---
title: "2D Ising Bootstrap Bound: Does the Kink Survive the Truncation? Interactive Lab | Sandbox Physics"
description: "Is the 2D Ising model really at the edge of the bootstrap bound, or is that an artefact of how the problem was cut down?"
canonical_url: "https://sandboxphysics.com/labs/bootstrap-bound-robustness/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/bootstrap-bound-robustness/"
last_modified: "2026-10-30"
---

FR37 · Conformal bootstrap

# 2D Ising Bootstrap Bound: Does the Kink Survive the Truncation?

**Is the 2D Ising model really at the edge of the bootstrap bound, or is that an artefact of how the problem was cut down?** It is real, as long as the truncation is sound. The bootstrap bound on the first scalar in two dimensions bends sharply exactly where the critical Ising model sits, and adding derivatives only pulls the bound down toward it. Keeping too few spins or too short a range of dimensions makes the search exclude the Ising model itself, which shows that truncation is broken.

Trace the 2D bootstrap bound on the first scalar and find the Ising point at its kink. Change derivatives, spins and dimension range to see what survives.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Crossing for four identical scalars in two dimensions is turned into a linear program. For each external dimension a bisection finds the largest scalar gap that no functional can exclude, using even spins up to a cutoff and sampled dimensions up to a chosen range. The 2D Ising spin and energy operators sit at one eighth and one. Published values from the TASI lectures (Table 1) are shown as reference points, not as data. Positivity is imposed on samples only, so an exclusion is numerical evidence, not a proof. Too few spins or too short a dimension range make the bound exclude the Ising model itself; that is a truncation artefact.

More derivatives approach the published bounds

Too few spins and the bound collapses

More numbers and how they are computed

- **Where the bounds come from**: $0$
- **Components of each crossing vector**: $0$
- **Even spins kept**: $0$
- **Dense dimension grid per spin**: $0$
- **Points traced on the bound**: $0$
- **Probe position**: $0$

Every bound on this page was solved in advance for each slider setting with this page's own linear programs and tolerances. The page stores each bisection step's verdict and replays it, so the numbers equal a live solve exactly. The 2D Ising check and the probe are solved live in your browser.

Rings in the first chart are the published values of Simmons-Duffin, TASI lectures, Table 1; they are references, not output of this page. A red point means every gap down to the bottom of the search window was excluded. Assumptions: four identical scalars, only even spins exchanged, unitarity, positivity on sampled dimensions.

Physics tutorial

## A kink, and how to tell it is real

**Background** In two dimensions the bootstrap bound on the first scalar, plotted against the external dimension, bends sharply at one point. That point is the critical Ising model, whose spin and energy dimensions are known exactly.

**Why it matters** A numerical bound is computed with finitely many derivatives, spins and sample points. Before trusting a feature, check that it does not move when those choices change.

Start with the essentials

- **Focus question**: Is the 2D Ising model really at the edge of the bootstrap bound, or is that an artefact of how the problem was cut down?
- **One-sentence intuition**: With enough spins and a wide enough dimension range, more derivatives only lower the bound toward the Ising value and sharpen the kink. Cutting spins or the range instead lets the search find functionals that exclude the Ising model itself, a clear sign of a broken truncation.

### Core mathematical model

#### Crossing with spin

$$
\sum_{\Delta,\ell}p_{\Delta,\ell}\,\vec F_{\Delta,\ell}=-\vec F_0,\qquad p_{\Delta,\ell}\ge0,\ \ \Delta\ge\ell
$$

Every exchanged operator carries a dimension and an even spin $\ell$. Unitarity requires $\Delta\ge\ell$ in two dimensions.

#### The Ising point

$$
\Delta_\sigma=\tfrac18,\qquad \Delta_\varepsilon=1
$$

These are exact results of the 2D Ising conformal field theory; the bound passes just above this point.

#### Published convergence

$$
\Delta_\varepsilon^{\max}\bigl(\Lambda=6,\,8,\,12\bigr)=1.020,\ 1.0027,\ 1.00053
$$

Simmons-Duffin, TASI lectures, Table 1. The Lab reproduces these with its own solver.

### Common difficulties

#### A lower bound is not always a better bound

**Typical misconception** If a truncation gives a lower bound, it is a stronger result.

**Better mental model** A lower bound can come from leaving operators out. Positivity is then not imposed where a real theory needs it, and the functional excludes theories that exist, such as the Ising model.

### Run the experiment

1. 01

   #### Find the kink

   Start from the default and wait for the curve to finish.

   **What to observe:** The bound rises steeply, bends at the white star, then rises slowly.

2. 02

   #### Break the truncation

   Choose the too-few-spins preset, then the range-too-short preset.

   **What to observe:** The star turns red: the search now excludes the Ising model, which cannot be right.

3. 03

   #### Add derivatives

   Return to the default and raise the derivative order to 12.

   **What to observe:** The convergence chart approaches the published points from above and the kink sharpens.

Physical conclusion

Trust a bootstrap feature only after it survives more derivatives, more spins and a wider range of dimensions.

## Related experiments and articles

Conformal bootstrap · Reading path

### From a critical chain to the kink in the 2D bound

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **The kink in the two-dimensional bound, and the Ising model at it**](https://sandboxphysics.com/stories/conformal-bootstrap/two-dimensional-ising-kink/)
2. [Article **How to read Table 1 and Figure 27 of the TASI bootstrap lectures**](https://sandboxphysics.com/stories/conformal-bootstrap/reading-the-tasi-bounds/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)