What a numerical bootstrap bound proves, and what it does not

The exact claim

A bound assumes a unitary theory, identical external operators of a given dimension, and no exchanged operator below the gap. Under those assumptions, a valid functional at order $\Lambda$ excludes every gap above $\Delta_*(\Lambda)$ .

Using fewer derivatives throws information away; it never adds a false constraint. So a bound at low order is weaker than the best possible bound $\Delta_*$ but still true, and adding derivatives can only lower it.

\Delta_{\mathrm{gap}}\le\Delta_*\le\Delta_*(\Lambda'),\qquad \Delta_*(\Lambda')\le\Delta_*(\Lambda)\ \ \text{for}\ \ \Lambda'>\Lambda

Any consistent theory sits below the optimal bound, which sits below every valid finite-order bound.

Source: D. Simmons-Duffin · D. Poland

Where a numerical bound can go wrong

Sampling. The linear programme asks for $\alpha(F_\Delta)\ge0$ only at sampled dimensions. Fewer constraints let more functionals pass, so a sampled bound can sit below the true one. The functional can dip negative between samples.

Truncation. In two dimensions the spins and the range of dimensions must be cut off. Leaving out operators again removes constraints and can make a bound look stronger than it is. The next chapter shows a case where this excludes the Ising model.

Precision. High derivative orders need many digits. In double precision the problem becomes ill-conditioned, which is why the Labs stop at moderate orders and published work uses high-precision solvers.

Source: S. Rychkov and N. Su · D. Poland

Worked example: auditing one exclusion

Take the default problem of the functional Lab: $\Delta_\psi=1/2$ , $\Lambda=13$ , trial gap $2.4$ . The linear programme returns a functional with $\alpha(F_0)=1$ , so the gap is excluded on the sampled grid.

Check 1, between the samples. On a six-times finer grid the smallest normalised value is $-2.72\times10^{-5}$ . Strictly, the functional fails there. The gap sits $0.25$ above the computed bound, so a dip this small is unlikely to change the verdict, but only a functional checked on the whole interval would settle it.

Check 2, how sampling moves the bound. On the base grid alone the bound is $2.15255$ . Adding the points where the functional dipped raises it to $2.15328$ . More constraints, higher bound: the continuous bound lies at or above $2.15328$ .

Check 3, the arithmetic. A separate script sums the hypergeometric series in 30-digit arithmetic and solves the same grid with a different solver, HiGHS. Across twelve bounds it agrees within $4.3\times10^{-6}$ . This tests the code, not the sampling or the assumptions.

\hat\alpha(\Delta)=\frac{\alpha(F_\Delta)}{|\alpha|\,|F_\Delta|},\qquad \min_\Delta\hat\alpha=-2.72\times10^{-5}

Dividing by both lengths makes the dip comparable between dimensions and between functionals.

Source: S. Rychkov and N. Su · D. Mazáč

Allowed is not the same as existing

Below the bound, the solver returns a spectrum with positive weights. At $\Lambda=3$ and gap $2.3$ it uses just two operators, at $2.3$ and $3.365$ . That satisfies two derivative equations and nothing more. It is not a theory.

So an allowed point means only that this search found no contradiction. Evidence that a real theory sits somewhere comes from elsewhere: an exact solution, a lattice model, or a feature such as a kink that sharpens as more derivatives are added.

Source: D. Poland · D. Simmons-Duffin

Try it in the Lab

Read the certificate check

At the default settings, find the certificate readout and note how far below zero the functional dips.
Predict whether raising the trial gap will make the dip larger or smaller, then try it.
Lower the gap until the verdict flips, and look at how few operators the allowed spectrum uses.

Exercises with explained answers

Positivity is imposed only on sampled dimensions. Does that make the computed bound higher or lower than the continuous one?
Show the answer
Lower or equal. Fewer constraints let more functionals qualify, so more gaps are excluded. That is why adding sample points can only raise the computed bound.
A bound computed at $\Lambda=9$ is $2.2316$ and one at $\Lambda=13$ is $2.1533$ . Can the first be correct if the second is?
Show the answer
Yes. A lower order uses less information, so its bound is weaker but still valid. Both are above the optimal value $2$ , as they must be.
An independent code reproduces every bound to six digits. Does that prove the bounds are rigorous?
Show the answer
No. It shows that the arithmetic is right on the chosen grid. The sampling, the truncations and the physical assumptions are the same in both codes, so neither can catch an error there.

Sources and what we recomputed

D. Simmons-Duffin, TASI Lectures on the Conformal Bootstrap, arXiv:1602.07982v1 (2016)
Version 1. Eqs. (157)-(158): the four-point function as a sum over conformal multiplets, one conformal block each. Eqs. (164)-(168): the Casimir equation for blocks and their closed form in two dimensions, built from the one-variable function k. Eqs. (194)-(199), Sections 10.5-10.7: the crossing equation for identical scalars as a sum of vectors, the derivative functionals and the assumptions behind the scalar-gap bound. Fig. 26: 2D crossing vectors. Fig. 27 and Table 1: 2D bounds for derivative cutoffs 6 to 28 with the kink at the Ising point. Fig. 28: the 3D bound with its kink.
D. Mazáč, Analytic bounds and emergence of AdS2 physics from the conformal bootstrap, arXiv:1611.10060v1; JHEP 04 (2017) 146
Version 1. Eqs. (2.12)-(2.17): the four-point function of identical operators on a line, the cross-ratio, the one-variable block and the crossing equation. Eq. (2.18) and Fig. 2: the numerical gap bound, which the text says converges to twice the external dimension plus one as derivatives are added. Eqs. (2.19)-(2.23): the generalized free fermion that saturates it and the zeros of the extremal functional. Fig. 1: the action of a typical extremal functional.
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, arXiv:1805.04405v3; Rev. Mod. Phys. 91, 015002 (2019)
Version 3. Section IV.A: linear programming on a discretised set of dimensions, which the review calls obsolete but pedagogical. Section IV.B: semidefinite programming, which imposes positivity for all dimensions through polynomial approximations, a spin cutoff, and solvers run above machine precision. Fig. 5: vectors that can or cannot sum to zero with positive coefficients, and the separating plane. Fig. 7: the 3D single-correlator bound with the Ising model near its kink. Figs. 12, 14 and 15: the 3D Ising island from mixed correlators. Table II: the stable operators of the 3D Ising model. The islands and the table are reading material only.
S. Rychkov and N. Su, New developments in the numerical conformal bootstrap, arXiv:2311.15844v3 (2024)
Version 3. Section 2: the solver SDPB uses arbitrary-precision arithmetic because the matrices are poorly conditioned. Section 3: the standard method, with derivatives at the symmetric point, polynomial approximations of block derivatives and a semidefinite feasibility problem. Footnote 18 in Section 4.2: approximations that remain not fully rigorous even in published error bars. Section 6.3.1: rigorous and non-rigorous error bars compared.