**1. Introduction.-**

A case of great interest is the impurity problem. The impurity problem refers to the physics of a magnetic atom in a non-magnetic host. If the host is metallic, the flow of electrons hybridizing with the atomic states lead to an effective flip of the magnetic moment of the atom. If the temperature is very low, the flips become coherent. The phase of the wavefunction is well defined and all electrons become correlated. The ground state of the system is a humongously unique wave function. This is the Kondo effect. The STM gives unprecedented insight into the Kondo effect by studying the conductance between a metallic tip position a few Ångströms away from the impurity and the holding substrate. Recently, we have shown that a group of impurities that individually cannot correlate the substrate’s electrons, when they entangle quantally, they create the Kondo many-body ground state.

**2. The Kondo effect.-**

Jun Kondo computed the resistivity due to scattering off magnetic impurities. These calculations proved that magnetic impurities were responsible for the resistivity minimum and they further showed that coherent spin-flip scattering was the physical process at work. As temperature was reduced, all excitations in the solid were quenched except the ones due to spin-flip scattering that gave rise to an increased scattering rate of conducting electrons. Due to the interaction between the magnetic moments of the electron and the impurity, the spin of an electron can flip at the same time as the magnetic moment of the impurity. The electron can change the magnetic moment in one unit of spin (two Bohr magnetons in magnetic moment units) and the impurity changes in minus one unit, keeping the magnetic moment constant during the electron-impurity scattering.

The calculation by Kondo was based on the s-d exchange model developed by Zener. This model considered the magnetic exchange interaction between itinerant electrons and a quantum impurity. The itinerant electrons were represented by the s-band of a free-electron-like host. The d-electrons took into account the localized spin of the impurity. The calculation by Kondo was based on perturbation theory. The result revealed that the second order of perturbation theory gave rise to a logarithmic term on the temperature. This is troublesome for the perturbation scheme because at low temperatures the second order grows as the temperature is reduced and the perturbation expansion is not converging. Nevertheless, the logarithmic behavior turned out to be correct. Kondo fit the experimental data using an expansion including logarithms giving very good agreement with experiments. In this way, he made a connection between electron scattering off impurity spins and the resistivity minimum.

Logarithmic scaling with interactions is ubiquitous in Kondo systems. They reflect complex many-body processes that cannot be solved with simple techniques. It basically implies that perturbational approaches will fail evaluating the ground state of the system because the new ground state is fundamentally different from the non-interacting one. It further means that all behavior with temperature will be smooth and slow, giving rise to cross-overs rather than phase transitions.

The new type of ground state is clearly seen by following a Renormalization-Group (RG) approach. The RG approach is based on the observation that to evaluate the ground state, higher-energy degrees of freedom can be effectively considered by a new Hamiltonian. This new Hamiltonian contains the correct ground state if the interactions are changed while removing the higher-energy states. As the higher-energy degrees of freedom are removed, the interactions are renormalized. The evolution or flow of the interactions as the number of available electronic states is reduced, gives clear information on the properties of the ground state.

The conditions to have a Kondo ground state are that the magnetic impurity contains internal degrees of freedom that are degenerate and can be switched by an external electron collision where the spin of the electron changes by 1. This is an elastic process where the electron produces a spin flip thanks to a change in the state of the impurity. Kramers theorem warranties that half-integer spin impurities will present a degenerate ground state. In the presence of a positive uniaxial magnetic anisotropy the ground state of the impurity will be +-1/2 and the spin-flip becomes possible.

**3. A physical picture of the Kondo effect:-**

Charge fluctuations play an important role in the Kondo effect. P. W. Anderson soon realized that the s-d exchange model of Yoshida and others could be explained in terms of charge fluctuations of the impurity. The occupation of the impurity level will change over time as a consequence of the hybridization with the substrate and the large intra-atomic repulsion.

If the level is very far from the Fermi energy, the charge state will be frozen and no Kondo effect is possible. But if the average charge state is not zero or one, the level occupation fluctuates between zero and one electron. During charge fluctuation the spin is initially in one state, say up. After a certain time the impurity gets empty. Then it captures another electron with spin up or down with equal probability for non-magnetic substrates. And this process is repeated over time. As a consequence, the spin on the impurity changes randomly over time.

If the average charge state is large, say 90%, then the impurity is 90% of the time occupied and for a brief 10% of the time, it is empty. An observer would see an impurity that is occupied but that changes randomly its spin due to its hybridization with the metal substrate. Effectively, this is a s-d exchange interaction: the spin on the impurity flips by flipping back a spin in the metal. P. W. Anderson showed that this effective interaction antialigns spins in the impurity and in the metal. Later on Schrieffer and Wolff set a rigorous framework to transform between the Anderson Hamiltonian (describing the impurity hybridization and charge fluctuations) and the s-d exchange or Kondo Hamiltonian. This is the cellebrated Schrieffer-Wolf transformation.