Auxiliary-Bath Numerical Renormalization Group Method and Successive Collective Screening in Multi-Impurity Kondo Systems

Abstract

We propose an auxiliary-bath algorithm for the numerical renormalization group (NRG) method to solve multi-impurity models with shared electron baths. The method allows us to disentangle the electron baths into independent Wilson chains to perform standard NRG procedures beyond the widely adopted independent bath approximation. Its application to the 2-impurity model immediately reproduces the well-known even- and odd-parity channels. For 3-impurity Kondo models, we find collective screening of cluster degrees of freedom depending on impurity configurations and clarify the false prediction of a non-Fermi liquid ground state for the $C_3$ symmetric case in previous literature due to improper treatment of disentanglement. Our work highlights the importance of nonlocal spatial correlations due to shared baths and reveals a generic picture of successive collective screening for entropy depletion that is crucial in real correlated systems. Our method greatly expands the applicability of the NRG and opens an avenue for its further development.

Figures (4)

• Figure 1: Illustration of different mapping strategies that disentangle the original shared bath to $N_A=3$ and 4 auxiliary baths for the $C_3$ symmetric 3-impurity model on a triangular lattice. The green (red) lines for $N_A=4$ indicate a positive (negative) sign of the dimensionless coupling parameter $w_{\mu p}$. Also compared are the densities of states (DOS) of the original bath and the auxiliary baths. The narrow negative $\tilde{\rho}_0(\omega)$ region (dashed line) of the auxiliary baths for $N_A=4$ is replaced by zero in practice supp. The bandwidth of the original bath is set to $D=1.5$ as the basic energy scale.
• Figure 2: Auxiliary-bath NRG results for the 3-impurity model with $C_3$ symmetry. (a) Comparison of the impurity entropy $S_{{\rm imp}}$, the spin susceptibility $\mu_{\rm eff}\equiv T\chi$, and the spin correlation function $\langle\boldsymbol{S}_{1}\boldsymbol{\cdot S}_{2}\rangle$ as functions of the temperature $T$ for $\alpha=0.05$, 0.1, 0.2. Also presented are the local ($\mu=\nu$) and nonlocal ($\mu\neq\nu$) spectra of the original bath at $T=10^{-10}$ and $10^{-6}$. The insets show the features around zero energy in an enlarged plot. (b) Evolution of $\langle\boldsymbol{S}_{1}\boldsymbol{\cdot S}_{2}\rangle$ and its derivative as a function of $\alpha$ at $T=10^{-10}$. (c) Comparison of the entropy and spin susceptibility for $\alpha=0.05$ using two different mapping strategies with $N_A=3$ and 4.
• Figure 3:
• Figure 4: Schematic plot of the multi-impurity states evolving successively from free local moments, to cluster spin states, to collective Kondo screening for moderate hybridization, and to individual screening at strong hybridization limit.