Past exam of the advanced Quantum Condensed Matter Physics course of the University of Cambridge / 2015 / Paper 1 / 3 / d

Use $|j⟩$ for the six printed basis vectors. The three singly occupied triplets areEach has energy zero: either Pauli exclusion prevents hopping to double occupancy or the two contributions cancel. DefineThe state $D_{-}$ has energy $U$. In the $S,D_{+}$ subspace the Hamiltonian is $\left(\begin{array}{cc}0& -2t\\ -2t& U\end{array}\right)$, withFor $U\gg |t|>0$, the fourth low-energy eigenstate is the normalized singletwhere $\mathcal{N}=[1+(E_{-}/(2t){)}^2{]}^{-1/2}$. Its energy isThe four lowest energies are $-4t^2/U,0,0,0$ to leading order, with the singlet and triplets just given. The upper pair lie near $U$ and $U+4t^2/U$.

The low-energy Hamiltonian can be writtenusing dimensionless spin-half operators. It places the singlet below the triplet by $J$, hence favors antiferromagnetic alignment. A lattice of such bonds can support antiferromagnetism, although its ordering pattern also depends on geometry and competing couplings. The physical energy gain comes from virtual charge fluctuations, not from a permanent double occupancy in the large-$U$ limit.