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.

Change one setting and watch the probabilities

The same density can give different neighbors.

Fine-tune the lattice

Try the three settings, then tune a control. The probability bars respond immediately.

On-site interaction

U/E_*

8

Tunneling

J/E_*

1

Temperature

k_B T/E_*

0

Filling, lattice & sampling

Linear tilt

F a/E_*

0

Middle bond multiplier

J_c/J

1

Central confinement

V/E_*

0

Sites

L

4

Up-spin atoms

N_\uparrow

2

Down-spin atoms

N_\downarrow

2

Preparations per batch

M

400

Sampling seed

s

5607

Changing a physical setting prepares the atoms again and clears the data. To compare, pin a model or save a sample first.

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 spinDown spinEmpty siteSchematic · 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 snapshotsSelect 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 Mazurenko et al. · 2017 Companion experiment

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 misconceptionA few sites are enough to locate a macroscopic phase transition.

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

Calculated curves and counted atoms

Typical misconceptionThe calculated curve is reconstructed from the counted atoms.

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

Run the experiment

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.
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.
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.