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Numerical renormalization group
The numerical renormalization group (NRG) is a technique devised by Kenneth Wilson to solve certain many-body problems where quantum impurity physics plays a key role.
==History==
The numerical renormalization group is an inherently non-perturbative procedure, which was originally used to solve the Kondo model.〔K. Wilson, Rev. Mod. Phys. 47, 773 (1975)〕 The Kondo model is a simplified theoretical model which describes a system of magnetic spin-1/2 impurities which couple to metallic conduction electrons (e.g. iron impurities in gold). This problem is notoriously difficult to tackle theoretically, since perturbative techniques break down at low-energy. However, Wilson was able to prove for the first time using the numerical renormalization group that the ground state of the Kondo model is a singlet state. But perhaps more importantly, the notions of renormalization, fixed points, and renormalization group flow were introduced to the field of condensed matter theory — it is for this that Wilson won the Nobel Prize in 1982. The complete behaviour of the Kondo model, including both the high-temperature 'local moment' regime and the low-temperature 'strong coupling' regime are captured by the numerical renormalization group; an exponentially small energy scale TK (not accessible from straight perturbation theory) was shown to govern all properties at low-energies, with all physical observables such as resistivity, thermodynamics, dynamics etc. exhibiting universal scaling. This is a characteristic feature of many problems in condensed matter physics, and is a central theme of quantum impurity physics in particular. In the original example of the Kondo model, the impurity local moment is completely screened below TK by the conduction electrons via the celebrated Kondo effect; and one famous consequence is that such materials exhibit a resistivity minimum at low temperatures, contrary to expectations based purely on the standard phonon contribution, where the resistivity is predicted to decrease monotonically with temperature.
The very existence of local moments in real systems of course presupposes strong electron-electron correlations. The Anderson impurity model describes a quantum level with an onsite Coulomb repulsion between electrons (rather than a spin), which is tunnel-coupled to metallic conduction electrons. In the singly occupied regime of the impurity, one can derive the Kondo model from the Anderson model, but the latter contains other physics associated with charge fluctuations. The numerical renormalization group was extended to deal with the Anderson model (capturing thereby both Kondo physics and valence fluctuation physics) by H. R. Krishnamurthy ''et al.''〔H. R. Krishnamurthy, J. W. Wilkins, K. G. Wilson, Phys. Rev. B 21, 1003 (1980)〕 in 1980. Indeed, various important developments have been made since: a comprehensive modern review has been compiled by Bulla ''et al.''〔R. Bulla, T. A. Costi and T. Pruschke, Rev. Mod. Phys. 80, 395–450 (2008) 〕
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