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Sandbox Physics

Q057 · Prepare / image / compare

Fermi–Hubbard Spin Correlations

Prepare two atoms of each spin. Compare repulsion, attraction and heating through joint site readings, double occupation and spin correlations.

Interactive modelFermi–Hubbard Spin Correlations
Accepted snapshots—\text{—}
Neighbor spin correlation · model—\text{—}

02 / TAKE A PICTURE

One preparation. Four sites.

Blue: spin up. Amber: spin down. Both colors: a doublon. Dots are decoded counts, not classical spin directions.

Neighbor spin correlation · recorded

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Double-occupation fraction · recorded

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03 / REPEAT TO SEE THE PATTERN

Which spins tend to oppose each other?

Select a site to inspect its count distribution.

Blue bars: recorded frequencies. Amber ticks: model probabilities. Thin bars: pointwise 95% Wilson intervals.

Longitudinal spin correlation · from the same snapshots

Blue: negative. Amber: positive. Zero: white. Select a cell for its recorded and model values.

MODEL DIAGNOSTIC · NOT RECONSTRUCTED FROM THE IMAGES

Virtual hopping favors opposite spins. The superexchange scale applies to strong positive interaction and temperatures below the charge gap.

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04 / KEEP A FAIR COMPARISON

Save. Change. Measure again.

Saving freezes this sample; the next capture starts a new one. Up to eight comparisons.

Scientific model & measurement boundaries

Four sites with open ends. Only nearest neighbors tunnel; the middle link can be weakened. The energy rises from left to right for positive tilt. One arbitrary reference energy sets every control. A complete canonical density matrix is used, including all eigenstates. At zero temperature, degenerate ground states are equally mixed.

Two fermions per spin; all 36 states in that number sector are retained. Fermionic exchange signs are included. Fixed zero total spin projection creates a correlation background even at high temperature. Negative correlations alone do not establish long-range order or entanglement.

Each trial starts from a new equilibrium preparation. Availability discards an entire image independently of its state; unread trials remain in the CSV. Ideal occupation readout differs from parity-only fluorescence microscopy. No optical point-spread function, light-assisted collisions, preparation dynamics or detector error is modeled.

The apparatus shows an enlarged four-site sample, collection objective and camera along a shared vertical axis. It illustrates readout geometry, not a calibrated optical design.

Zhang & Dong · 2011 · Cheuk et al. · 2016 · Bakr et al. · 2009

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Physics tutorial

Build correlations from atom snapshots

BackgroundFour sites expose many-body statistics without a thermodynamic-limit claim.

Why it mattersA convincing picture is not yet a correlation measurement.

Start with the essentials

Focus question
Do colder neighbors have stronger opposite-spin correlations?
One-sentence intuition
Every histogram and recorded correlation uses the same joint snapshots.

Core mathematical model

Fermi–Hubbard Hamiltonian

H=−∑i=13∑σJi(ciσ†ci+1,σ+h.c.)+U∑ini↑ni↓+∑iViniH=-\sum_{i=1}^3\sum_\sigma J_i(c_{i\sigma}^\dagger c_{i+1,\sigma}+\mathrm{h.c.})+U\sum_i n_{i\uparrow}n_{i\downarrow}+\sum_i V_i n_i

Each spin has two particles. The ordered spin orbitals include the fermionic exchange signs.

Read both spins

Siz=ni↑−ni↓2,Cijzz=⟨SizSjz⟩S_i^z=\frac{n_{i\uparrow}-n_{i\downarrow}}2,\quad C_{ij}^{zz}=\langle S_i^zS_j^z\rangle

Multiply site spin readings within each image, then average across fresh images. A doublon and an empty site both have zero spin projection.

Virtual hopping scale

Jex≃4J2UU2−F2(U−∣F∣≫J, U>0)J_{\rm ex}\simeq\frac{4J^2U}{U^2-F^2}\quad(U-|F|\gg J,\ U>0)

For the outer links, including the tilt correction. The middle-link scale is multiplied by the square of its strength factor. This is a strong-coupling interpretation, not the solver used here.

Prepare equilibrium

ρ=Z−1e−H/(kBT),Vi=F(i−5/2),J1=J3=J, J2=rJ\rho=Z^{-1}e^{-H/(k_BT)},\quad V_i=F(i-5/2),\quad J_1=J_3=J,\ J_2=rJ

Canonical ensemble in the fixed number sector. Zero temperature equally mixes degenerate ground states; changing controls prepares a new state, not a dynamical ramp.

A whole-image probability

P(n)=⟨n∣ρ∣n⟩P(\mathbf n)=\langle\mathbf n|\rho|\mathbf n\rangle

Sample a joint Fock configuration, never independent site marginals. The full image either reads successfully or is marked unread.

Common difficulties

Finite size

Typical misconceptionA four-site result is a bulk phase diagram.

Better mental modelIt is a finite-system crossover. No thermodynamic critical point, long-range order or superconductivity is inferred.

Readout contract

Typical misconceptionThe colored tiles reproduce a calibrated microscope image.

Better mental modelThey encode ideal number-resolved readings, with spin resolution for fermions. Real parity imaging, optical resolution and light-assisted loss require additional models.

Run the experiment

  1. 01

    Prepare

    Start with the first preset, then choose its contrasting partner.

    What to observe: Each preset includes 200 labeled simulated shots.
  2. 02

    Inspect

    Select a snapshot, a site and a correlation cell.

    What to observe: Single-shot configurations differ even at identical settings.
  3. 03

    Compare

    Save, adjust temperature or interaction, collect again and export.

    What to observe: The CSV preserves unread trials and captured settings.