Three-Dimensional Ising Bootstrap Benchmark

The three-dimensional Ising CFT is the numerical bootstrap’s strongest precision benchmark: mixed-correlator crossing, unitarity, discrete-symmetry, and spectral assumptions confine key CFT data to extraordinarily small regions that agree with Monte Carlo and fuzzy-sphere calculations. The certification is conditional, however. A solver can certify infeasibility of a finite approximation; it does not by itself prove that the assumed isolated island contains a unique, existing continuum CFT.

Evidence cutoff. 11 August 2026. Reassess by 11 February 2027 or after a materially tighter island, an independent implementation, or a discrepancy with a non-bootstrap method.

Required background. Mixed-correlator islands explains how spectral assumptions isolate the Ising region; benchmark provenance and reproduction fixes what must be preserved for a numerical claim to be rerun.

Helpful background. Solver certificates and verification distinguishes a numerical optimizer report from a checked functional; correlated uncertainty propagation clarifies errors in comparison methods; criticality evidence and model discrimination prevents universality-class agreement from being confused with microscopic identity.

The benchmark claim

The target is the parity-invariant, unitary, local three-dimensional CFT with global $\mathbb Z_2$ symmetry associated with the Ising critical point. The most precisely compared observables are the leading odd and even scalar dimensions, $\Delta_\sigma$ and $\Delta_\epsilon$, selected OPE coefficients, and the stress-tensor central charge. The brief excludes nonunitary continuations, boundary critical behavior, and claims that crossing alone constructs the CFT.

Crossing for correlators built from $\sigma$, $\epsilon$, and, in the 2025 stress-tensor analysis, $T_{\mu\nu}$ becomes a semidefinite feasibility problem after expanding conformal blocks and truncating derivative and spin data. Rigorous language must retain all three layers: exact crossing and positivity; the finite representation of those constraints; and the additional gaps or uniqueness assumptions used to carve out an island.

Evidence matrix

Source and method	Relation to the bounded claim	Independence	Result and stated uncertainty	Main limitation
Kos et al., 2016, mixed $\sigma$–$\epsilon$ correlators and semidefinite optimization	establishes the classic precision island	one bootstrap pipeline and block representation; subsequent studies share much of the formal setup	$\Delta_\sigma=0.5181489(10)$ and $\Delta_\epsilon=1.412625(10)$, with precise leading OPE coefficients	island size is conditional on the assumed low-lying spectrum and finite numerical cutoff
Chang et al., 2025, mixed correlators including the stress tensor	supports and greatly sharpens the benchmark	new correlators and improved software, but partly continuous in authorship and methodology with the earlier program	$\Delta_\sigma=0.518148806(24)$ and $\Delta_\epsilon=1.41262528(29)$, plus stress-tensor OPE data	precision of the isolated estimate should not be read as an unconditional existence theorem
Poland, Prilepina, and Tadić, 2025, truncated mixed five-point correlators	qualifies and extends the comparison to new OPE data	a different correlator topology, but it imports low-lying four-point data and uses a truncation ansatz	previously unknown OPE coefficients consistent with fuzzy-sphere predictions	this is not a standalone island or positivity certificate; truncation and disconnected-correlator approximations control the inference
Simmons-Duffin, 2015, SDPB	supports reproducible high-precision feasibility tests	software infrastructure used by many later analyses, so publications using it are not implementation-independent	arbitrary-precision primal-dual semidefinite optimization with residual and duality diagnostics	a converged solver result still depends on correct problem generation and certificate checking
Hasenbusch, 2010, improved-lattice Monte Carlo	independently supports the Ising universality-class exponents	different regulator and inference pipeline; shares universality assumptions, not bootstrap numerics	critical exponents consistent with dimensions inferred from the bootstrap	finite-size scaling, action tuning, and correction-to-scaling systematics limit precision
Zhu et al., 2023, fuzzy-sphere regularization	qualifies and independently supports the low-lying spectrum	Hamiltonian geometry and finite-size extrapolation differ from both bootstrap and cubic-lattice Monte Carlo	emergent conformal multiplets and operator data consistent with the 3D Ising CFT	small accessible system sizes and operator identification give broader uncertainties

What is genuinely certified

A dual functional with the required positivity properties can certify that a tested point or region is inconsistent with the finite crossing problem. Residuals, precision, primal/dual feasibility, polynomial approximations, and sampled positivity all matter. Re-running the same input with the same SDPB executable is reproducibility, but not full independence; regenerating blocks, changing cutoffs and bases, and checking certificates with separate code is stronger.

The tiny Ising island is more than a numerical coincidence because it persists and contracts as constraint sets are strengthened, while independent lattice-based methods land in the same universality-class neighborhood. Yet the central-value notation compresses several distinct uncertainties. The interval at a chosen cutoff is not automatically a frequentist or Bayesian coverage statement, and convergence across a finite sequence does not supply an error theorem for the infinite-dimensional problem.

Established result, assumption, and interpretation

• Established within the computation: declared points outside a solver-certified allowed region violate the finite set of positivity and crossing constraints.
• Additional assumption: gaps, uniqueness of relevant scalars, stress-tensor normalization, and other spectral conditions used to identify the island describe the Ising CFT and exclude alternatives.
• Interpretation: the island is the continuum 3D Ising fixed point. Agreement with Monte Carlo and fuzzy-sphere spectra makes this interpretation compelling, but the bootstrap calculation alone does not construct the theory or prove uniqueness.

The appropriate assessment is therefore “exceptionally precise conditional benchmark,” not “numerically proved CFT.”

What would change the assessment

Confidence would increase with independently generated conformal blocks, archived input and output records, machine-checkable dual functionals, and stable contraction under substantially larger derivative and spin cutoffs. A persistent mismatch with high-control Monte Carlo or fuzzy-sphere determinations after common normalization and finite-size analyses would weaken the universality-class identification. An analytic construction or convergence theorem would change the logical status more profoundly than another decimal place.

Source selection

The finite set includes the foundational mixed-correlator island, the 2025 stress-tensor extension, a higher-point truncated-correlator cross-check, the primary solver paper, and two regulator-distinct comparisons. Papers that simply import the same island as an input were excluded. The quoted bootstrap intervals are authors’ allowed-region determinations, not recomputed uncertainties.

Related research pages

• Analytic and numerical conformal bootstrap for the full approximation and certificate chain.
• Numerical bootstrap certification for the unresolved infinite-dimensional inference problem.
• Crossing solutions and actual CFTs for existence and reconstruction limits.

References

• Chang, Cyuan-Han, et al. “Bootstrapping the 3D Ising Stress Tensor.” Journal of High Energy Physics 03 (2025): 136. DOI; arXiv.
• Hasenbusch, Martin. “Finite Size Scaling Study of Lattice Models in the Three-Dimensional Ising Universality Class.” Physical Review B 82 (2010): 174433. DOI.
• Kos, Filip, David Poland, David Simmons-Duffin, and Alessandro Vichi. “Precision Islands in the Ising and $O(N)$ Models.” Journal of High Energy Physics 08 (2016): 036. DOI.
• Poland, David, Valentina Prilepina, and Petar Tadić. “Mixed Five-Point Correlators in the 3D Ising Model.” Journal of High Energy Physics 10 (2025): 237. DOI.
• Simmons-Duffin, David. “A Semidefinite Program Solver for the Conformal Bootstrap.” Journal of High Energy Physics 06 (2015): 174. DOI.
• Zhu, Wei, Chao Han, Emilie Huffman, Johannes S. Hofmann, and Yin-Chen He. “Uncovering Conformal Symmetry in the 3D Ising Transition: State–Operator Correspondence from a Quantum Fuzzy Sphere Regularization.” Physical Review X 13 (2023): 021009. DOI.