Superexchange and the t–J Projection

At large positive $U$, virtual double occupancy lowers an antiferromagnetic bond singlet but not the Pauli-blocked triplet, generating $J=4t^2/U$. With doping, the same Schrieffer–Wolff transformation gives projected hopping, exchange, three-site terms, and transformed observables; discarding the latter two is an additional approximation.

Required background. Use the Hubbard model and its strong-coupling sector. Helpful background. Integrating out heavy fields supplies the general matching logic.

Schrieffer–Wolff canonical transformation

Split $H=H_U+T$ and let $P$ project onto states without double occupancy. Choose an anti-Hermitian $S=O(t/U)$ so that

$$[H_U,S]=PTQ+QTP, \qquad Q=1-P.$$

Then

$$H_{\mathrm{eff}}=Pe^SHe^{-S}P =PTP-PTQ\frac1{H_U-E_0}QTP+O(t^3/U^2).$$

At half filling $PTP=0$. On one bond, the intermediate doublon costs $U$. The triplet cannot access the symmetric doubly occupied state, while the singlet is lowered by $4t^2/U$. Hence

$$H_{J}=J\sum_{\langle ij\rangle} \left(\mathbf S_i\cdot\mathbf S_j-\frac14n_in_j\right), \qquad J=\frac{4t^2}{U}.$$

The sign and factor are independently checked by the exact two-site Hubbard spectrum.

Doped sector and operator matching

When holes are present, projected hopping survives:

$$H_{tJ}=-t\sum_{\langle ij\rangle\sigma} (\widetilde c_{i\sigma}^\dagger\widetilde c_{j\sigma}+\mathrm{h.c.}) +H_J+H_{3\mathrm s}, \qquad \widetilde c_{i\sigma}=c_{i\sigma}(1-n_{i\bar\sigma}).$$

$H_{3\mathrm s}=O(t^2/U)$ describes correlated motion through an intermediate site and is of the same formal order as exchange. It may be small for a chosen observable and doping, but cannot be omitted by power counting alone.

Every observable must also be transformed: $\mathcal O_{\mathrm{eff}}=Pe^S\mathcal Oe^{-S}P$. For example, an electron operator acquires virtual doublon components that carry spectral weight across the Hubbard gap. Matching only $H$ cannot reproduce high-energy spectra. MacDonald, Girvin, and Yoshioka systematize this expansion in MacDonald, Girvin, and Yoshioka 1988, pp. 9753–9759.

Validity

The expansion requires $t/U\ll1$ and energies well below the charge gap. Doping increases real charge fluctuations; longer-range hopping changes both projected motion and exchange paths. Near degeneracies or smaller $U$, use the original Hubbard model or retain more sectors.

Exercises

Why is the density term $-Jn_in_j/4$ necessary in the projected bond Hamiltonian?

Solution

For two occupied sites, $\mathbf S_i\cdot\mathbf S_j$ is $1/4$ in a triplet and $-3/4$ in a singlet. Subtracting $n_in_j/4$ makes the triplet shift zero and the singlet shift $-J=-4t^2/U$, matching the two-site calculation. It also vanishes when either site is empty.

References

• Allan H. MacDonald, Steven M. Girvin, and David Yoshioka, “$t/U$ Expansion for the Hubbard Model,” Physical Review B 37 (1988) 9753–9759, doi:10.1103/PhysRevB.37.9753.