1 Wilson’s Numerical Renormalization Group

The idea of the numerical renormalization group (NRG) for a quantum-mechanical system with Hamiltonian H is to obtain the many-body eigen-states and eigenvalues on all energy scales ω 1 > ω 2 >. .. in a sequence of steps, with each step corresponding to a distinct energy or length scale [1]. This is achieved by a formal procedure of tracing out high energy states to (RG) transformation R relates effective Hamiltonians describing the physics on successive energy scales : H N +1 = R[H N ]. Although the idea is straightforward , its practical implementation turns out to be a difficult task, especially if one wants the RG transformation to give the eigenstates and eigenvalues of H accurately down to the lowest energies. A complete implementation of the method for a quantum mechanical problem was given by Wilson, who succeeded in constructing a non-perturbative RG transformation for the Kondo model valid on all energy scales [1]. This gave the first convincing calculation of the crossover from the high temperature weak coupling regime of the Kondo model to the low temperature strong coupling regime. The same transformation has since been applied to a range of similar quantum impurity models with equal success [2]. On the other hand, it has proven more difficult to develop similar RG transformations for quantum lattice models. For example, calculations using an approximate RG transformation for the one-dimensional Hubbard model [3] showed that the spectra became inaccurate after 4 RG iterations, corresponding to a maximum system size of 16 lattice sites, clearly not sufficient to access the lowest energies of interest. These difficulties with the real space approaches provided the motivation for the development of the density matrix renormalization group (DMRG) [4,5] described in this book. In this chapter we describe a specific implementation of the NRG idea, Wilson's non-perturbative NRG method for the Kondo model [1]. This may serve as a useful background for the later chapters on the DMRG. A detailed description of the formalism of the NRG, the analysis of fixed points and full details on the calculation of thermodynamic quantities can be found in [1,2]. We also describe the calculation of dynamic and transport properties of quantum impurity models via the NRG. The outline is as follows: Sect. 1 introduces the Anderson and Kondo models [6], which are used in subsequent sections to illustrate various aspects of the NRG. The basic ideas of …