A new approach to the Kondo effect: sophomore's dream function and Lee-Yang zeros

Abstract

We develop an analytical approach to the Kondo problem based on an effective partition function that generalizes the sophomore's dream integral. The Kondo crossover is represented as a passage through an approximately equidistant ladder of its Lee-Yang zeros in the complex plane, rather than as a singularity on the real axis. Summation over this ladder yields the Gamma function $Γ(α+ T_K/(2 πT))$ as the universal crossover backbone, where the offset $α=1/2$ is fixed by the Friedel sum rule. The Euler-Maclaurin expansion connects the discrete zero structure to the Nozieres Fermi-liquid theory through Bernoulli numbers. A two-cutoff formula provides a compact, essentially one-parameter description of the full crossover in terms of a temperature-space Fano lineshape which is a thermal average of the energy-space Fano resonance observed in STM. Drift corrections are treated. The approach extends to $\mathrm{SU}(N)$ Coqblin-Schrieffer, underscreened and Anderson impurity models, and Kondo lattices.