 square lattice model We provide data in Supplementary Fig. 1 as evidence that th...

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For iPEPS,m2 is the square of the per-site Néel ampli- tudem= 12|⟨SA⟩−⟨S B⟩|. For QMC,m2(L) =⟨ ¯m· ¯m⟩, with ¯m=L −2∑i(−1)xi+yiSi, on periodicL×Llattices at zero or sufficiently low temperature. The QMC reference extends toL= 96for energy andL= 256form 2. The iPEPS fits use only the leading corrections in1/ξ3 and1/ξ, respectively; the QMC fits include the...

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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J. Demmel, L. Grigori, M. Hoemmen, and J. Langou, Communication-optimal parallel and sequential QR and LUfactorizations,SIAMJournalonScientificComputing 34, A206 (2012). 12 Supplementary Information Supplementary Note 1: Plaquette-order convergence atJ2/J1 = 0.55for theJ 1-J2 square lattice model We provide data in Supplementary Fig. 1 as evidence that th...

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For iPEPS,m2 is the square of the per-site Néel ampli- tudem= 12|⟨SA⟩−⟨S B⟩|. For QMC,m2(L) =⟨ ¯m· ¯m⟩, with ¯m=L −2∑i(−1)xi+yiSi, on periodicL×Llattices at zero or sufficiently low temperature. The QMC reference extends toL= 96for energy andL= 256form 2. The iPEPS fits use only the leading corrections in1/ξ3 and1/ξ, respectively; the QMC fits include the...

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