Trails in Quantum Mechanics and Surroundings, Como, Italy
We consider a hierarchical version of the Kondo model, which consists in a 1-dimensional chain of spin-1/2 Fermions interacting magnetically with a single impurity. We approach the model using Wilson Renormalization Group techniques, in which we write the fields appearing in the corresponding Quantum Field Theory as a superposition of fields on different scales, and consider a hierarchical version of the model by assuming that a field on some scale does not fluctuate on smaller scales. Due to the Fermionic nature of the system, the hierarchical model is exactly solvable, in that the renormalization group flow equations are finite. In contrast to non-hierarchical models in which the flow equations are usually written as power series which only converge if the interaction is weak, the finite nature of the flow equations enables us to investigate the non-perturbative regime of the hierarchical model, i.e. when the renormalized interaction is not small. We show that if the interaction is anti-ferromagnetic, then the renormalization group flow tends to a non-Gaussian fixed point, even for very small values of the interaction. This means that the behavior of the interacting system is qualitatively different from the non-interacting (Gaussian) one, which we illustrate by showing that the 0-temperature magnetic susceptibility of the impurity, which is infinite in the non-interacting case, is finite for arbitrarily small anti-ferromagnetic interactions.
Joint work with Giuseppe Benfatto and Giovanni Gallavotti.
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