Fermi-Hubbard dimer operator (Jordan-Wigner encoded)

The single-band Fermi-Hubbard model on a lattice with hopping $t$ and on-site interaction $U$ is

$H = -t\sum_{\langle i,j\rangle,\sigma} \big(c^\dagger_{i\sigma} c_{j\sigma} + c^\dagger_{j\sigma} c_{i\sigma}\big) + U\sum_i n_{i\uparrow} n_{i\downarrow}, \qquad n_{i\sigma} = c^\dagger_{i\sigma}c_{i\sigma}.$

Two-site dimer. With 2 sites and 2 spins there are 4 fermionic modes $(1{\uparrow},1{\downarrow},2{\uparrow},2{\downarrow})$, mapped to 4 qubits. Using the Jordan-Wigner transform with mode ordering $1{\uparrow},1{\downarrow},2{\uparrow},2{\downarrow}$,

$c^\dagger_k = \Big(\prod_{l<k} Z_l\Big)\frac{X_k - iY_k}{2}, \qquad c_k = \Big(\prod_{l<k} Z_l\Big)\frac{X_k + iY_k}{2},$

so a hopping term $c^\dagger_k c_{k+1} + \text{h.c.}$ becomes $\tfrac12(X_kX_{k+1}+Y_kY_{k+1})$ acting on adjacent-index modes (with an extra $Z$-string factor if the modes are not adjacent in the chosen ordering), and each density term $n_{i\sigma} = \tfrac12(I - Z_{i\sigma})$ is diagonal.

Particle-number conservation. Both the hopping and interaction terms conserve total fermion number $N=\sum_{i\sigma} n_{i\sigma}$: hopping moves a fermion between sites without creating or destroying one, and the interaction term is purely diagonal in occupation. Consequently $[H, N] = 0$, which block-diagonalizes the Hamiltonian by total filling and is the symmetry sector structure any correct encoding must reproduce.

Physical content. For $U/t \to 0$ the model reduces to free fermions hopping on the lattice (band physics); for $U/t \to \infty$ double occupancy is forbidden and, at half filling, the low-energy physics maps onto the antiferromagnetic Heisenberg model via a Schrieffer-Wolff-type superexchange argument ($J_{\text{eff}} = 4t^2/U$) — directly connecting this operator to the Heisenberg XXZ operator elsewhere in this catalog. The model is believed (though not proven in 2D) to host a $d$-wave superconducting regime near intermediate coupling, which is the primary reason it remains a central target for near-term quantum simulation.