Exact criterion for choosing the hopping operator in the four-slave-boson approach

Abstract

We consider the N-component generalization of the four-slave-boson approach to the Hubbard model, where 1/N acts as the small parameter that controls the fluctuations about the saddle point, and address the problem of the appropriate choice of the bosonic hopping operator ${\mathit{z}}_{\mathit{i}\mathrm{\ensuremath{\sigma}}}$. By suitably reorganizing the Fock space, we show that the square-root form for ${\mathit{z}}_{\mathit{i},\mathrm{\ensuremath{\sigma}}}$ (originally introduced by Kotliar and Ruckenstein) reproduces the exact independent-fermion (U=0) results not only at the mean-field (N=\ensuremath{\infty}) level but also to all orders in the 1/N expansion, provided one relaxes the usually adopted normal-ordered prescription for ${\mathit{z}}_{\mathit{i},\mathrm{\ensuremath{\sigma}}}$. This ensures that ${\mathit{z}}_{\mathit{i},\mathrm{\ensuremath{\sigma}}}$ need not be modified at successive orders in the fluctuation expansion, and implies that all correlation functions are correctly recovered in the U=0 limit, a nontrivial result for the slave-boson approach. In addition, it provides a stringent requirement on the form of ${\mathit{z}}_{\mathit{i},\mathrm{\ensuremath{\sigma}}}$, which might be also generalized to alternative slave-boson formalisms (like the spin-rotation-invariant formulation).