---
title: "The kink in the two-dimensional bound, and the Ising model at it"
description: "In two dimensions the scalar gap bound bends sharply at the Ising model. This chapter measures the kink and tests which truncations would fake it."
canonical_url: "https://sandboxphysics.com/stories/conformal-bootstrap/two-dimensional-ising-kink/"
language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/conformal-bootstrap/two-dimensional-ising-kink/"
last_modified: "2026-10-29"
---

[Conformal bootstrap](https://sandboxphysics.com/stories/conformal-bootstrap/)

L4 · Chapter 7

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

Sandbox Physics · 2026-10-29

Run the same functional search in two dimensions and plot the largest allowed scalar gap against the external dimension. The curve is not smooth: it bends sharply at one point, and that point is the two-dimensional Ising model. This chapter measures the bend, watches it sharpen with more derivatives, and checks which numerical shortcuts would move or fake it.

## Before you begin

Chapters 3 to 6: blocks, crossing vectors, linear functionals and the caveats of a numerical bound. The Ising operator dimensions from chapter 2 are used as the target.

- State the two-dimensional crossing equation and the assumptions behind the bound on the leading scalar.
- Measure the kink at the Ising point from bound data, and use truncation tests to decide whether a bound can be trusted.

## 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](https://sandboxphysics.com/labs/bootstrap-bound-robustness/)

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

## Exercises with explained answers

1. 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 $\Delta\ge\ell$.

2. 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.

3. 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 $1$.

[Previous: What a numerical bootstrap bound proves, and what it does not](https://sandboxphysics.com/stories/conformal-bootstrap/what-a-bound-proves/) [Next: How to read Table 1 and Figure 27 of the TASI bootstrap lectures](https://sandboxphysics.com/stories/conformal-bootstrap/reading-the-tasi-bounds/)

## Sources and what we recomputed

1. [D. Simmons-Duffin, TASI Lectures on the Conformal Bootstrap, arXiv:1602.07982v1 (2016)](https://arxiv.org/abs/1602.07982v1)

   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.

2. [S. Rychkov and A. Vichi, Universal constraints on conformal operator dimensions, arXiv:0905.2211v2; Phys. Rev. D 80, 045006 (2009)](https://arxiv.org/abs/0905.2211v2)

   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.

3. [D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, arXiv:1805.04405v3; Rev. Mod. Phys. 91, 015002 (2019)](https://arxiv.org/abs/1805.04405v3)

   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.