Collective Charge Excitations of Strongly Correlated Electrons, Vertex Corrections and Gauge Invariance
Résumé
We consider the collective, long wavelength charge excitations in correlated media in presence of short and long range forces. As an example for the case of a short range interaction, we examine the two-dimensional Hubbard model within dynamical mean-field theory (DMFT). It is shown explicitly that the DMFT susceptibility including vertex corrections respects the Ward identity and yields a manifestly gauge invariant response in finite dimensions. For computing the susceptibility, we use a different expression and establish its formal equivalence to the standard DMFT formula. It allows for a more stable analytical continuation. We find a zero-sound mode expected for short range forces. The relation between the vertex corrections, gauge invariance and the appearance of the collective modes is discussed. Long range forces are treated within extended dynamical mean-field theory. In order to obtain a gauge invariant response it is necessary to additionally incorporate some non-local vertex corrections into the polarization. In doing so, we obtain plasmons in the three-dimensional Hubbard model. The plasma frequency is determined by the (single-particle) density distribution as a consequence of gauge invariance. We compare this result with the plasma frequency extracted from the analytical continuation of the susceptibility. It is in good agreement with the prediction from the gauge invariance condition.