Numerical Renormalization Group Calculations for Impurity ...

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3. NUMERICAL RENORMALIZATION GROUP
APPROACH
3.1 Kondo problem and invention of NRG
Wilson originally developed the numerical renormalization group method (NRG)
for the solution of the Kondo problem (Wilson 1975). The history of this problem
(Hewson 1993) goes back to the 1930’s when a resistance minimum was
found at very low temperatures in seemingly pure metals (de Haas, de Bör and
van den Berg 1934). This minimum, and the strong increase of the resistance
ρ(T) upon further lowering of the temperature, has been later found to be caused
by magnetic impurities (such as iron). Kondo successfully explained the resistance
minimum within a perturbative calculation for the s-d (or Kondo) model
(Kondo 1964), a model for magnetic impurities in metals. However, Kondo’s result
implies a divergence of ρ(T) for T → 0, in contrast to the saturation found
experimentally. The numerical renormalization group method, where the concept
of poor man’s scaling (Anderson 1970) is adopted into the numerical diagonalization
procedure, succeeded to obtain many-particles spectra with extremely high
energy-resolution and to explain the finite value of resistance ρ(T) for T → 0.
The detailed strategy is discussed in the following section.
3.2 Summary of the Basic Techniques
The fact that a proper description of T → 0 limit is achieved only after thermodynamic
limit (N → ∞) is taken into account makes it difficult for the usual
numerical approaches on impurity models to pursue the T → 0 limit. For example,
substituting a continuous band with a finite set of discrete states yields
a finite size of mesh δε in energy-space, with which one can describe thermodynamics
of the continuous system only for the temperature T larger than δε. In
this sense, a given temperature T makes a criterion for discretization,
δε ≪ T. (3.1)
Assuming that an impurity couples to an electronic bath with a band-width
D, the number of degrees of freedom of the discretized system (N) is roughly
estimated as
N ∝ D δε ≫ D T . (3.2)