Renormalization-group study of a concentrated fluctuating-valence model

The ground state and gap properties of a spinless model of $f$ and $d$ electrons with hybridization $V$ and $f-d$ interaction $U_{\mathrm{fd}}$ are studied in one dimension using a real-space renormalizationgroup (RG) method. To understand the dependence of the RG results on the cell size chosen, we first study the problem for $U_{\mathrm{fd}}=0\mathrm{}$ and arbitrary cell size $N$. It is found that even-site cells give qualitatively incorrect results for the gap, while odd-site cells (as small as $N=3\mathrm{}$) reproduce very accurately the exact results for the gap and give a reasonable upper bound for the groundstate energy. On the basis of this result the phase diagram is studied as a function of arbitrary $V$ and $U_{\mathrm{fd}}$ using three site cells. We consider the half-filled band situation for which the ground state is found to be insulating for all values of $U_{\mathrm{fd}}\left(\ne 0\right)$. The $f$-electron occupation number as a function of $E_f$ changes continuously, confirming recent mean-field results. This result indicates that the interacting $f-d$ system does not scale to a single impurity model as argued in previous studies. Using a Jordan-Wigner transformation from spin to fermion operators, the Kondo necklace model is mapped onto a fluctuating valence model and the transition from ferromagnetic to Kondo-state behavior is numerically studied using $N=3\mathrm{}$.