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arXiv:2609.40109

Probing Collective and Individual Kondo Screening: Multi-Stage, Multi-Channel Kondo Effects in a $C_3$-Symmetric Four-Impurity Model

Danqing Hu, Jiangfan Wang, Zixiang Hu, Yi-feng Yang

Published Sep 30, 2026• 0 citations
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Abstract

Collective screening has been proposed to underlie the basic physics of multi-impurity and lattice Kondo systems. But how to establish this picture and distinguish it from individual (local) Kondo screening remains a grand challenge. The recently developed auxiliary-bath numerical renormalization group (AUNRG) method provides a key step towards resolving this issue. Its application to the $C_3$-symmetric three-impurity Kondo (3IK) model with a shared electron bath reveals fully screened Fermi liquid ground states arising from collective screening of cluster spin degrees of freedom at small Kondo coupling $J_{\rm K}$ and individual (local) screening of local impurities at large $J_{\rm K}$. Here we design a four-impurity Kondo (4IK) model where the probe impurity couples only to the original three Kondo impurities to detect the nature of the screened states. We show that the collective and individual Kondo screenings give rise to emergent multi-stage, two-channel Kondo effect and unstable three-channel Kondo effect, respectively. This suggests a special helicity structure of the screened states and confirms the idea of collective screening in multi-impurity Kondo systems. Our work also demonstrates that the same auxiliary-bath construction can be readily extended to models with additional impurities to explore novel many-body quantum states with emergent cluster degrees of freedom.

DOI
10.48550/arxiv.2609.40109

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Probing Collective and Individual Kondo Screening: Multi-Stage, Multi-Channel Kondo Effects in a C3C_3C3​-Symmetric Four-Impurity Model
Collective screening has been proposed to underlie the basic physics of multi-impurity and lattice Kondo systems. But how to establish this picture and distinguish it from individual (local) Kondo screening remains a grand challenge. The recently developed auxiliary-bath numerical renormalization group (AUNRG) method provides a key step towards resolving this issue. Its application to the C3C_3C3​-symmetric three-impurity Kondo (3IK) model with a shared electron bath reveals fully screened Fermi liquid ground states arising from collective screening of cluster spin degrees of freedom at small Kondo coupling JKJ_{\rm K}JK​ and individual (local) screening of local impurities at large JKJ_{\rm K}JK​. Here we design a four-impurity Kondo (4IK) model where the probe impurity couples only to the original three Kondo impurities to detect the nature of the screened states. We show that the collective and individual Kondo screenings give rise to emergent multi-stage, two-channel Kondo effect and unstable three-channel Kondo effect, respectively. This suggests a special helicity structure of the screened states and confirms the idea of collective screening in multi-impurity Kondo systems. Our work also demonstrates that the same auxiliary-bath construction can be readily extended to models with additional impurities to explore novel many-body quantum states with emergent cluster degrees of freedom.
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 C3C_3C3​ 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.
Kitaev's honeycomb-lattice compass model describes a spin liquid with emergent fractionalized excitations. Here, we study the physics of isolated magnetic impurities coupled to the Kitaev spin-liquid host. We reformulate this Kondo-type problem in terms of a many-state quantum impurity coupled to a multichannel bath of Majorana fermions and present the numerically exact solution using Wilson's numerical renormalization group technique. Quantum phase transitions occur as a function of Kondo coupling and locally applied field. At zero field, the impurity moment is partially screened only when it binds an emergent gauge flux, while otherwise it becomes free at low temperatures. We show how Majorana degrees of freedom determine the fixed-point properties, make contact with Kondo screening in pseudogap Fermi systems, and discuss effects away from the dilute limit.
We analyze the single-channel Kondo model using the recently developed unitary renormalization group (URG) method, and obtain a comprehensive understanding of the Kondo screening cloud. The fixed-point low-energy Hamiltonian enables the computation of a plethora of thermodynamic quantities (specific heat, susceptibility, Wilson ratio, etc.) as well as spectral functions, all of which are found to be in excellent agreement with known results. By integrating out the impurity, we obtain an effective Hamiltonian for the excitations of the electrons comprising the Kondo cloud. This is found to contain both k-space number-diagonal (Fermi liquid) as well off-diagonal four-fermion scattering terms. Our conclusions are reinforced by a URG study of the two-particle entanglement and many-body correlations among members of the Kondo cloud and impurity. The entanglement between the impurity and a cloud electron, as well as between any two cloud electrons, is found to increase under flow towards the singlet ground state at the strong-coupling fixed point. Both the number-diagonal and off-diagonal correlations within the conduction cloud are also found to increase as the impurity is screened under the flow, and the latter are found to be responsible for the macroscopic entanglement of the Kondo-singlet ground state. The unitary RG flow enables an analytic computation of the phase shifts suffered by the conduction electrons at the strong-coupling fixed point. This reveals an orthogonality catastrophe between the local moment and strong-coupling ground states, and is related to a change in the Luttinger volume of the conduction bath. Our results offer fresh insight on the nature of the emergent many-particle entanglement within the Kondo cloud, and pave the way for further investigations in more exotic contexts such as the fixed point of the over-screened multi-channel Kondo problem.
We analyze the single-channel Kondo model using the recently developed unitary renormalization-group (URG) method and obtain a comprehensive understanding of the Kondo screening cloud. The fixed-point low-energy Hamiltonian enables the computation of a plethora of thermodynamic quantities (specific heat, susceptibility, Wilson ratio, etc.) as well as spectral functions, all of which are found to be in good agreement with known results. By integrating out the impurity, we obtain an effective Hamiltonian for the excitations of the electrons comprising the Kondo cloud. This is found to contain both kkk-space number diagonal (Fermi liquid) and off-diagonal four-fermion scattering terms. Our conclusions are reinforced by a URG study of the two-particle entanglement and many-body correlations among members of the Kondo cloud and impurity. The entanglement between the impurity and a cloud electron, as well as between any two cloud electrons, is found to increase under flow towards the singlet ground state at the strong-coupling fixed point. Both the number diagonal and off-diagonal correlations within the conduction cloud are also found to increase as the impurity is screened under the flow, and the latter are found to be responsible for the macroscopic entanglement of the Kondo-singlet ground state. The URG flow enables an analytic computation of the phase shifts suffered by the conduction electrons at the strong-coupling fixed point. This reveals an orthogonality catastrophe between the local-moment and strong-coupling ground states and is related to a change in the Luttinger volume of the conduction bath under the crossover to strong coupling. Our results offer fresh insight into the nature of the emergent many-particle entanglement within the Kondo cloud and pave the way for further investigations in more exotic contexts such as the fixed point of the overscreened multichannel Kondo problem.
The Kondo effect is a hallmark of strongly-correlated systems, where an impurity's local degrees of freedom are screened by conduction electrons, forming a many-body singlet. With increasing degrees of freedom in the impurity, theoretical studies face significant challenges in accurately identifying and characterizing the underlying mechanisms that screen the impurity. In this work, we introduce a straightforward yet powerful methodology for identifying the formation of Kondo singlets and their screening mechanisms, by utilizing the numerical renormalization group (NRG) combined with the matrix product states (MPS) technique. We demonstrate the effectiveness of our method on the single and two-level Anderson impurity models (AIM). Furthermore, we discuss potential generalizations of the method to multichannel and multiorbital Kondo impurities. Harnessing advanced tensor network techniques, our approach extends to complex impurity systems, offering a robust and versatile framework for studying Kondo physics.
Geometrically frustrated quantum impurities coupled to metallic leads have been shown to exhibit rich behavior with a quantum phase transition separating Kondo screened and local moment phases. Frustration in the quantum impurity can alternatively be introduced via Kitaev couplings between different spins of the impurity cluster. We use the numerical renormalization group (NRG) to study a range of systems where the quantum impurity comprising a Kitaev cluster is coupled to a bath of noninteracting fermions. The models exhibit a competition between Kitaev and Kondo dominated physics depending on whether the Kitaev couplings are greater or less than the Kondo temperature. We characterize the ground-state properties of the system and determine the temperature dependence of the crossover scale for the emergence of fractionalized degrees of freedom in the model. We also demonstrate qualitatively as well as quantitatively that in the Kondo limit, the complex impurity can be mapped to an effective two-impurity system, where the emergent spin 1/21/21/2 comprises of both Majorana and flux degrees of freedom. For a tetrahedral-shaped Kitaev cluster, an extra orbital degree of freedom closely related to a flux degree of freedom remains unscreened even in the presence of both Heisenberg and Kondo interactions.