[challenge]: Certified energy-density bounds vs. Bethe ansatz

Released by

Jie Wang (AMSS, Chinese Academy of Sciences) & Jin-Guo Liu (Hong Kong University of Science and Technology (Guangzhou))

Contact email

cacate0129@gmail.com

Method

Other

Challenge issue

Difficulty: ★ (rated by Jie Wang)

Background

Modern certified many-body pipelines combine renormalization-group compression of translation-invariant relaxations (arXiv:2212.03014), Pauli-algebra convergence rates (arXiv:2605.29959), and symmetry+sparsity-structured NC polynomial optimization (arXiv:2604.01555). Bethe-ansatz-solvable chains (spin-1/2 Heisenberg, XXZ) provide exact thermodynamic-limit energy densities — a perfect, ungameable ground truth against which no systematic calibration of the modern certified stack exists.

Research objective

Push certified two-sided bounds on the thermodynamic ground-state energy density of the Heisenberg and XXZ chains as tight as possible (SU(2)/U(1) symmetry reduction + term sparsity; upper bounds from inner hierarchies, arXiv:2402.02126, or MPS), and test whether observed level-wise errors match the proved rates.

Verification plan

• Success gate: the certified interval contains the Bethe value at every level (hard correctness check), and the interval width at the top computable level improves on the best previously published rigorous bound for the Heisenberg chain — a numeric comparison against fixed literature constants.
• Hope signal: intervals valid but wide — profiling which constraint families tighten fastest is a useful calibration dataset on its own.
• Pivot signal: certified intervals exclude the Bethe value anywhere — a built-in alarm; if it persists after audit, the symmetry reduction is unsound and that finding takes priority.

Why this may lead to research output

Both a benchmark paper (the field's first systematic calibration against exact interacting models) and, wherever certified error bars beat prior rigorous bounds (Anderson-type, arXiv:2601.07800, cond-mat/0110486), a record with a proof.

References

1. Kull, Schuch et al., Lower bounds on ground-state energies of local Hamiltonians through the renormalization group, arXiv:2212.03014.
2. Wang et al., Scalable ground-state certification of quantum spin systems, arXiv:2604.01555.
3. Klep et al., Quantitative semidefinite certificates for ground-state energies of Pauli Hamiltonians, arXiv:2605.29959.
4. Upper bound hierarchies for noncommutative polynomial optimization, arXiv:2402.02126.

No repository files, tests, or entry points are identified in the issue. Start by locating the existing certified-bound pipeline and its Heisenberg or XXZ benchmark support, then establish the Bethe-ansatz reference values and the highest computable hierarchy level. Done means certified intervals contain the Bethe value at every level and the top-level Heisenberg interval improves on the cited rigorous bounds, or the requested calibration and failure analysis is documented.