On an exact criterion for choosing the hopping operator in the four-slave-boson approach

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 $z_{i\g}$. By suitably reorganizing the Fock space, we show that the square-root form for $z\is$ (originally introduced by Kotliar and Ruckenstein) reproduces the exact independent-fermion ($U=0$) results not only at the mean-field ($N=\infty$) level but also to {\evi all orders in the $1/N$ expansion}, provided one relaxes the usually adopted normal-ordered prescription for $z\is$. This ensures that $z\is$ needs 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 $z\is$, which might be also generalized to alternative slave-boson formalisms (like the spin-rotation-invariant formulation).