Unveiling the Kondo cloud: unitary RG study of the Kondo model Anirban Mukherjee, Abhirup Mukherjee, N. S. Vidhyadhiraja, A. Taraphder, Siddhartha LalKondo effect  impurity models  MERG  
Feb 2, 2022  PRB 105, 085119 arXiv:2111.10580

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...
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.
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The Kondo problem has a special place in the history of quantum condensed matter physics. While it was first discovered as an intriguing minimum in the resistivity of certain metallic samples at very low temperatures by people like de Haas in the 1930s, a concrete theoretical explanation was first offered by Jun Kondo in 1964 using many-body techniques. Kondo showed that the upturn in the resistivity was caused by spin-flip scattering of conduction electrons from magnetic impurities. Several landmark studies were carried out in the 1970s for the simplest setting of a single spin-1/2 magnetic impurity, and showed that the spin-flip scattering leads eventually to the binding of some of the conduction electrons together with the impurity into a spin singlet at low-energies. This emergent screening of the magnetic impurity then ensures that the other conduction electrons (i.e., those not participating in the screening) in the metallic host no longer suffer spin-flip scattering from the impurity. The conduction electrons that participate in the screening, and the formation of the singlet state, are collectively called the Kondo cloud.

It is quite remarkable that a host of very powerful analytical and numerical methods have been devoted towards understanding this problem, and several stalwarts such as Philip Anderson have participated in this journey. Indeed, the development of the numerical renormalisation group method by Kenneth Wilson in the 1970s was devoted to the study of the Kondo problem. His impressive demonstration of the ideas of scaling and renormalisation (at play in the evolution of the Kondo screening phenomenon as temperature is lowered) led to widespread acclaim for our present understanding of critical phenomena, and culminated in his winning the Nobel prize for physics in 1982. Even today, the Kondo problem retains its relevance in the form of (i) a simple model for a qubit coupled to a noisy electronic environment, and (ii) the heart of the metal-insulator transition in strongly correlated electronic systems as developed by the dynamical mean-field theory approach.

Despite such a rich history, a puzzle at the very heart of the Kondo problem has remained open till now: what is the structure of the Kondo cloud (e.g., the many- particle entanglement encoded within it), and how should we describe it in terms of an effective theory? We have provided some answers to these questions by employing an analytic non-perturbative renormalisation group method that we have developed recently. Our approach unveils how spin-flip scattering off the magnetic impurity by the conduction electrons leads to their entanglement, and the formation of the Kondo cloud. The effective Hamiltonian obtained by us has been used to compute several thermodynamic quantities, and these are found to be in very good agreement with results obtained in the past. In this way, our work opens the door to further investigations of other paradigmatic quantum impurity problems and beyond.

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