---
title: "Quantum Critical Ising Chain: The Same Pattern at Every Scale Interactive Lab | Sandbox Physics"
description: "Why does a critical spin chain look the same on a ring of 16 spins and a ring of 256? Because at the critical field nothing sets a length."
canonical_url: "https://sandboxphysics.com/labs/critical-chain-scaling/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/critical-chain-scaling/"
last_modified: "2026-10-30"
---

FR37 · Conformal bootstrap

# Quantum Critical Ising Chain: The Same Pattern at Every Scale

**Why does a critical spin chain look the same on a ring of 16 spins and a ring of 256?** Because at the critical field nothing sets a length. Spin correlations fall as a power of distance, so after rescaling by a fixed power of the ring size, rings of 16 and 256 spins show the same pattern. The energy levels, scaled by the ring size, settle on the numbers of the Ising conformal field theory. Away from the critical field a correlation length appears and the rings no longer match.

Compare spin correlations on Ising rings of 16 to 256 sites. At the critical field every ring shows the same rescaled pattern and the Ising CFT spectrum.

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

Model assumptions

Five periodic transverse-field Ising rings, from 16 to 256 spins, are solved exactly as free fermions. Each site is coloured by its correlation with the top spin, multiplied by the fourth root of the ring size. At the critical field this product depends only on the fraction of the ring between the two spins, so every ring shows the same colours. The rescaled energy gaps approach the Ising CFT dimensions one eighth, one and nine eighths, and the ground-state energy gives the central charge one half. Off the critical field the rings disagree. All results are computed; no measured data are shown.

Every ring on one curve at the critical point

Energy levels land on Ising CFT dimensions

More numbers and how they are computed

- **Ring used for the readouts**: $0$
- **Probe separation in sites**: $0$
- **First even level, rescaled (Ising: 1)**: $0$
- **Levels shown below 2.4**: $0$

The dashed curve in the first chart is the conformal shape, normalised once to the 256-site critical ring at half the ring; it does not change with the field. Energy gaps are divided by the critical velocity 2J at every field, so off the critical point the rescaled numbers are not dimensions.

Physics tutorial

## A critical chain has no preferred length

**Background** In the transverse-field Ising chain a field across the spins competes with the coupling that aligns neighbours. A weaker field leaves the spins ordered; a stronger one disorders them. When the two are equal the chain is critical.

**Why it matters** A critical system has no correlation length, so nothing sets a scale. The conformal bootstrap starts from that symmetry; this Lab shows it in a model solved exactly.

Start with the essentials

- **Focus question**: Why does a critical spin chain look the same on a ring of 16 spins and a ring of 256?
- **One-sentence intuition**: At the critical field, correlations fall as a power of distance, so multiplying by a fixed power of the ring size makes every ring identical. The energy levels, rescaled by the ring size, settle on the numbers of the Ising conformal field theory.

### Core mathematical model

#### Power law on a ring

$$
\langle\sigma^x_0\sigma^x_r\rangle\propto\Bigl[\tfrac{N}{\pi}\sin\tfrac{\pi r}{N}\Bigr]^{-2\Delta_\sigma},\qquad 2\Delta_\sigma=\eta=\tfrac14
$$

On a ring the distance that enters is the chord. Multiplying by $N^{1/4}$ leaves a function of $r/N$ alone.

#### Levels are scaling dimensions

$$
E_n-E_0\simeq\frac{2\pi v}{N}\,\Delta_n,\qquad \Delta_n\in\{\tfrac18,\,1,\,\tfrac98,\,2,\dots\}
$$

A critical ring of $N$ sites is a CFT on a cylinder, so each gap is a scaling dimension times $2\pi v/N$, as derived in Cardy's lectures; here $v=2J$.

#### Central charge from the ground energy

$$
E_0=Ne_\infty-\frac{\pi v c}{6N}+\dots,\qquad c=\tfrac12
$$

The same cylinder formula puts the central charge into the $1/N$ term of the ground energy. The exact ring energies give $c=0.500002$ at $N=256$.

### Common difficulties

#### Ordered rings also look alike

**Typical misconception** If correlations look similar at every size, the chain must be critical.

**Better mental model** In the ordered phase correlations approach a constant, so the raw values also agree. The test is the rescaled pattern: with the factor of the ring size included, ordered rings grow brighter with size.

### Run the experiment

1. 01

   #### Compare the rings

   Start from the critical preset and follow the probe around.

   **What to observe:** At every angle the five rings have the same colour; the curves in the first chart lie on top of each other.

2. 02

   #### Leave the critical point

   Move the field to 0.8, then to 1.2.

   **What to observe:** The rings separate: brighter outer rings when ordered, fading colour when disordered. The central charge drops to zero.

3. 03

   #### Read the spectrum

   Return to the critical field and look at the second chart.

   **What to observe:** Rescaled levels sit on the dashed lines at $\tfrac18,\ 1,\ \tfrac98$ and $2$.

Physical conclusion

A critical point is scale invariant, and its data are a short list of scaling dimensions and a central charge. The bootstrap tries to find such lists without a model.

## 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 **Why a critical chain looks the same at every scale**](https://sandboxphysics.com/stories/conformal-bootstrap/same-correlation-at-every-scale/)
2. [Lab **Quantum Spin Chain Quench** The same chain, short and out of equilibrium](https://sandboxphysics.com/labs/quantum-spin-chain-quench/)
3. [Article **Conformal symmetry reduces a critical point to a list of operators**](https://sandboxphysics.com/stories/conformal-bootstrap/conformal-symmetry-and-operators/)

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)