---
title: "Quantum Spin Chain Quench Interactive Lab | Sandbox Physics"
description: "If you flip one spin in a chain, where does its influence go? The difference from an unflipped control spreads through neighboring couplings."
canonical_url: "https://sandboxphysics.com/labs/quantum-spin-chain-quench/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/quantum-spin-chain-quench/"
last_modified: "2026-10-30"
---

Q058 · Follow the effect of one flip

# Quantum Spin Chain Quench

**If you flip one spin in a chain, where does its influence go?** The difference from an unflipped control spreads through neighboring couplings. A global quench also creates correlations, so comparing the two preparations isolates the influence of the initial flip.

Choose a site to flip and compare its chain with an unflipped control. Watch the mean spins evolve, then save joint snapshots at one observation time.

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

Arrows show local mean readouts of an isolated open spin chain.

## Keep one measurement time fixed

A seeded sample is loaded on entry.

Frozen observation time: $t=1.20\,\hbar/E_0$

Up readout Down readout White dashes: model at the saved time

Rows are independent chains; whiskers show one standard error.

## Compare a local flip with a global quench

Positive Negative Gold line: live time Violet line: saved time

More shots and saved-time correlations

Each row is one simultaneous readout of an independently prepared chain. The mean and its standard error use all recorded chains at this single time.

Additional shots use the saved settings and saved observation time, even while the live scene moves. A new acquisition replaces the current sample; saved samples remain available for replay.

Correlation estimates use the joint outcomes and exclude the reference site itself. No confidence interval is assigned to these plug-in estimates.

Model, units and scientific sources

The model is an isolated open chain of four to eight spins with nearest-neighbor Ising coupling and a transverse field. Arrow lengths show local mean readouts; the saved sample stays at its labeled observation time.

$$
H(t)=-J(t)\sum_{j=1}^{L-1}X_jX_{j+1}-h(t)\sum_{j=1}^{L}Z_j
$$

Four to eight spin-half sites have open ends. Readouts are Pauli eigenvalues along the field direction; arrow lengths show their local expectations. Time is measured in the reduced Planck constant divided by the reference energy, and site spacing defines one length unit.

Every preparation starts with all sites up along the readout axis, with an optional local flip. Neighbor coupling starts at zero; a sudden switch or linear ramp sets the final drive. Midpoint Strang splitting evolves the full state vector with maximum step 0.005; step halving and independent finite-chain references check the numerical approximation.

$$
C_{ij}=\langle Z_iZ_j\rangle-\langle Z_i\rangle\langle Z_j\rangle,\qquad S_A=-\operatorname{Tr}(\rho_A\log_2\rho_A)
$$

The time map and entropy are state-vector model references. Single-basis snapshots provide the displayed magnetization and correlation estimates. Finite chains recur; the picture does not establish a thermodynamic transition or a relativistic propagation speed.

$S_A=1.22\,\mathrm{bit}$ $\Delta t\le0.005\,\hbar/E_0$

[Calabrese, Essler & Fagotti · 2012 · Eq. 1 and section 1.1](https://arxiv.org/abs/1204.3911v3)

That paper studies a periodic chain prepared in a ground state. Here the open finite chain starts in a specified product state; the shared Ising Hamiltonian motivates the comparison, and the paper’s asymptotic formulas are not used as this sample’s measurement data.

Physics tutorial

## A local disturbance and the correlations created by a global quench

**Background** Neighboring spins interact after a coupling is switched on. One preparation has a single flipped spin; the control has no flip and follows the same drive.

**Why it matters** A correlation between two spins can grow in both preparations. Comparing them tells us which change came from the local intervention.

Start with the essentials

- **Focus question**: Which sites change because of the initial flip?
- **One-sentence intuition**: The difference from the unflipped control measures the effect of the flip; connected correlation measures a different feature of the quantum state.

### Core mathematical model

#### Open-chain transverse-field Ising Hamiltonian

$$
H(t)=-J(t)\sum_{j=1}^{L-1}X_jX_{j+1}-h(t)\sum_{j=1}^{L}Z_j
$$

The neighbor coupling acts along one spin axis and the field along the perpendicular readout axis. Both preparations start as specified product states. The full state vector is evolved by midpoint Strang splitting, with a maximum time step of 0.005 in the displayed time unit.

#### Change caused by the local preparation

$$
\Delta_j(t)=\langle Z_j(t)\rangle_{\mathrm{flip}}-\langle Z_j(t)\rangle_{\mathrm{control}}
$$

The drive, chain length and observation time are the same in the two model calculations. Their difference isolates the effect of the one changed initial spin.

#### Connected correlation at one observation time

$$
C_{ij}=\langle Z_iZ_j\rangle-\langle Z_i\rangle\langle Z_j\rangle
$$

Joint outcomes supply both terms in a sampled estimate. The displayed correlation estimate uses one frozen observation time; records from different times are kept separate.

### Common difficulties

#### An arrow is an average

**Typical misconception** Each arrow traces a spin following a definite path.

**Better mental model** The arrow shows the mean readout component of the local quantum state. A single joint snapshot instead contains an up or down outcome at every site.

#### Correlation and influence differ

**Typical misconception** Any new correlation must have come from the flipped spin.

**Better mental model** Switching the entire chain can create correlations even when no spin is flipped. Compare the locally changed preparation with its matched control.

#### A saved time stays fixed

**Typical misconception** Letting the scene play adds measurements from each new time to the same mean.

**Better mental model** The saved record keeps its own preparation, observation time and seed. Additional shots use those saved settings. “Record the live time” starts a new sample from the currently displayed state.

### Run the experiment

1. 01

   #### Find how a change spreads

   Choose “A local disturbance travels”, then drag the gold ring to a different site.

   **What to observe:** The two chains start differently at that site. Their model difference subsequently develops at neighboring sites.

2. 02

   #### Separate a global quench

   Choose “Quench without a local flip” and view “Connected correlation”.

   **What to observe:** The two mean-spin chains agree, but connected correlations can still grow. The drive changed the whole chain.

3. 03

   #### Freeze one measurement time

   Pause, choose a time and select “Record the live time”. Save that sample, move the live time, then replay the saved sample.

   **What to observe:** The replay restores the original parameters and raw shots without resampling. The shot chart is compared with the model at the record’s own observation time.

4. 04

   #### Test the role of interaction

   Choose “Disconnected neighbors”, then compare with “Switch on gradually”.

   **What to observe:** Without neighbor coupling the flip affects only its original site. A gradual switch changes the driving history and the resulting spreading pattern.

Physical conclusion

Use matched preparations to identify a local influence, and use joint outcomes at one time to estimate correlations.

## Related experiments and articles

Condensed matter · Reading path

### Magnetism, disorder and topological response

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Condensed matter · Reading path

### Cold liquids and superconducting order

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

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Fields, scales and critical phenomena · Reading path

### From field modes to critical phenomena

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

1. [Article **What a correlation function predicts**](https://sandboxphysics.com/learn/field-theory/correlators-to-scattering/)
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1 more group

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 Critical Ising Chain: The Same Pattern at Every Scale** Collapse the rings](https://sandboxphysics.com/labs/critical-chain-scaling/)

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/)
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5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
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[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)