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.

Largest scalar gap at

\Delta_\phi=1/8

$\Delta_0^{\max}=1.0026$

2D Ising spin and energy $(\tfrac18,\,1)\ \text{allowed}$

Probe point you drag $\text{excluded}$

Evidence at the probe $\min_\Delta\hat\alpha=0$

Stress-test the bound

Derivative order

8

\Lambda

Largest spin kept

20

\ell_{\max}

Dimension range sampled

36

\Delta_{\mathrm{span}}

Drag the probe point anywhere in the plane, or pick a preset that breaks the truncation.

Model, method and sources

$\sum_{\ell\le\ell_{\max}}\sum_{\Delta}p_{\Delta,\ell}\,\vec F_{\Delta,\ell}=-\vec F_0,\quad \Delta\ge\Delta_0\ (\ell=0),\ \Delta\ge\ell$

Each crossing vector holds the derivatives of order up to the chosen limit at the symmetric point. About 120 scalar and 60 spinning dimensions per spin are sampled up to the chosen range, plus five far points. Bounds are bisected to one part in ten thousand at the Ising dimension and one in a thousand elsewhere. They were solved in advance for every slider setting with this same code, and the page replays the recorded steps. Point checks run live and add cutting-plane refinement on a finer grid.

Simmons-Duffin, TASI lectures (2016), §10 · Rychkov & Vichi (2009)

Read about the Ising kink

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.

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 misconceptionIf a truncation gives a lower bound, it is a stronger result.

Better mental modelA 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

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.
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.
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.

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

A kink, and how to tell it is real

Core mathematical model

Crossing with spin

The Ising point

Published convergence

Common difficulties

A lower bound is not always a better bound

Run the experiment

Find the kink

Break the truncation

Add derivatives

Related experiments and articles

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