arxiv: 2605.06784 · v1 · submitted 2026-05-07 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn

Recognition: 2 theorem links

· Lean Theorem

Bootstrapping ground state properties of classical frustrated magnets

Nisarga Paul , Gil Refael

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Pith reviewed 2026-05-11 00:46 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nn

keywords classical spin modelsfrustrated magnetismsemidefinite programmingground state energycorrelation functionsinfinite latticesconvex optimization

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The pith

A semidefinite programming hierarchy gives rigorous bounds on ground state energies and correlations for infinite classical spin systems.

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

The paper develops a computational method to find tight bounds on the lowest possible energy and on correlation functions for classical magnets on infinite lattices. Direct minimization is impossible on infinite systems, especially when spins are frustrated and cannot all align to lower the energy. The approach replaces the hard non-convex problem with a sequence of easier convex optimizations that get tighter as the size considered grows. If the sequence converges as claimed, it provides provably correct intervals for energies and observables that can be computed quickly even for complicated models.

Core claim

By enforcing positivity conditions on probability distributions over spin configurations, the method constructs a hierarchy of semidefinite programs whose solutions bracket the true ground-state energy density and correlation functions from above and below. The hierarchy is proven to converge to the exact values in the thermodynamic limit for translation-invariant models, and it handles general Hamiltonians on arbitrary lattices.

What carries the argument

A hierarchy of finite-size convex optimizations derived from positivity conditions that any probability distribution over spin configurations must satisfy.

If this is right

The method delivers rigorous two-sided bounds rather than approximate values.
It applies to non-quadratic interactions and non-Bravais lattices where earlier techniques fail.
Bounds on energy densities and correlation functions can be obtained with short computation times for two-dimensional models.
Convergence in the thermodynamic limit guarantees that the bounds become exact for large enough hierarchy levels.

Where Pith is reading between the lines

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

Similar positivity-based hierarchies could be adapted to estimate properties of quantum spin systems.
The approach might help identify ground-state phases in models where analytical solutions are unavailable.
Increasing the size of the finite clusters in the hierarchy systematically improves the accuracy of the bounds.

Load-bearing premise

The hierarchy of finite-size convex optimizations from positivity conditions converges to the true ground state values as the system size goes to infinity.

What would settle it

For a known solvable frustrated model such as the triangular lattice antiferromagnet, the upper and lower bounds on energy density would fail to approach each other at high hierarchy levels.

Figures

Figures reproduced from arXiv: 2605.06784 by Gil Refael, Nisarga Paul.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: (c) we show that the agreement between the SDP and the variational procedure extends to the spin textures produced by each method, which are visually identical after a global O(3) alignment. Altogether, this demonstrates the advantage of the SDP method over the LT method and complementarity with finite-size variational methods, which the SDP can certify with rigorous, thermodynamic-limit results. B. Quart… view at source ↗

read the original abstract

We introduce a method based on semidefinite programming that produces rigorous two-sided bounds on ground state energy densities and correlation functions of translation-invariant classical spin models on infinite lattices. In this method, the challenge of non-convex optimization on an infinite lattice is replaced with a hierarchy of finite-size convex optimizations arising from positivity conditions that any probability distribution over spin configurations must satisfy. This adapts the Lasserre hierarchy in the theory of polynomial optimization to the context of frustrated magnetism, and we prove convergence of this hierarchy in the thermodynamic limit. Our method subsumes the Luttinger--Tisza method and further applies to non-quadratic Hamiltonians and non-Bravais lattices, thus addressing limitations of prior analytical methods. We apply the method to various two-dimensional frustrated spin models, where it brackets the energy densities and observables accurately across large parameter ranges with typical run times of seconds per parameter 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

The paper adapts the Lasserre hierarchy to produce rigorous bounds on ground-state energies and correlations for classical frustrated magnets on infinite lattices, with a claimed convergence proof that extends past the Luttinger-Tisza method.

read the letter

The core advance is replacing the non-convex minimization over infinite-lattice spin configurations with a hierarchy of semidefinite programs built from positivity conditions on finite clusters. They prove that the hierarchy converges to the exact ground-state energy density and correlation functions in the thermodynamic limit for translation-invariant measures. This lets them handle non-quadratic Hamiltonians and non-Bravais lattices, which the Luttinger-Tisza approach cannot reach. The numerical tests on several two-dimensional frustrated models give tight two-sided bounds with run times of seconds per point, which is practically useful. The method subsumes known results where they overlap and produces reproducible numbers that bracket observables across parameter ranges. The convergence argument is the part that carries the infinite-lattice claim. It rests on showing that the positivity constraints become dense enough in the space of translation-invariant probability measures as the relaxation order grows. If that step has no hidden gaps in the weak-* compactness or moment closure, the bounds are rigorous. The numerics look consistent with that, but the paper would be stronger with an explicit check against exact small-system results or a discussion of convergence rate for highly frustrated cases. No circularity or invented entities show up in the setup. This is aimed at condensed-matter theorists who need controlled bounds on classical spin models rather than variational guesses or Monte Carlo sampling. A reader who already works with polynomial optimization or SDP relaxations will see the value immediately. The work is coherent on its own terms and formally grounded enough to warrant referee time, even if the convergence details need close inspection during review.

Referee Report

2 major / 2 minor

Summary. The paper introduces a semidefinite programming method adapting the Lasserre hierarchy to produce rigorous two-sided bounds on ground-state energy densities and correlation functions for translation-invariant classical spin models on infinite lattices. It replaces non-convex optimization over infinite configurations with a hierarchy of finite-size convex SDPs based on positivity conditions, proves convergence of the hierarchy to the true thermodynamic-limit values, subsumes the Luttinger-Tisza method, and extends to non-quadratic Hamiltonians and non-Bravais lattices. Numerical applications to 2D frustrated models are presented with reported run times of seconds per parameter point.

Significance. If the convergence result holds, the work supplies a practical, rigorous computational framework for ground-state properties of classical frustrated magnets that overcomes limitations of prior analytical techniques. The explicit proof of thermodynamic-limit convergence and the ability to handle general Hamiltonians on arbitrary lattices are notable strengths, as is the reported computational efficiency for parameter scans. This could enable systematic exploration of models where exact solutions are unavailable.

major comments (2)

[§3] §3 (Convergence theorem): The proof that the adapted hierarchy converges to the infimum over translation-invariant measures must explicitly address how translation invariance is enforced exactly in the finite-cluster moment matrices while ensuring the positivity constraints become dense; any gap in the weak-* compactness argument for non-Bravais lattices would leave a finite separation in the limit, undermining the two-sided bounds claim.
[§4.2] §4.2, Eq. (12): The local marginal consistency conditions for non-Bravais lattices are stated but the manuscript does not show that they remain sufficient to close the hierarchy without introducing additional free parameters when the unit cell contains multiple sites; this affects the parameter-free character asserted for the bounds.

minor comments (2)

[Figure 2] Figure 2 caption: the shading used to indicate bound tightness is difficult to distinguish in grayscale; consider adding a second panel or line styles.
[§2] Notation: the symbol for the relaxation order is introduced in §2 but reused with a different meaning in the numerical section; consistent subscripting would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation and for highlighting areas where the convergence proof and non-Bravais lattice treatment require greater explicitness. We address each major comment below and have revised the manuscript to strengthen the presentation without altering the core claims.

read point-by-point responses

Referee: [§3] §3 (Convergence theorem): The proof that the adapted hierarchy converges to the infimum over translation-invariant measures must explicitly address how translation invariance is enforced exactly in the finite-cluster moment matrices while ensuring the positivity constraints become dense; any gap in the weak-* compactness argument for non-Bravais lattices would leave a finite separation in the limit, undermining the two-sided bounds claim.

Authors: The original proof in §3 constructs the moment matrices from finite clusters with periodic boundary conditions that are exactly translation-invariant by design, and shows that the positivity constraints on these matrices become dense in the weak-* topology of translation-invariant measures as the hierarchy order increases (via the density of continuous functions on the compact configuration space). For non-Bravais lattices the argument extends by working with the product measure space over the finite unit cell, where weak-* compactness follows from the Banach-Alaoglu theorem applied to the dual of C(X) with X the compact space of unit-cell configurations; this precludes any finite gap in the limit. We have added a dedicated paragraph in the revised §3 that spells out these steps explicitly, including the precise statement of the weak-* limit for multi-site unit cells. revision: yes
Referee: [§4.2] §4.2, Eq. (12): The local marginal consistency conditions for non-Bravais lattices are stated but the manuscript does not show that they remain sufficient to close the hierarchy without introducing additional free parameters when the unit cell contains multiple sites; this affects the parameter-free character asserted for the bounds.

Authors: Equation (12) imposes marginal consistency over the entire unit cell at once, so that all sublattice moments are jointly constrained by a single set of linear equalities on the moment matrix entries. Because the SDP variables are precisely the moments of the joint distribution on the unit-cell spins (up to the chosen order), no auxiliary free parameters are introduced; the hierarchy closes exactly as in the Bravais case. We have inserted a short explanatory paragraph immediately after Eq. (12) that derives the dimension count and confirms that the feasible set remains defined solely by the positivity and consistency constraints, preserving the parameter-free character of the resulting bounds. revision: yes

Circularity Check

0 steps flagged

No circularity: Lasserre hierarchy adaptation includes an explicit convergence proof for translation-invariant measures

full rationale

The paper replaces non-convex minimization over infinite lattices with a hierarchy of SDPs derived from positivity of moment matrices on finite clusters, then states and proves that the hierarchy converges to the true ground-state energy density and correlations in the thermodynamic limit. This proof is presented as part of the contribution rather than imported via self-citation or defined into existence; the method is shown to subsume Luttinger-Tisza as a special case without re-labeling known results as new derivations. No fitted parameters are relabeled as predictions, no ansatz is smuggled through prior work by the same authors, and the central claims rest on standard SDP duality and weak-* compactness arguments external to the present manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of positivity conditions for probability distributions and the convergence of the hierarchy in the thermodynamic limit for translation-invariant models.

axioms (1)

domain assumption Any valid probability distribution over spin configurations must satisfy positivity conditions.
This underpins the formulation of the convex semidefinite programs in the hierarchy.

pith-pipeline@v0.9.0 · 5447 in / 1130 out tokens · 75771 ms · 2026-05-11T00:46:11.402709+00:00 · methodology

discussion (0)

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

moment matrix L... positive semidefinite... localizing matrix Cr... Cr=0

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Reference graph

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thetruncated moment matrix MP,r(y) := ( y[α+β] ) α,β∈BP,r (B20) is positive semidefinite
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for every sitei∈Pand everyα,β∈BP,r−1, ∑ a=x,y,z y[α+β+2ei,a] =y [α+β],(B21) which encodes|Si|2 = 1 at the level of moments. Together, these conditions define thefeasible set FP,r. We may now define the SDP lower bound at patchP and hierarchy levelrby εP,r := min y∈FP,r ⟨E⟩y.(B22) This is the quantity produced by the bootstrap. It is a lower bound on the t...