Numerical Renormalization Group Calculations for Impurity ...

4.2. Results from perturbative RG 33
symmetric SC fixed point [This “peak” can be viewed as a generalization of the
Kondo resonance in the standard case (r = 0), and scaling of this peak is observed
upon approaching the SC-LM phase boundary (Logan and Glossop 2000, Bulla,
Pruschke and Hewson 1997, Bulla et al. 2000).]
At the critical point, non-trivial behavior corresponding to a fractional moment
can be observed: S imp = k B C s (r), χ imp = C χ (r)/(k B T) with C s , C χ being
universal functions of r. The spectral functions at the quantum critical points
display an ω −r power law (for r < 1) with a remarkable “pinning” of the critical
exponent.
Apart from the static and dynamic observables described above, the NRG
provides information about the many-body excitation spectrum at each fixed
point. The non-trivial character of the quantum critical points are prominent in
this case, too. For the strong-coupling and local-moment fixed points, a detailed
understanding of the NRG levels is possible since the fixed point can be described
by non-interacting electrons. Intermediate-coupling fixed point at the quantum
critical points have a completely different NRG level structure, i.e., smaller degeneracies
and non-equidistant levels. They cannot be cast into a free-particle
description.
In this chapter, we demonstrate that a complete understanding of the NRG
many-body spectrum of critical fixed points is actually possible, by utilizing renormalized
perturbation theory around a non-interacting fixed point. In the soft-gap
Anderson model, this approach can be employed near certain values of the bath
exponent which can be identified as critical dimensions. Using the knowledge
from perturbative RG calculations, which yield the relevant coupling constants
being parametrically small near the critical dimension, we can construct the entire
quantum critical many-body spectrum from a free-Fermion model supplemented
by a small perturbation. In other words, we shall perform epsilon-expansions
to determine a complete many-body spectrum (instead of certain renormalized
couplings or observables). Conversely, our method allows us to identify relevant
degrees of freedom and their marginal couplings by carefully analyzing the NRG
spectra near critical dimensions of impurity quantum phase transitions.
This chapter is organized as follows. In Section 4.2 we summarize the recent
results from perturbative RG for both the soft-gap Anderson and Kondo models.
In Section 4.3, we discuss (i) the numerical data for the structure of the quantum
critical points and (ii) the analytical description of these interacting fixed points
close to the upper (lower) critical dimension r = 0 (r = 1/2).
4.2 Results from perturbative RG
The Anderson model (4.1) is equivalent to a Kondo model when charge fluctuations
on the impurity site are negligible. The Hamiltonian for the soft-gap Kondo