Numerical Renormalization Group Calculations for Impurity ...
Numerical Renormalization Group Calculations for Impurity ...
Numerical Renormalization Group Calculations for Impurity ...
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Contents<br />
iii<br />
CONTENTS<br />
1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1<br />
2. Introduction to Quantum Phase Transitions . . . . . . . . . . . . . 3<br />
2.1 The scaling limit and universality . . . . . . . . . . . . . . . . . . 4<br />
2.2 Quantum phase transitions and quantum critical points . . . . . . 6<br />
2.3 <strong>Impurity</strong> quantum phase transitions . . . . . . . . . . . . . . . . . 10<br />
3. <strong>Numerical</strong> <strong>Renormalization</strong> <strong>Group</strong> Approach . . . . . . . . . . . . . 15<br />
3.1 Kondo problem and invention of NRG . . . . . . . . . . . . . . . 15<br />
3.2 Summary of the Basic Techniques . . . . . . . . . . . . . . . . . . 15<br />
3.2.1 Logarithmic discretization . . . . . . . . . . . . . . . . . . 16<br />
3.2.2 Iterative diagonalization of a semi-infinite chain . . . . . . 21<br />
3.3 Flow diagrams and Fixed points . . . . . . . . . . . . . . . . . . . 25<br />
4. Soft-Gap Anderson Model . . . . . . . . . . . . . . . . . . . . . . . . 31<br />
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31<br />
4.2 Results from perturbative RG . . . . . . . . . . . . . . . . . . . . 33<br />
4.2.1 RG near r=0 . . . . . . . . . . . . . . . . . . . . . . . . . 34<br />
4.2.2 RG near r=1/2 . . . . . . . . . . . . . . . . . . . . . . . . 34<br />
4.3 Structure of the quantum critical points . . . . . . . . . . . . . . 35<br />
4.3.1 Perturbation theory close to r = 0 . . . . . . . . . . . . . . 38<br />
4.3.2 Perturbation theory close to r = 1/2 . . . . . . . . . . . . 43<br />
5. Spin-Boson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49<br />
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49<br />
5.2 Quantum phase transitions in the sub-ohmic Spin-Boson model . 51<br />
5.2.1 Localized/Delocalized fixed points . . . . . . . . . . . . . . 53<br />
5.2.2 Quantum critical fixed points . . . . . . . . . . . . . . . . 56<br />
6. Bosonic Single-<strong>Impurity</strong> Anderson Model . . . . . . . . . . . . . . . 61<br />
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61<br />
6.2 Quantum phase transitions in the bosonic single-impurity Anderson<br />
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63<br />
6.2.1 BEC phase . . . . . . . . . . . . . . . . . . . . . . . . . . 64<br />
6.2.2 Mott phase . . . . . . . . . . . . . . . . . . . . . . . . . . 67
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