The Kondo effect is a neat piece of physics, an archetype of a problem involving strong electronic correlations and entanglement, with a long and interesting history and connections to bulk materials, nanostructures, and important open problems.

First, some stage setting. In the late 19th century, with the development of statistical physics and the kinetic theory of gases, and the subsequent discovery of electrons by JJ Thomson, it was a natural idea to try modeling the electrons in solids as a gas, as done by Paul Drude in 1900. Being classical, the Drude model misses a lot (If all solids contain electrons, why aren’t all solids metals? Why is the specific heat of metals orders of magnitudes lower than what a classical electron gas would imply?), but it does introduce the idea of electrons as having an elastic mean free path, a typical distance traveled before scattering off something (an impurity? a defect?) into a random direction. In the Drude picture, as (T \rightarrow 0), the only thing left to scatter charge carriers is disorder ("dirt"), and the resistivity of a conductor falls monotonically and approaches (\rho_{0}), the "residual resistivity", a constant set in part by the number of defects or impurities in the material. In the semiclassical Sommerfeld model, and then later in nearly free electron model, this idea survives.

One small problem: in the 1930s (once it was much easier to cool materials down to very low temperatures), it was noticed that in many experiments (here and here, for example) the electrical resistivity of metals did not seem to fall and then saturate at some (\rho_{0}). Instead, as (T \rightarrow 0), (\rho(T)) would go through a minimum and then start increasing again, approximately like (\delta \rho(T) \propto - \ln(T/T_{0})), where (T_{0}) is some characteristic temperature scale. This is weird and problematic, especially since the logarithm formally diverges as (T \rightarrow 0).

Over time, it became clear that this phenomenon was associated with magnetic impurities, atoms that have unpaired electrons typically in (d) orbitals, implying that somehow the spin of the electrons was playing an important role in the scattering process. In 1964, Jun Kondo performed the definitive perturbative treatment of this problem, getting the (\ln T) divergence.

[Side note: many students learning physics are at least initially deeply uncomfortable with the idea of approximations (that many problems can’t be solved analytically and exactly, so we need to take limiting cases and make controlled approximations, like series expansions). What if a series somehow doesn’t converge? This is that situation.]

The Kondo problem is a particular example of a "quantum impurity problem", and it is a particular limiting case of the Anderson impurity model. Physically, what is going on here? A conduction electron from the host metal could sit on the impurity atom, matching up with the unpaired impurity electron. However (much as we can often get away with ignoring it) like charges repel, and it is energetically very expensive (modeled by some "on-site" repulsive energy (U)) to do that. Parking that conduction electron long-term is not allowed, but a virtual process can take place, whereby a conduction electron with spin opposite to the localized moment can (in a sense) pop on there and back off, or swap places with the localized electron. The Pauli principle enforces this opposed spin restriction, leading to entanglement between the local electron and the conduction electron as they form a singlet. Moreover, this process generally involves conduction electrons at the Fermi surface of the metal, so it is a strongly interacting many-body problem. As the temperature is reduced, this process becomes increasingly important, so that the impurity’s scattering cross section of conduction electrons grows as (T) falls, causing the resistivity increase.

The eventual (T = 0) ground state of this system is a many-body singlet, with the localized spin entangled with a "Kondo cloud" of conduction electrons. The roughly (\ln T) resistivity correction rolls over and saturates. There ends up being a sharp peak (resonance) in the electronic density of states right at the Fermi energy. Interestingly, this problem actually can be solved exactly and analytically (!), as was done by Natan Andrei in this paper in 1980 and reviewed here.

This might seem to be the end of the story, but the Kondo problem has a long reach! With the development of the scanning tunneling microscope, it became possible to see Kondo resonances associated with individual magnetic impurities (see here). In semiconductor quantum dot devices, if the little dot has an odd number of electrons, then it can form a Kondo resonance that spans from the source electrode through the dot and into the drain electrode. This leads to a peak in the conductance that grows and saturates as (T \rightarrow 0) because it involves forward scattering. (See here and here). The same can happen in single-molecule transistors (see here, here, here, and a review here). Zero-bias peaks in the conductance from Kondo-ish physics can be a confounding effect when looking for other physics.

Of course, one can also have a material where there isn’t a small sprinkling of magnetic impurities, but a regular lattice of spin-hosting atoms as well as conduction electrons. This can lead to heavy fermion systems, or Kondo insulators, and more exotic situations.

The depth of physics that can come out of such simple ingredients is one reason why the physics of materials is so interesting.