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arxiv: 2606.03836 · v1 · pith:JWPVS4OVnew · submitted 2026-06-02 · 🪐 quant-ph · cond-mat.stat-mech· math-ph· math.MP

The bulk spectral gap is semi-decidable: a convergent family of certified upper bounds

Xiangling Xu , Matthias Sch\"otz , Jie Wang , Victor Magron , Igor Klep , Omar Fawzi , Marc-Olivier Renou This is my paper

Pith reviewed 2026-06-28 09:41 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechmath-phmath.MP
keywords bulk spectral gapsemi-decidabilitysemidefinite programmingquantum many-body systemsthermodynamic limittranslation-invariant Hamiltonianskagome lattice
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The pith

The bulk spectral gap of quantum many-body systems is semi-decidable.

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

The paper introduces a family of semidefinite programs whose solutions are certified upper bounds on the bulk spectral gap of translation-invariant quantum Hamiltonians. These bounds can be made arbitrarily tight by increasing the computational resources allocated to larger programs in the family. This construction establishes that the bulk spectral gap is semi-decidable. The result applies to general models without special structure or boundary conditions and stands in contrast to known undecidability results for gap notions that rely on sequences of finite systems.

Core claim

A complete family of semidefinite programs produces certified upper bounds on the bulk spectral gap that converge to the true value from above; solving successively larger programs in the family therefore decides, to any desired precision, whether the infinite-volume gap is positive.

What carries the argument

A convergent hierarchy of semidefinite programs whose feasible sets encode certified upper bounds on the infinite-volume bulk spectral gap.

If this is right

  • The bulk spectral gap can be approximated from above to arbitrary accuracy by solving a sequence of SDPs.
  • The method supplies the first nontrivial certified upper bounds for the spin-1/2 kagome Heisenberg antiferromagnet.
  • Semi-decidability holds without requiring special lattice symmetries or boundary conditions.
  • Alternative gap notions based on finite systems with prescribed boundaries remain undecidable.

Where Pith is reading between the lines

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

  • The hierarchy could be adapted to compute certified bounds on other bulk quantities such as correlation lengths or entanglement measures.
  • Implementation on specific models may reveal whether the gap decision problem is decidable in practice for physically relevant Hamiltonians.
  • The contrast with undecidable gap variants suggests that the choice of thermodynamic limit definition can determine computational tractability.

Load-bearing premise

The semidefinite programs are assumed to produce valid certified upper bounds on the infinite-volume bulk gap for arbitrary translation-invariant Hamiltonians.

What would settle it

A concrete translation-invariant Hamiltonian together with a numerical computation showing that the SDP bounds fail to approach the independently known bulk gap from above.

Figures

Figures reproduced from arXiv: 2606.03836 by Igor Klep, Jie Wang, Marc-Olivier Renou, Matthias Sch\"otz, Omar Fawzi, Victor Magron, Xiangling Xu.

Figure 1
Figure 1. Figure 1: A portion of the kagome lattice. From “Kagome [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Certified upper bounds for the transverse-field Ising chain with spin-flip symmetry imposed. Largest fea￾sible γ for g = 0.5, 1.0, 1.5, 2.0 along the relaxation levels (L, d) = (n + 1, n), n = 2, 3, 4. As the relaxations are nested, these certified upper bounds decrease monotoni￾cally in L and d. In the disordered phase (g = 1.5, 2.0) the bounds approach the known gap value 2(g − 1) from above. At critical… view at source ↗
Figure 3
Figure 3. Figure 3: Certified upper bounds for the spin- 1 2 kagome lattice Heisenberg antiferromagnet. Largest feasible gap parameter versus patch radius L, where L is the graph distance from a chosen reference site. We show four cases: the hierarchy with no symmetry restriction at d = 2 for L = 1, 2; the hierarchy with π-rotation spin symmetries at d = 2 for L = 1, 2; the same symmetries plus finite spin-isotropy and time-r… view at source ↗
read the original abstract

Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. Existing rigorous methods are largely limited to special settings, while variational numerical approaches typically provide estimates rather than certified bounds. Here we introduce a complete family of certified upper bounds on the bulk spectral gap of quantum many-body systems. These upper bounds are obtained by solving a series of semidefinite programs and they become arbitrarily tight at the cost of more computational resources. This shows that the bulk spectral gap is semi-decidable, in contrast to undecidability results for alternative notions of spectral gap based on sequences of finite systems with prescribed boundary conditions. As a proof of principle, we apply our algorithm to the spin-$\frac{1}{2}$ kagome lattice Heisenberg antiferromagnet and obtain, to our knowledge, the first nontrivial certified upper bounds on its bulk spectral gap.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a hierarchy of semidefinite programs (SDPs) whose optimal values furnish certified upper bounds on the bulk spectral gap of translation-invariant quantum many-body Hamiltonians. These bounds are proven to converge from above to the true infinite-volume gap as the hierarchy level increases, establishing semi-decidability of the bulk gap. This is contrasted with undecidability results that rely on sequences of finite systems with fixed boundary conditions. The approach is illustrated on the spin-1/2 kagome Heisenberg antiferromagnet, yielding the first nontrivial certified upper bounds for that model.

Significance. If the central claims hold, the work supplies a computationally grounded method for rigorous upper bounds on bulk gaps without requiring special structure or boundary conditions, which is a notable advance over purely variational estimates. The semi-decidability result clarifies the distinction between bulk and boundary-dependent gap notions. The SDP construction provides an explicit, convergent family of certificates, which strengthens the result beyond existence proofs.

minor comments (3)
  1. [§3] §3 (or the section defining the SDP hierarchy): the proof that the SDP value is a certified upper bound on the bulk gap should explicitly reference the translation-invariance assumption and how the local Hamiltonian terms are embedded; a short remark on why no boundary terms appear would aid clarity.
  2. [kagome application] Figure 1 (or the kagome application section): the reported numerical upper bounds lack an accompanying table of SDP sizes, solver tolerances, and duality gaps; without these, reproducibility of the 'first nontrivial' claim is harder to assess.
  3. Notation: the symbol for the bulk gap (likely Δ_bulk or similar) is introduced without an immediate comparison to the finite-system gap Δ_N; adding one sentence contrasting the two definitions would prevent reader confusion with existing undecidability literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report does not contain any listed major comments, so there are no specific points requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs a hierarchy of semidefinite programs to generate certified upper bounds on the bulk spectral gap of translation-invariant Hamiltonians, with a proof that these bounds converge from above to the true gap. This construction relies on standard SDP duality and relaxation techniques from optimization theory rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation. The semi-decidability conclusion follows directly from the existence of this convergent hierarchy without reducing to the paper's own inputs. The distinction from undecidability results for finite-system gaps with fixed boundary conditions is preserved by the explicit definition of the bulk gap used. The derivation is self-contained against external benchmarks in convex optimization.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The central claim rests on the unshown validity of the SDP hierarchy for the bulk gap.

pith-pipeline@v0.9.1-grok · 5707 in / 1023 out tokens · 18029 ms · 2026-06-28T09:41:55.443297+00:00 · methodology

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