---
title: "Fermi–Hubbard · Read the Spin Neighbors Interactive Lab | Sandbox Physics"
description: "Repulsion keeps two atoms off the same site, but opposite-spin neighbors can still lower their energy by briefly hopping onto one site and back."
canonical_url: "https://sandboxphysics.com/labs/fermi-hubbard-spin-correlations/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/fermi-hubbard-spin-correlations/"
last_modified: "2026-10-27"
---

Quantum matter / Fermions

# Fermi–Hubbard · Read the Spin Neighbors

**Why do neighbors prefer opposite spins?** Repulsion keeps two atoms off the same site, but opposite-spin neighbors can still lower their energy by briefly hopping onto one site and back. The Pauli principle forbids that for parallel spins. This small exchange energy favors antiferromagnetic correlations, with neighbors pointing opposite ways, and temperature works against it.

Count atoms of two spin species site by site in a small lattice. Tune the repulsion, tunneling, filling and temperature, and build up the double occupancy and connected spin correlations from complete snapshots.

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

The Lab computes exact canonical equilibrium in an open chain of two to four sites, with ideal detection of spin and atom number. Fixing the number of atoms in each spin state can create correlations by itself. The results are not a two-dimensional phase diagram or a superconductivity calculation.

Site-by-site view of the lattice

$L=4$

Up spin Down spin Empty site Schematic · one orbital per site

Each shot starts with a freshly prepared state.

Sites with two or more atoms $0\%$

Neighbors with opposite spins $0\%$

Model probabilities · averaged over every site or bond.

From prediction to recorded outcomes

## Which patterns keep appearing?

Select a row to inspect it above. Gold is the model probability; teal is its frequency in your snapshots.

Each snapshot comes from a fresh preparation

## The same state gives different outcomes each time.

Recent snapshots Select to replay

Read across a snapshot from left to right. A gap is a real empty site; two dots share an orbital. These are independent outcomes, not frames of atoms moving.

Measured site by site

## Count first. Average second.

Teal bars: recorded mean. Whiskers: one standard error. Gold marks: exact model expectation.

$M=0$ $\text{—}$ $\text{—}$

Pair count is averaged per site; the neighbor readout is the raw spin product, separate from connected covariance below. Estimates use raw rows only. Error bars describe independent preparations, not independent sites. A single preparation has no estimated standard error.

Per-site numerical values

| Site | Measured mean | Sample variance | Model mean |
| --- | --- | --- | --- |

Measured spin correlations

## What correlation survives the averaging?

$\begin{gathered}\widehat C_{ij}=\frac{1}{M-1}\sum_r\\(S_{i,r}^z-\bar S_i^z)(S_{j,r}^z-\bar S_j^z)\end{gathered}$

Coral is negative, teal positive; gray means fewer than two snapshots. Values are unbiased sample covariances. Fixed total spin creates a background correlation even at high temperature. No long-range-order claim follows from a small negative matrix.

## Keep a comparison

| Trial | Readout | Result |
| --- | --- | --- |

Latest twelve records shown. CSV retains every outcome and setting; fermion occupations list all up sites, then all down sites. Save up to six samples and select one to restore it.

Exact small system · explicit limits

$D=0$

All number configurations in the selected sector are included. There is no per-site occupation truncation beyond the fixed total boson number. The residual checks numerical diagonalization, not the accuracy of the single-band physical model.

$\begin{gathered}x_i=i-(L-1)/2\\V_i=Vx_i^2+Fa,x_i\end{gathered}$

Sites run left to right in spacing units. The middle-bond multiplier changes the central bond of four sites, or the first bond of a shorter chain. All energies use a fixed reference energy. Open boundaries; no loading ramp, higher bands, imaging point-spread function, reservoir or two-dimensional bulk limit. At zero temperature, degenerate ground states are equally mixed.

$\text{—}$

The exchange estimate appears only at half filling, with uniform bonds, no trap or tilt, and repulsion at least eight times the hopping. It is a strong-repulsion reference, not a transition temperature.

[Greiner et al. · 2002](https://doi.org/10.1038/415039a) [Mazurenko et al. · 2017](https://arxiv.org/html/1612.08436v1) [Companion experiment](https://sandboxphysics.com/labs/bose-hubbard-lattice/)

Physics tutorial

## Fermi–Hubbard · Read the Spin Neighbors

**Background** Count atoms of two spin species site by site in a small lattice. Tune the repulsion, tunneling, filling and temperature, and build up the double occupancy and connected spin correlations from complete snapshots.

**Why it matters** Prepare a few sites again and again, count the atoms each time, and the counts reveal many-body physics.

Start with the essentials

- **Focus question**: Why do neighbors prefer opposite spins?
- **One-sentence intuition**: Repulsion keeps two atoms from sharing a site. Virtual hopping then favors opposite spins on neighbors, but the exchange energy behind it is small, so temperature competes with it.

### Core mathematical model

#### Open-chain Hamiltonian

$$
\begin{gathered}H=-J\sum_{i,\sigma}(c^\dagger_{i\sigma}c_{i+1,\sigma}+\mathrm{h.c.})\\+U\sum_i n_{i\uparrow}n_{i\downarrow}+\sum_i V_i n_i\end{gathered}
$$

Each site has a single orbital. Energies are given in a fixed reference unit, so changing the hopping does not change the unit.

#### Connected spin correlation

$$
C_{ij}=\langle S_i^zS_j^z\rangle-\langle S_i^z\rangle\langle S_j^z\rangle
$$

The sample covariance subtracts the product of the mean spins. Because each spin population is fixed, even distant sites still affect each other.

#### Exchange scale

$$
J_{\mathrm{ex}}\simeq\frac{4J^2}{U}
$$

This is a reference value for strong repulsion at half filling, not an exact transition temperature at finite temperature.

### Common difficulties

#### A few sites are not an infinite system

**Typical misconception** A few sites are enough to locate a macroscopic phase transition.

**Better mental model** These exact calculations on small systems show gradual crossovers and correlations, not thermodynamic critical points.

#### Calculated curves and counted atoms

**Typical misconception** The calculated curve is reconstructed from the counted atoms.

**Better mental model** Reference curves come straight from the density matrix. The bars, means and covariances use only the raw samples.

### Run the experiment

1. 01

   #### Start with a control

   Acquire the free case and save it. Increase repulsion without changing the filling.

   **What to observe:** Compare the double occupancy and the spin-correlation matrix with the saved free case.

2. 02

   #### Break the pattern

   Warm the spins, then remove one up-spin atom.

   **What to observe:** The same repulsion need not produce the same correlations at another filling or trap.

3. 03

   #### Test a limit

   Set tunneling to zero. Compare two and four sites and export complete records.

   **What to observe:** The degenerate ground states are mixed with equal weights, so no single ordered snapshot is picked.

Physical conclusion

The Lab computes exact canonical equilibrium in an open chain of two to four sites, with ideal detection of spin and atom number. Fixing the number of atoms in each spin state can create correlations by itself. The results are not a two-dimensional phase diagram or a superconductivity calculation.

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)