Quantum theory from small to large scales lecture notes of the Les Houches Summer School Volume 95 August 2010 1st Edition by Jurg Frohlich, Manfred Salmhofer, Vieri Mastropietro, Wojciech De Roeck, Leticia Cugliandolo 019965249X 9780199652495

Quantum theory from small to large scales lecture notes of the Les Houches Summer School Volume 95 August 2010 1st Edition by Jurg Frohlich, Manfred Salmhofer, Vieri Mastropietro, Wojciech De Roeck, Leticia Cugliandolo – Ebook PDF Instant Download/Delivery: 019965249X, 9780199652495

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ISBN 10: 019965249X

ISBN 13: 9780199652495

Author: Jurg Frohlich, Manfred Salmhofer, Vieri Mastropietro, Wojciech De Roeck, Leticia F. Cugliandolo

This book collects lecture courses and seminars given at the Les Houches Summer School 2010 on “Quantum Theory: From Small to Large Scales”. Fundamental quantum phenomena appear on all scales, from microscopic to macroscopic. Some of the pertinent questions include the onset of decoherence, the dynamics of collective modes, the influence of external randomness and the emergence of dissipative behaviour. Our understanding of such phenomena has been advanced by the study of model systems and by the derivation and analysis of effective dynamics for large systems and over long times. In this field, research in mathematical physics has regularly contributed results that were recognized as essential in the physics community. During the last few years, the key questions have been sharpened and progress on answering them has been particularly strong. This book reviews the state-of-the-art developments in this field and provides the necessary background for future studies. All chapters are written from a pedagogical perspective, making the book accessible to master and PhD students and researchers willing to enter this field.

Table of contents:

Part I: Long lectures
1 Lecture notes on quantum Brownian motion
Preface
1.1 Overview of the rigorous derivations of diffusions
1.2 Some facts on stochastic processes
1.3 Classical mechanics of a single particle
1.4 Quantum mechanics of a single particle
1.5 Random Schrödinger operators
1.6 Main result
1.7 Feynman graphs (without repetition)
1.8 Key ideas of the proof of the diffusion (Theorems 1.13 and 1.14)
1.9 Integration of general Feynman graphs
1.10 Feynman graphs with repetitions
1.11 Computation of the main term
1.12 Conclusions
1.13 Appendix: Inequalities
1.14 Appendix: Power counting for integration of Feynman diagrams
References

2 The temporal ultraviolet limit
2.1 Introduction
2.2 Motivation for the stationary phase approximation
2.3 Bounds on the stationary phase approximation
2.4 Functional integrals
2.5 A simple high temperature expansion
2.6 Appendix: Complex Gaussian integrals
References

3 Locality in quantum systems
Preface
3.1 Introduction and notation
3.2 Locality and Lieb–Robinson bounds
3.3 Correlation decay
3.4 Topological order
3.5 Fourier transforms
3.6 Quasi-adiabatic continuation
3.7 Lieb–Schultz–Mattis in higher dimensions
3.8 What is a phase?
3.9 Stability of topologically ordered phase
Acknowledgements
References

4 Entropic fluctuations in quantum statistical mechanics—an introduction
4.1 Introduction
4.2 Prologue: a thermally driven classical harmonic chain
4.3 Algebraic quantum statistical mechanics of finite systems
4.4 Entropic functionals and fluctuation relations of finite quantum systems
4.5 Open quantum systems
4.6 The thermodynamic limit and the large time limit
4.7 Fermionic systems
4.8 Appendix: Large deviations
4.9 Vitali convergence theorem
References

5 Quantum phase transitions: introduction and some open problems
Preface
5.1 Quantum criticality and scaling
5.2 Field driven QPT as Bose–Einstein condensation
5.3 Order parameter theories and perturbative renormalization group
5.4 Beyond the standard models
Acknowledgements
References

6 Cold quantum gases and Bose–Einstein condensation
6.1 Introduction
6.2 Quantum many-body systems
6.3 BEC for interacting systems
6.4 Dilute Bose gases
6.5 Dilute Bose gases in traps
Acknowledgements
References

7 SUSY statistical mechanics and random band matrices
7.1 An overview
7.2 Classical models of quantum dynamics
7.3 Introduction to random matrix ensembles and SUSY
7.4 Averaging Det(E[ε] — H)⁻¹
7.5 The density of states for GUE
7.6 Statistical mechanics, sigma models, and Goldstone modes
7.7 Hyperbolic symmetry
7.8 Phase transition for a SUSY hyperbolic sigma model
7.9 Efetov’s sigma model
7.10 Appendix: Gaussian and Grassmann integration
7.11 Appendix: Formal perturbation theory
7.12 Appendix: Bounds on Green’s functions of divergence form
Acknowledgements
References

Part II: Short lectures
8 Mass renormalization in nonrelativistic quantum electrodynamics
8.1 Mass renormalization in NR QED—overview of results
8.2 Renormalization group based on the smooth Feshbach–Schur map
Acknowledgements
References

9 Long-time diffusion for a quantum particle
9.1 Introduction
9.2 The models
9.3 Weak coupling limit
9.4 Long-time behaviour
References

10 The ground state construction of the two-dimensional Hubbard model on the honeycomb lattice
10.1 Introduction
10.2 The model and the main results
10.3 The noninteracting system
10.4 Perturbation theory and Grassmann integration
10.5 The determinant expansion
10.6 The multi-scale integration: the ultraviolet regime
10.7 The multi-scale integration: the infrared regime
10.8 Conclusions
10.9 Appendix: Dimensional estimates of the propagators
10.10 Appendix: Truncated expectations and determinants
10.11 Appendix: Symmetry properties
Acknowledgements
References

11 On transport in quantum devices
Preface
11.1 Quantum pumps
11.2 Transport statistics
References

12 Renormalization group and problem of radiation
12.1 Overview
12.2 Nonrelativistic QED
12.3 Resonances
12.4 Existence of the ground and resonance states
12.5 Generalized Pauli–Fierz transformation
12.6 Renormalization group map
12.7 A Banach space of Hamiltonians
12.8 Action of renormalization map
12.9 Renormalization group
12.10 Related results
12.11 Conclusion
12.12 Comments on literature
12.13 Appendix: Hamiltonian of the standard model
12.14 Translationally invariant Hamiltonians
12.15 Proof of Theorem 12.5
12.16 Smooth Feshbach–Schur Map
12.17 Pull-through formulae
12.18 Supplement: creation and annihilation operators
Acknowledgements
References

13 Bell inequalities

14 Universality of generalized Wigner matrices
14.1 Introduction
14.2 Local semicircle law
14.3 Universality of Gaussian divisible ensembles via Dyson Brownian motion
14.4 Green function comparison theorem
References

15 Topological insulators and superconductors

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Tags: Jurg Frohlich, Manfred Salmhofer, Vieri Mastropietro, Quantum