Low- temperature properties of the Kondo Hamiltonian

Abstract

The equilibrium properties of a magnetic impurity in a metal are discussed in the long-time approximation introduced by Nozières and de Dominicis for the x-ray threshold problem. It is shown that the free energy satisfies an exact homogeneity condition, from which it is possible to display explicitly the structure of the singularities of perturbation theory in the exchange coupling $J$ as the magnetic field $H\to 0$ and the temperature $T\to 0$. The nature of the Kondo problem is made clear and it is shown that, for antiferromagnetic coupling, when $H=0\mathrm{}$ and $T=0\mathrm{}$, physical properties are expected to be nonanalytic functions of $J$. To resolve those problems, the partition function is first shown to be exactly the same as that of a spin-1/2 interacting with a certain boson field. The ground-state properties are then studied by finding a new division of the spin-boson Hamiltonian into two parts, such that perturbation theory is finite term by term. For antiferromagnetic coupling, the moment vanishes at $T=0\mathrm{}$ and for weak ferromagnetic coupling, the free moment is only slightly renormalized. The partition function is also shown to be equivalent to that of an unusual one-dimensional Ising model. For $T\ne 0$, the free energy is intensive. For $T=0\mathrm{}$, the free energy is extensive and the interaction follows an inverse-square law at large distances but is infinitely strong for neighboring spins. The method of solving the spin-boson problem is worked out explicitly for the Ising model and in this way extended to finite temperatures.