Superexchange from a strong-coupling expansion

At exactly one electron per site and U much larger than the hopping magnitude, define a projector onto the subspace with no empty or doubly occupied sites. A single hop leaves that subspace by creating a neighboring hole and doublon. Consequently the projected hopping vanishes to first order. The intermediate configuration costs U in the simple uniform model, so a second hop can return to the low-energy subspace with an energy shift of order hopping squared over U.

Eliminating the higher-energy sector gives a negative second-order operator because its energy denominators are positive relative to the low-energy reference. That negative sign represents energy lowering through virtual admixture. The operators still retain fermionic signs and spin exchange; treating the process as classical random jumps loses the relative signs that distinguish the singlet from the triplet. The expansion applies to low energies and temperatures compared with the charge gap, with corrections organized in powers of hopping over interaction.

For two sites and two electrons, the spin triplet cannot couple to an on-site pair because one orbital with two electrons supports only a spin singlet. The low-energy singlet does mix with doubly occupied states and gains energy. Rotational invariance therefore fixes the projected bond interaction to an antiferromagnetic Heisenberg form plus a constant. Use dimensionless spin operators with eigenvalues one half, and count each undirected bond once; these choices fix the numerical coefficient.

The resulting exchange is four times squared hopping magnitude over U. Keeping the density offset makes the triplet bond energy zero and the singlet bond energy minus the exchange, directly matching the Hubbard reference energies. Dropping that constant changes absolute energies but not their separation. For complex spin-independent hopping the leading magnitude remains squared modulus, while spin-orbit-dependent hopping can produce anisotropic exchange. Multiple orbitals, Hund coupling and direct exchange can change the sign or structure and are absent here.

The Hubbard dimer can be diagonalized exactly in its singlet sector. Its lower singlet energy contains a square root that expands to the second-order result plus higher even powers of hopping. This exact benchmark reveals the leading error without invoking a thermodynamic phase transition. Agreement for a small hopping ratio establishes the effective bond scale, but does not solve a frustrated quantum spin lattice or establish its ordered state.

At finite doping the low-energy subspace also contains mobile holes, so a spin-only Hamiltonian is insufficient and projected hopping must remain. Even at half filling, longer virtual loops generate additional interactions at higher order. Exchange sets a useful temperature scale after division by Boltzmann constant, but an ordering temperature also depends on dimension, anisotropy and fluctuations. The two-level lab can illustrate singlet-state superposition algebra only after a basis is declared; it cannot perform the strong-coupling projection or represent an interacting electron lattice.