Numerical Renormalization Group Approach to the Two-Impurity Kondo Hamiltonian

Anna I. Tóth

Ian K. Affleck, Andreas W. W. Ludwig, Barbara Jones

Numerical renormalization-group and conformal-field-theory work indicate that the two-impurity Kondo Hamiltonian has a non-Fermi-liquid critical point separating the Kondo-screening phase from the interimpurity singlet phase when particle-hole (P-H) symmetry is maintained. We clarify the circumstances under which this critical point occurs, pointing out that there are two types of P-H symmetry. Only one of them guarantees the occurrence of the critical point. Much of the previous numerical work was done on models with the other type of P-H symmetry. We analyze this critical point using the boundary conformal-field-theory technique. The finite-size spectrum is presented in detail and compared with about 50 energy levels obtained using the numerical renormalization group. Various Green's functions, general renormalization group behavior, and a hidden SO(7) are analyzed.

Mikito Koga, M. Matsumoto

A connection between competing Fermi and non-Fermi-liquid states is studied by a Kondo Hamiltonian for a triplet impurity, keeping cubic symmetry in mind. A uniaxial anisotropy stabilizes either state, corresponding to a local field in the two-channel Kondo model, which reflects on the local charge redistribution in the singlet-doublet configuration of the impurity. The numerical renormalization group and SU(2) group theoretical analyses show that the critical point is described by threefold degenerate terms of the two-channel Kondo Hamiltonian, one of which derives the non-Fermi-liquid state and the other two lead to the Fermi liquid.

Walter Hofstetter

Rukhsan Ul Haq

Renormalization group methods become natural for quantum many-body systems where there is an emergence of new energy scales like as Kondo scale in case of Kondo model. It was the Numerical Renormalization Group which gave the full quantitative understanding of the strong coupling fixed point of Kondo model and also of Anderson Impurity model. However, it was Andersons poor man scaling method which gave the analytical insights into the scaling behaviour of quantum impurity models. That makes poor man scaling method very important, and hence it has been used extensively in carrying out renormalization studies of quantum impurity models. This method has been generalized and also applied to other quantum many-body Hamiltonians. In this paper, we give a detailed presentation of poor man scaling method and apply it to study the scaling behaviour of Kondo model and Anderson impurity model as well. Though there are many reviews of perturbative renormalization of Kondo model, in this paper we have also presented a detailed discussion about scaling analysis of Anderson impurity model. We also present another method called flow equation method which is a generalization of poor man scaling method and has recently been applied to study the quantum many-body dynamics both in equilibrium and out-of equilibrium. We also present a numerical procedure to extract Kondo scale from flow equations of Kondo model. The importance of this procedure is that it can be applied to calculate the Kondo scale for situations where poor mans scaling method can not be applied.

Kenneth Geddes Wilson