Numerical Renormalization Group Calculations for Impurity ...

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1. OVERVIEW
The thesis presents the results of the Numerical Renormalization Group (NRG)
approach to three impurity models centered on the issues of impurity quantum
phase transitions.
We start from introducing general concepts of quantum phase transitions
and address the relevant physical questions of the impurity models (Chapter 2).
Chapter 3 consists of the technical details of the NRG, where we discuss how the
NRG tracks down possible fixed points that govern the universal behavior of the
system at low temperature.
All the three impurity models studied show second order quantum phase transitions
and quantum critical points but the levels of understanding of each case,
particularly to the issues of quantum phase transitions, are quite different for
historical reasons. The soft-gap Anderson model (Withoff and Fradkin 1990) is
one of the most well-established cases in the contexts of impurity quantum phase
transitions and various analytic and numerical methods examined the physical
properties of the quantum critical phase as well as the stable phases on both sides
of the transition point. Our contribution is made to the former case by analyzing
the NRG many-particle spectrum of critical fixed points, with which we can see
how the impurity contribution of the thermodynamic quantities have fractional
degrees of freedom of charge and spin.
The quantum phase transition of the spin-boson model has a long history
(Leggett, Chakravarty, Dorsey, Fisher, Garg and Zwerger 1987) but most of
achievements were reached for the ohmic dissipation 1 . In the ohmic case, a delocalized
and a localized phase are separated by a Kosterlitz-Thouless transition
at the critical coupling α = 1. 2 The new development of the NRG treating the
bosonic degrees of freedom broadened the range of the parameter space to include
the sub-ohmic case and, as a result, second order phase transitions were found
for the bath exponent 0 < s < 1 (Bulla, Tong and Vojta 2003) as we discuss in
Chapter 5.
The bosonic single-impurity Anderson model (bsiAm) is a very new model and
there is no precedent work on it. Nonetheless, the NRG approach to the bsiAm
shows that the zero temperature phase diagrams are full of interesting physics
such as the enhancement or the suppression of the Bose-Einstein condensation by
the impurity and the existence of quantum critical points. The works presented
1 The s = 1 case where s is the exponent of the bath spectral function.
2 for the unbiased case of ɛ = 0.