Numerical renormalization group
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The Numerical Renormalization Group is a method devised by Kenneth Wilson et al. to solve many-body problems. It is a numerical non-perturbative procedure, which was used in 1975 to solve the Kondo model for low temperature[1]. It was applied to the Anderson model by Krishnamurthy et al.[2] in 1980.
It is an iterative procedure, which is an example of a Renormalization group technique.
The technique consists of first dividing the conduction band into logarithmic intervals (i.e. intervals which get smaller exponentially as you move closer to the Fermi energy). One conduction band state from each interval is retained, this being the totally symmetric combination of all the states in that interval. The conduction band has now been "logarithmically discretized". The Hamiltonian is now in a position to be transformed into so-called linear chain form, in which the impurity is coupled to only one conduction band state, which is coupled to one other conduction band state and so on. Crucially, these couplings decrease exponentially along the chain, so that, even though the transformed Hamiltonian is for an infinite chain, one can consider a chain of finite length and still obtain useful results.
Once the Hamiltonian is in linear chain form, one can begin the iterative process. First the isolated impurity is considered, which will have some characteristic set of energy levels. One then considers adding the first conduction band orbital to the chain. This causes a splitting in the energy levels for the isolated impurity. One then considers the effect of adding further orbitals along the chain, doing which splits the hitherto derived energy levels further. Because the couplings decrease along the chain, the successive splittings caused by adding orbitals to the chain decrease.
When a particular number of orbitals have been added to the chain, we have a set of energy levels for that finite chain. This is obviously not the true set of energy levels for the infinite chain, but it is a good approximation to the true set in the temperature range where: the further splittings caused by adding more orbitals is negligible, and we have enough orbitals in the chain to account for splittings which are relevant in this temperature range. The results of this is that the results derived for a chain of any particular length are valid only in a particular temperature range, a range which moves to lower temperatures as the chain length increases. This means that by considering the results at many different chain lengths, one can build up a picture of the behavior of the system over a wide temperature range.
The Hamiltonian for a linear chain of finite length is an example of an effective Hamiltonian. It is not the full Hamiltonian of the infinite linear chain system, but in a certain temperature range it gives similar results to the full Hamiltonian.