The kink in the two-dimensional bound, and the Ising model at it

From a line to a plane

In two dimensions four points leave two cross-ratios, $u$ and $v$ , or equivalently $z$ and $\bar z$ . Exchanged operators now carry a spin $\ell$ as well as a dimension, and each block is built from two of the one-dimensional blocks of chapter 3.

Take four identical scalars $\sigma$ that are odd under a sign flip, as the Ising spin is. Then only even operators with even spin appear in $\sigma\times\sigma$ , and unitarity requires $\Delta\ge\ell$ . The bound concerns the lightest scalar, $\varepsilon$ in the Ising model.

\sum_{\Delta,\ell}p_{\Delta,\ell}\Bigl[v^{\Delta_\sigma}g_{\Delta,\ell}(u,v)-u^{\Delta_\sigma}g_{\Delta,\ell}(v,u)\Bigr]=0,\qquad p_{\Delta,\ell}\ge0

The structure is the same as on a line: a sum of known functions with non-negative weights that must vanish.

g_{\Delta,\ell}=\frac{k_{h}(z)\,k_{\bar h}(\bar z)+k_{\bar h}(z)\,k_{h}(\bar z)}{1+\delta_{\ell0}},\qquad h=\tfrac{\Delta+\ell}{2},\ \ \bar h=\tfrac{\Delta-\ell}{2}

Here

k_h(x)=x^h\,{}_2F_1(h,h;2h;x)

is the one-dimensional block with weight

h

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

A bound with a corner

For each external dimension $\Delta_\sigma$ the search gives the largest allowed $\Delta_\varepsilon$ . These two-dimensional bounds first appeared in a 2009 paper by Rychkov and Vichi. The curve bends sharply near $\Delta_\sigma=1/8$ , where the exact Ising values are $\Delta_\sigma=1/8$ and $\Delta_\varepsilon=1$ .

Nothing in the search knows about the Ising model, so the bend is a finding, not an input. A corner that stays put while the bound tightens is one of the clearest signs that a bound is tracking a real theory.

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 · D. Poland

Worked example: measuring the kink

At $\Lambda=8$ with spins up to $20$ , the bound is $0.7454$ at $\Delta_\sigma=0.1$ , $1.0026$ at $0.125$ and $1.0866$ at $0.15$ . The slope drops from $10.3$ to $3.36$ across the Ising point, a factor of $3.06$ .

With four derivatives the same slopes are $3.98$ and $4.00$ : no kink at all. With twelve derivatives they are $13.7$ and $2.91$ , a factor of $4.70$ . The corner sharpens as information is added, which is what a real feature does.

The bound at the Ising point falls from $1.0265$ to $1.0026$ to $1.0004$ as $\Lambda$ goes from 4 to 8 to 12. It approaches the exact $\Delta_\varepsilon=1$ from above and never crosses it, as it must not, because the Ising model exists.

\Lambda=8:\ \ \frac{1.0026-0.7454}{0.025}=10.3,\qquad \frac{1.0866-1.0026}{0.025}=3.36

Finite differences over the same step on each side of the Ising point.

Source: D. Simmons-Duffin · D. Poland

Testing the truncation

A two-dimensional search keeps a finite set of spins and a finite range of dimensions. Keep spins only up to $4$ at $\Lambda=8$ , and the search excludes every gap down to $0.30$ , the bottom of the scanned window. The Ising model is excluded with everything else.

Sample only up to $2$ units above the gap, and the bound at $\Delta_\sigma=1/8$ becomes $0.9717$ , again below the Ising value. Raising the spin cutoff to $8$ or the range to $6$ restores $1.0026$ . A trustworthy bound is one that stops moving when the truncations are relaxed.

Source: D. Simmons-Duffin · D. Poland

Try it in the Lab

Find the kink and try to fake it

Look at the curve at the default preset and find where it bends. Read the bound at the Ising point.
Compare the few-derivatives and more-derivatives presets. Predict which one has the sharper corner before you look.
Select the too-few-spins preset and explain why excluding the Ising model proves the truncation is broken.

Exercises with explained answers

In the Ising model $\sigma\times\sigma$ contains the identity, $\varepsilon$ and the stress tensor with spin $2$ . Which of them does the scalar gap bound constrain, and what is its dimension?
Show the answer
The lightest non-identity scalar, $\varepsilon$ , with $\Delta_\varepsilon=1$ . The stress tensor has spin $2$ ; in this search spinning operators only need $\Delta\ge\ell$ .
A proposed theory has $\Delta_\sigma=0.15$ and its lightest scalar at $1.1$ . Is it allowed at $\Lambda=8$ ?
Show the answer
No. The bound there is $1.0866$ , below $1.1$ , so the search excludes it under the stated assumptions.
One truncation gives a bound of $0.97$ at $\Delta_\sigma=1/8$ . Without running anything else, what can you conclude?
Show the answer
The truncation is wrong. The Ising model is a consistent unitary theory with $\Delta_\varepsilon=1$ at this point, so any valid bound must be at least $1$ .

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.