How to read Table 1 and Figure 27 of the TASI bootstrap lectures

What Figure 27 plots

The horizontal axis is the external dimension $\Delta_\phi$ from about $0.05$ to $0.30$ . The vertical axis is $\Delta_0$ , the dimension of the lightest scalar in $\phi\times\phi$ . Above each curve is disallowed, below it allowed. The assumptions are those of Eq. (199): a sign-flip symmetry, even spins and the two-dimensional unitarity bound.

The curves are for derivative cutoffs $\Lambda=6,8,12,16,20,28$ , the darkest being $\Lambda=28$ . The caption says this uses a 105-dimensional space of functionals. The black dots are the unitary minimal models, the first of which is the Ising model, and the dashed line is a free-boson family.

N(\Lambda)=\tfrac12\Bigl\lfloor\tfrac{\Lambda+1}{2}\Bigr\rfloor\Bigl(\Bigl\lfloor\tfrac{\Lambda+1}{2}\Bigr\rfloor+1\Bigr),\qquad N(8)=10,\ \ N(12)=21,\ \ N(28)=105

The number of independent derivatives, from the footnote to Eq. (198). It grows roughly as the square of the cutoff.

(\Delta_\phi,\Delta_0)=\Bigl(\tfrac12-\tfrac{3}{2(m+1)},\ 2-\tfrac{4}{m+1}\Bigr),\qquad m=3,4,5,6

The minimal-model points in the caption. The case

m=3

is the Ising model at

(1/8,\,1)

Largest allowed

\Delta_\varepsilon

against

\Delta_\sigma

in two dimensions, recomputed by this course with four derivatives (dashed) and eight derivatives (solid). Dots: unitary minimal models, the first being Ising at

(1/8,\,1)

. Above a curve is excluded.

Source: D. Simmons-Duffin · S. Rychkov and A. Vichi

What Table 1 lists

Table 1 gives the bound at $\Delta_\phi=1/8$ for each cutoff: $1.020$ , $1.0027$ , $1.00053$ , $1.000043$ , $1.0000070$ and about $1.0000005$ for $\Lambda=6$ up to $28$ . The text notes that the last is within $5\times10^{-7}$ of the exact value.

Two things are worth noticing. The bound approaches the Ising value from above and never crosses it. And the approach is steady: at each step in the table the remaining distance shrinks by a factor between 5 and 14. That steady convergence is what lets readers treat the kink as a real feature.

Source: D. Simmons-Duffin

Worked example: reproducing three entries

The robustness Lab and the benchmark script use the same assumptions, a discretised linear programme with cutting planes, and spins up to a cutoff. They give $1.019827$ , $1.002621$ and $1.000443$ at $\Lambda=6$ , $8$ and $12$ .

The differences from the table are $-1.7\times10^{-4}$ , $-7.9\times10^{-5}$ and $-8.8\times10^{-5}$ . The first is within the rounding of $1.020$ . All three are negative, the direction chapter 6 predicts: positivity checked only on samples lets slightly more functionals through, so a sampled bound sits a little below the continuous one.

The same data can be checked against the minimal-model dots. At $\Lambda=8$ the course bound sits $0.0026$ above the Ising point, $0.049$ above the $m=4$ point at $(0.2,\,1.2)$ and $0.10$ above the $m=6$ point. Every dot is allowed, and the Ising model is by far the closest to the boundary.

\Lambda=6:\ 1.0198\ \text{vs}\ 1.020,\qquad \Lambda=8:\ 1.00262\ \text{vs}\ 1.0027,\qquad \Lambda=12:\ 1.00044\ \text{vs}\ 1.00053

Course reproduction first, published Table 1 value second.

Source: D. Simmons-Duffin · D. Poland

What this course does not reproduce

The course solver works in double precision and stops at $\Lambda=12$ in two dimensions; the lectures compute every curve with the semidefinite solver SDPB. Figure 28 shows the three-dimensional bound, computed with $\Lambda=24$ , with a kink near $(0.518,\,1.412)$ . The course quotes it but does not compute it.

The small islands of allowed dimensions in later work come from several correlators at once and from an extra assumption, that only a few scalars are relevant. The review by Poland, Rychkov and Vichi shows them. Read them as published results; nothing in this course checks them.

Source: D. Simmons-Duffin · D. Poland

Try it in the Lab

Hold the Lab against Table 1

Read the bound at the Ising point at the default settings and compare it with the Table 1 entry for the same cutoff.
Pick the more-derivatives preset. Predict the sign of the difference from the table before you read it.
Drag the probe to the point of the next minimal model and check that it is allowed.

Exercises with explained answers

How many independent derivatives does a two-dimensional functional use at $\Lambda=20$ ?
Show the answer
$\lfloor21/2\rfloor=10$ , so $N=\tfrac12\times10\times11=55$ .
Where does the $m=5$ minimal model sit in Figure 27?
Show the answer
At $\Delta_\phi=1/2-3/12=0.25$ and $\Delta_0=2-4/6\approx1.333$ . The course bound there at $\Lambda=8$ is $1.411$ , so the point is allowed.
A reproduction gives bounds slightly above the published ones at every cutoff. What would you suspect first?
Show the answer
Not the sampling: sampling pushes a bound down. Look for stronger assumptions, extra constraints, or a different normalisation of the external dimension, and compare the stated assumptions line by line.

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.
S. Rychkov and A. Vichi, Universal constraints on conformal operator dimensions, arXiv:0905.2211v2; Phys. Rev. D 80, 045006 (2009)
Version 2. Eq. (2.8): the crossing sum rule written with conformal blocks. Section 3, Eqs. (3.1)-(3.2): the positivity argument, a linear functional of derivatives at the symmetric point that rules out a spectrum. Table 2: the two-dimensional bounds of that paper.
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.