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arXiv:2501.10746 · Physical review. B./Physical review. B

Unraveling screening mechanisms in Kondo impurities using an NRG-MPS-based method

Lidia Stocker, Oded Zilberberg

Published Apr 1, 2026• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1103/f88w-hg3j

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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.
The Kondo effect is a paradigmatic model of strongly correlated physics, where a magnetic impurity forms a many-body singlet with a fermionic environment. The properties of this singlet depend sensitively on correlations within the environment, which are particularly pronounced in one dimension, where interactions give rise to Luttinger liquid behavior. A promising platform to study these effects is provided by cold gases of ytterbium (Yb) atoms, where an atomic impurity is coupled to an interacting one-dimensional fermionic environment via both magnetic and potential scattering. These two scattering mechanisms compete with each other, raising the question of how robust Kondo screening remains in such systems. Here, we show that potential scattering can suppress Kondo screening in one-dimensional Yb gases. Remarkably, however, Kondo physics persists in well-defined regimes determined by the interactions in the environment. Combining analytical renormalization-group methods with density-matrix renormalization-group simulations, we identify a transition from a strongly to a weakly entangled impurity as potential scattering increases. The two approaches are in excellent agreement across the parameter regimes considered. Our results provide a quantitative criterion for the emergence of Kondo screening in one-dimensional Yb gases and delineate experimentally accessible regimes for its realization in cold-atom platforms.
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.
A magnetic moment in a metal or in a quantum dot is, at low temperatures, screened by the conduction electrons through the mechanism of the Kondo effect. This gives rise to spin-spin correlations between the magnetic moment and the conduction electrons, which can have a substantial spatial extension. We study this phenomenon, the so-called Kondo cloud, by means of the density matrix renormalization group method for the case of the single-impurity Anderson model. We focus on the question whether the Kondo screening length, typically assumed to be proportional to the inverse Kondo temperature, can be extracted from the spin-spin correlations. For several mechanisms---the gate potential and a magnetic field---which destroy the Kondo effect, we investigate the behavior of the screening cloud induced by these perturbations.
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.
Quantum impurity models describe interactions between some local degrees of freedom and a continuum of non-interacting fermionic or bosonic states. The investigation of quantum impurity models is a starting point towards the understanding of more complex strongly correlated systems, but quantum impurity models also provide the description of various correlated mesoscopic structures, biological and chemical processes, atomic physics and describe phenomena such as dissipation or dephasing. Prototypes of these models are the Anderson impurity model, or the single- and multi-channel Kondo models. The numerical renormalization group method (NRG) proposed by Wilson in mid 70's has been used in its original form for a longtime as one of the most accurate and powerful methods to deal with quatum impurity problems. Recently, a number of new developments took place: First, a spectral sum-conserving density matrix NRG approach (DM-NRG) has been developed, which has also been generalized for non-Abelian symmetries. In this manual we introduce some of the basic concepts of the NRG method and present recently developed Flexible DM-NRG code. This code uses user-defined non-Abelian symmetries dynamically, computes spectral functions, expectation values of local operators for user-defined impurity models. The code can also use a uniform density of states as well as a user-defined density of states. The current version of the code assumes fermionic bath's and it uses any number of U(1), SU(2) charge SU(2) or Z(2) symmetries. The Flexible DM-NRG code can be downloaded from http://www.phy.bme.hu/~dmnrg