Numerical Renormalization Group method
The Numerical Renormalization Group (NRG) method is a non-perturbative numerical method which can be applied to various quantum impurity problems. Quantum impurity problems are defined as problems, where a small system with few degrees of freedom (the "impurity") interacts with a very large system with very many degrees of freedom (the "bath"). Both subsystems, impurity and bath (which can consist of either fermionic particles, bosonic particles or a mixture of both of them), are treated quantum mechanically. The method was first designed to calculate "static" thermodynamic quantities in the limit of zero temperature, such as expectation values for certain quantum mechanical operators or specific heats for these systems. Within the last decades it has been extended to calculate "dynamic" quantities like spectral functions at finite temperatures or the real-time evolution of impurity magnetizations after suddenly switching a magnetic field to name only a few. In our group the method is used for studying quantum transport in nanostructures and serves as an impurity solver for the dynamical mean-field theory (DMFT) in various projects in ultracold gases.
It was invented in the early 1970ies by Kenneth G. Wilson [1] and is still widely used due to the revived interest in the Kondo effect [2] due to technological advances in recent years. A complete presentation of the method, its strengths, weaknesses and all its applications is far beyond the scope of this short article, but we refer the interested reader to a recent review article [3]. We will here only outline the main four steps characteristic for all flavors of the numerical renormal group method.
Reducing the problem to one dimension
In general, one has to take into account that the quantum mechanical eigenstates of the bath particles and their coupling to the impurity subsystem depend on the full three dimensional momentum vector k. Since this would lead to a very complicated problem, which is hard to deal with numerically, the problem is simplified to a one dimensional one, letting the eigenenergies of the free bath particles as well as the coupling between bath and impurity subsystem depend on the modulus of the momentum vector k=|k|. This is by the way equivalent to expand the scattering cross section in spherical harmonics and only considering the dominant contribution of the s-wave eigenstates.
Logarithmic discretization of the "bath" continuum
In order to deal with the problem numerically we have to choose a finite number of quantum mechanical eigenstates in the bath continuum. A smart choice for quantum impurity problems, where so called "infrared divergencies" occur, is to discretize the bath logarithmically introducing a discretization parameter <math> \Lambda \approx 2</math>. In each of the intervals <math> [\Lambda^{-n},\Lambda^{-(n+1)}] </math> set up by this discretization only a single eigenstate is chosen. This is a crucial simplification within the NRG method, but it proved to be sufficient to stick to a single eigenstate in each of these intervals, since the intervals themselves are chosen such that in the low-energy sector (i.e. close to the Fermi edge for fermionic baths) more states are taken into account, resulting in a better energy resolution for the physically more relevant regime.
Mapping the model to a semi-infinite Wilson chain
The next step within the numerical renormalization group method is to map the problem to a semi-infinite chain, which is commonly referred to as the "Wilson chain". What does this mean? Although we have simplified the problem by a great amount, the impurity subsystem is still directly coupled to all bath degrees of freedom, as depicted in the left part of the figure to the right (where the green rectangle stands for the impurity part and the blue circles represent the bath degrees of freedom). The goal is now to find a smart eigenbasis describing the bath particles such that the impurity only couples to a single bath degree of freedom (or quasiparticle, if you prefer). Such a basis can be found easily, but once we apply the unitary transformation to the original basis, we end up with a new set of bath eigenstates which are coupled with each other. Choosing the new eigenbasis such that the bath eigenstates are only coupled to their next neighbors, we end up with a Hamiltonian, which can be illustrated by a semi-infinite chain of particles the Wilson chain (see right part of the figure).
Iterative diagonalization
We have not yet commented on a very important aspect of the steps performed so far: the eigenenergies of the quantum mechanical states of the Wilson chain decrease with increasing length of the Wilson chain. Why is this detail so important? It implies that excitations of the full system with relatively high excitation energies can be determined to very good accuracy using a chain of finite length. Since excitations with a different eigenenergies are "connected" to Wilson chains of different lengths, one refers to this as having achieved "energy scale separation".
The separation of different energy scales helps solving the problem numerically. We can write the Hamiltonian of the full (semi-infinite) Wilson chain as a series of Hamiltonians each describing a finite Wilson chain of fixed length N. The problem can now be solved iteratively by first setting up a matrix describing the impurity degrees of freedom, diagonalizing the resulting matrix and adding the first "bath" degree of freedom. One thereby enlarges the size of the matrix by a fixed factor (which is "4" in case of a fermionic bath of particles considering their spin). This matrix can now be diagonalized again and one is free to iterate this procedure, until the size of the matrices becomes too large to handle.
Now the "trick" is to only consider a fixed number of eigenstates with lowest eigenenergy in this iteration and "discard" the rest. The size of the matrices one has to diagonalize thus becomes feasible again and since the correction the "discarded" eigenstates would have got, if we would have considered them further is small (because of the energy scale separation), the error made by this "truncation" of high-energy eigenstates is not too bad
References
[1] Kenneth G. Wilson, "The renormalization group: Critical phenomena and the Kondo problem", Rev. Mod. Phys. 47, 773-840 (1975)
[2] Leo Kouwenhoven, Leonid Glazman "Revival of the Kondo effect", Physics World 14, 33-38 (2001)
[3] Ralf Bulla, Theo A. Costi, Thomas Pruschke, "Numerical renormalization group method for quantum impurity systems", Rev. Mod. Phys. 80, 395–450 (2008)
