A strong-coupling expansion for the Hubbard model

We reconsider the strong-coupling expansion for the Hubbard model recently introduced by Sarker and Pairault {\it et al.} By introducing slave particles that act as projection operators onto the empty, singly occupied and doubly occupied atomic states, the perturbation theory around the atomic limit distinguishes between processes that do conserve or do not conserve the total number of doubly occupied sites. This allows for a systematic $t/U$ expansion that does not break down at low temperature ($t$ being the intersite hopping amplitude and $U$ the local Coulomb repulsion). The fermionic field becomes a two-component field, which reflects the presence of the two Hubbard bands. The single-particle propagator is naturally expressed as a function of a $2 \times 2$ matrix self-energy. Furthermore, by introducing a time- and space-fluctuating spin-quantization axis in the functional integral, we can expand around a ``non-degenerate'' ground-state where each singly occupied site has a well defined spin direction (which may fluctuate in time). This formalism is used to derive the effective action of charge carriers in the lower Hubbard band to first order in $t/U$. We recover the action of the t-J model in the spin-hole coherent-state path integral. We also compare our results with those previously obtained by studying fluctuations around the large-$U$ Hartree-Fock saddle point.