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The Kondo effect is the anomalous increase of resistivity at low temperatures in metals containing dilute magnetic impurities. Instead of the expected monotonic decrease (phonon scattering diminishes as T falls), the resistivity reaches a minimum and then rises logarithmically: rho ~ rho_0 - c ln(T/T_K), where T_K is the Kondo temperature. Below T_K, the impurity spin is screened by a cloud of conduction electrons forming a many-body singlet state, and the impurity behaves as a strong (unitary) scatterer. The Kondo problem was the first example in condensed matter of a renormalization group flow between weak-coupling and strong-coupling fixed points, solved exactly by Wilson's numerical RG (1975).
The Kondo effect has a remarkable history. In the 1930s, experimentalists noticed that some metals showed an unexpected resistivity minimum at low temperatures: instead of the expected monotonic decrease from phonon freezeout, the resistivity turned upward below ~10-30 K. The effect was traced to dilute magnetic impurities (a few ppm of iron in gold, for example), but its theoretical explanation eluded physicists for thirty years. In 1964, Jun Kondo showed that third-order perturbation theory in the exchange coupling J between the impurity spin and conduction electrons produces a logarithmic correction: delta rho proportional to J3 N(0)2 ln(k_BT/D), which diverges as T goes to 0 — explaining the resistivity upturn but also signaling the breakdown of perturbation theory.
The resolution came from Kenneth Wilson's numerical renormalization group (1975), which mapped the Kondo problem onto an equivalent one-dimensional chain that could be solved iteratively by keeping only the lowest-energy states at each step. Wilson showed that the physics crosses over smoothly between two limits. Above the Kondo temperature T_K = D exp(-1/JN(0)), the impurity spin is essentially free: it contributes a Curie susceptibility chi proportional to 1/T and scatters conduction electrons weakly. Below T_K, the conduction electrons form a many-body singlet state with the impurity spin — a "Kondo cloud" of radius xi_K ~ hbar v_F/k_BT_K that collectively screens the impurity moment to zero.
The screened impurity at T << T_K is a remarkable object. It has no magnetic moment (the susceptibility becomes Pauli-like), but it scatters conduction electrons at the maximum possible rate — the unitarity limit. The impurity behaves as an infinitely strong potential scatterer, contributing a residual resistivity proportional to sin^2(delta_0)/E_F where delta_0 = pi/2 (the phase shift is maximal). The crossover from free spin to screened singlet is completely smooth — no phase transition occurs — and is captured by a single energy scale T_K.
The Kondo effect has become a paradigm for strong-coupling many-body physics. Its mathematical structure — a logarithmic divergence in perturbation theory leading to a non-perturbative energy scale T_K — parallels the BCS problem and asymptotic freedom in QCD. The Kondo effect extends far beyond dilute impurities: Kondo lattice systems (where every site carries a magnetic moment, as in heavy-fermion compounds) are among the most complex many-body systems in condensed matter. And in the quantum dot context, a single quantum dot connected to leads acts as an artificial magnetic impurity, allowing the Kondo effect to be studied with unprecedented control — tuning T_K with gate voltages and directly observing the Kondo resonance in the differential conductance.