Quantum Impurity Problems in the Framework of Natural Orbitals: A Comprehensive Study (Springer Theses)

✍ Scribed by Maxime Debertolis

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 170
• Category: Library

✦ Synopsis

This book presents a complete study of natural orbitals in quantum impurity problems, revealing a certain simplicity in these interacting many-body problems. These systems consist of a few localized degrees of freedom that undergo strong interactions and hybridize with a larger system of free particles; they are central in the study of strongly correlated systems.

In a first step, the standard non-perturbative numerical renormalization group method is employed to demonstrate the hierarchical structure of correlations unveiled by natural orbitals. This simplification brought new insights for simulating quantum impurity problems, and a new algorithm is developed to generate an optimized subset of natural orbitals independently of existing methods, going beyond their usual limitations. This algorithm is presented in detail in the book, and a careful benchmark on known results is carried out to guarantee the validity of the method. It is then used to study spatial entanglement structures under various conditions that were not accessible with previous methods, such as representing the electron bath by a realistic 2D square lattice or taking account of static disorder in the metallic host.

In the last chapter, the non-interacting problem in the presence of disorder is studied through random matrix theory, reproducing some of the results presented in the previous chapters. The main original result of this chapter lies in the analytical calculation of the joint distribution of one-particle orbitals energies and amplitudes of the impurity, which makes it possible to calculate any disordered averaged local correlation functions. Starting from this result, calculations in the large-N limit are compared with numerical simulations.

✦ Table of Contents

Supervisor's Foreword
Abstract
Publications related to this thesis
Acknowledgements
Contents
Part I Introduction
1 The Quantum Impurity Problem
1.1 The Many-Body Problem in Condensed Matter
1.2 The Kondo Effect in a Nutshell
1.3 Models of Quantum Impurities
1.3.1 Single Impurity Anderson Model
1.3.2 Atomic Limit of the SIAM
1.3.3 Schrieffer-Wolff Transformation and the Kondo Model
1.3.4 Bosonization and the Interacting Resonant Level Model
References
2 IRLM and Kondo Correlations
2.1 Non-interacting Limit upper U equals 0U=0
2.2 Numerical Renormalization Group
2.2.1 Writing the Hamiltonian on a Logarithmic Grid
2.2.2 Iterative Diagonalization of the Chain
2.2.3 Calculating Observables
2.3 Quantum Phase Transition in the IRLM
2.4 The Kondo Screening Cloud
2.5 Kondo Correlations in a Disordered Host
References
Part II Quantum Impurity Problems and Natural Orbitals
3 Few-Body Nature of Kondo Correlated Ground States
3.1 Motivations
3.2 One-Body Density Matrix
3.2.1 Single Slater Determinant
3.2.2 Simple Structure of Entanglement: Bell-Like State
3.2.3 Some General Properties of the upper QQ Matrix
3.3 IRLM and SIAM Correlation Spectra
3.4 Convergence to the Thermodynamic Limit
3.5 Few-Body Ansatz from Natural Orbitals
3.6 One-Body Picture and Spatial Dispersion of Natural Orbitals
References
4 Recursive Generation of Natural Orbitals
4.1 Motivations
4.2 Recursive Generation of Natural Orbitals (RGNO)
4.2.1 General Idea
4.2.2 Initial Guess of Correlated Orbitals
4.2.3 Iterative Diagonalization
4.2.4 Sweep Until Convergence
4.3 Benchmark on the Wilson Chain
4.4 Real Space Simulations and Kondo Screening Cloud
4.5 Two-Dimensional Screening Cloud
4.5.1 No Reduction to One-Dimensional Chains
References
Part III Quantum Impurities in a Disordered Environment
5 RGNO Study of Screening Clouds in Disordered Environments
5.1 Statistics on Disordered Quantum Impurities
5.2 Distribution of Kondo Temperatures
5.3 Effect of Disorder on Spatial Correlations
5.3.1 Screening Clouds
5.3.2 Cloud Amplitude
5.3.3 Spatial Dispersion of Natural Orbitals
5.4 Local Charge Distribution
5.5 Localization Length: One-Body Perspective
5.6 Charge and Spin Disorder in the SIAM
References
6 Random Matrix Impurity Model
6.1 Motivations
6.2 Random Matrix Resonant Level Model
6.2.1 Weak Coupling Regime: upper V divided by sigma tilde 1 divided by StartRoot 2 upper N EndRootV/σsim1/sqrt2N
6.2.2 Diluted Regime: upper V divided by sigma less than StartRoot 2 upper N EndRootV/σ< sqrt2N
6.2.3 Bound States Regime: upper V divided by sigma greater than StartRoot 2 upper N EndRootV/σ> sqrt2N
6.2.4 Participation Ratio
6.3 Toy Model for upper P left parenthesis n Subscript d Superscript Baseline right parenthesisP(nd)
6.4 Full PDF of the Perturbed Problem
6.4.1 Definition of the Eigenvalue Problem
6.4.2 Change of Basis
6.4.3 Distribution
6.5 Large upper NN Calculations of Distribution Functions
6.5.1 Simplifications for upper P left parenthesis backslash nd right parenthesisP(nd)
6.5.2 Remarks on the Weak Coupling Regime
6.5.3 upper P left parenthesis upper E 0 Superscript Baseline comma z 0 Superscript Baseline right parenthesisP(E0,z0) in the Bound States Regime
References
Part IV Conclusions and Perspectives
7 Conclusion and Perspectives
Reference
Appendix A NRG—Implementation of the IRLM
A.1 Flow of the Hamiltonian
A.2 Computing an Observable
A.3 One-Body Density Matrix
A.3.1 Annihilation Operators
A.4 Initialization
A.4.1 Hamiltonian
A.4.2 upper QQ Matrix and cc Vector
Appendix B NRG—Implementation of the SIAM
B.1 Flow of the Hamiltonian
B.2 One-Body Density Matrix
B.3 Initialization
B.3.1 Hamiltonian
B.3.2 upper QQ Matrix and cc Vector
Appendix C RGNO—Symmetry Breaking by the Few-Body Ansatz
Appendix D RGNO—Details on the Exact Diagonalization
D.1 Basis Representation and Fock Space
D.2 Sparse Representation: Masks
D.3 Construction of the Hamiltonian
D.4 One-Body Density Matrix
D.5 Variance of the Hamiltonian
Appendix E RMT—Calculating Distribution Functions in Simpler Models
E.1 Random Model Without Impurity
E.2 Random Model with Onsite Potential
References