Excitons and Cooper Pairs

Two Composite Bosons in Many-Body Physics

Monique Combescot and Shiue-Yuan Shiau

Oxford Graduate Texts

Friendly introduction to two major fields of condensed matter physics.
Detailed derivations and numerous appendices.
Contains all four approaches to Cooper pairs.
Illustrated with an abundance of diagrams.

Excitons and Cooper Pairs

Two Composite Bosons in Many-Body Physics

Monique Combescot and Shiue-Yuan Shiau

Description

This book bridges a gap between two major communities of Condensed Matter Physics, Semiconductors and Superconductors, that have thrived independently. Using an original perspective that the key particles of these materials, excitons and Cooper pairs, are composite bosons, the authors raise fundamental questions of current interest: how does the Pauli exclusion principle wield its power on the fermionic components of bosonic particles at a microscopic level and how this affects their macroscopic physics? What can we learn from Wannier and Frenkel excitons and from Cooper pairs that helps us understand "bosonic condensation" of composite bosons and its difference from Bose-Einstein condensation of elementary bosons? The authors begin with a solid mathematical and physical foundation to derive excitons and Cooper pairs. They further introduce Shiva diagrams as a graphic support to grasp the many-body physics induced by fermion exchange in the absence of fermion-fermion interaction - a novel mechanism not visualized by standard Feynman diagrams. Advanced undergraduate or graduate students in physics with no specific background will benefit from this book. The developed concepts and formalism should also be useful for current research on ultracold atomic gases and exciton-polaritons, and quantum information.

Excitons and Cooper Pairs

Two Composite Bosons in Many-Body Physics

Monique Combescot and Shiue-Yuan Shiau

Oxford Graduate Texts

1 Introduction
1.1 Technical aspects
1.2 On the possible ways to draw diagrams

Part I Excitons
2 The Exciton Concept
2.1 The physical picture
2.2 Relevant Coulomb processes
2.3 Exciton-photon coupling
2.4 Many-body effects
2.5 Thermal effects
2.6 The semiconductor Hamiltonian
3 Wannier Excitons
3.1 Phenomenological approach
3.2 Microscopic derivation
3.3 One Wannier exciton
3.4 Many-body effects
4 Frenkel Excitons
4.1 Atomic states and the tight-binding approximation
4.2 The second quantization formulation
4.3 One Frenkel exciton
4.4 Spin and orbital degrees of freedom
4.5 Many-body effects
5 Elementary Bosons, Wannier Excitons, and Frenkel Excitons
5.1 The physical picture
5.2 Commutation relations and Pauli scatterings
5.3 Interaction scatterings
5.4 Closure relations
5.5 Normalization factors
5.6 Many-body parameters
5.7 Hamiltonian mean values

Part II Cooper Pairs
6 The Cooper Pair Problem
6.1 The four main approaches to BCS superconductivity
6.2 Effective attraction between two electrons
7 The Bardeen-Cooper-Schrieffer Approach
7.1 The Cooper problem
7.2 The BCS problem
7.3 The BCS approach to the BCS problem
7.4 Hamiltonian mean value
7.5 Mean value minimization
7.6 Ground-state energy
7.7 The physical meaning of condensation energy
7.8 The energy gap
8 The Bogoliubov Approach
8.1 The Bogoliubov procedure
8.2 Diagonalization of the Bogoliubov Hamiltonian
8.3 Eigenstates of the Bogoliubov Hamiltonian
8.4 The ground-state energy of the BCS Hamiltonian
8.5 The ground-state wave function of the BCS Hamiltonian
8.6 Discussion
9 The Gorkov Approach
9.1 The mean-field Hamiltonian
9.2 The Gorkov equations for T = 0
9.3 The energy gap
9.4 The Gorkov equations and the energy gap for T is not equal to 0
10 The Richardson-Gaudin Exact Solution
10.1 Commutator formalism for zero-momentum fermion pairs
10.2 One-pair eigenstates (The Cooper problem)
10.3 Two-pair eigenstates
10.4 Three-pair eigenstates
10.5 Richardson-Gaudin equations for N pairs
10.6 The analytical solution of the Richardson-Gaudin equations
10.7 Hints on the analytical resolution of the Richardson-Gaudin equations
10.8 The many-body parameter for Cooper pairs
11 Links Between Cooper Pairs and Excitons
11.1 Degrees of freedom
11.2 Potentials
11.3 One composite boson
11.4 Two composite bosons

Excitons and Cooper Pairs

Two Composite Bosons in Many-Body Physics

Monique Combescot and Shiue-Yuan Shiau

Oxford Graduate Texts

Author Information

Monique Combescot, Directeur de Recherche CNRS Emeritus, Institute de NanoSciences de Paris, Universite Pierre-et-Marie-Curie,Shiue-Yuan Shiau, Assistant Research Fellow, Department of Physics and National Center for Theoretical Sciences, National Cheng Kung University

M. Combescot is a former student of the Ecole Normale Superieure in Paris, France (Mathematics and Physics majors). In 1973, she obtained her Ph.D. in condensed matter many-body theory from the University of Paris, France (adviser Prof. Ph. Nozières). After two years of postdoc at Cornell University, USA, she returned as a research member at CNRS in Paris, permanently. Yet, she likes teaching very much -- she has been first at the "Aggregation de Physique", a competitive exam for high school teachers. For many years, she taught undergraduate courses at universities and engineering schools in Paris. Over the last two decades, she also gave various courses at the graduate level, in Paris and in many other places in the world. This book is based on one of these lectures.

Shiue-Yuan Shiau obtained B.S. in 1998 and M.S. in 2002 from National Taiwan University. He received his Ph.D in physics in 2007 from the University of Wisconsin-Madison. After postdoctoral positions in INAC/SPSM Commissariat a l'Energie Atomique de Grenoble, France and Academia Sinica, Taipei, Taiwan, he is currently assistant research fellow at National Cheng Kung University, Taiwan.

Excitons and Cooper Pairs

Two Composite Bosons in Many-Body Physics

Monique Combescot and Shiue-Yuan Shiau

Oxford Graduate Texts

Reviews and Awards

"In this book a new and fascinating light is shined on an old question. It is must reading for anyone interested in this subject." --Leon Cooper, Brown University, USA

"This new book provides a thorough and interesting account of the microscopic theories behind superconductors, excitons and Bose condensates, drawing parallels between all three areas. It promises to be a useful textbook for postgraduates as well as a stimulating monograph for current practitioners in the fields of condensed matter physics." --Derek Lee, Imperial College London, UK

"I strongly recommend this clear, accessible book gathering for the first time two aspects of pair formation in condensed matter, namely excitons in semiconductors and Cooper pairs in superconductors." --Claude Cohen-Tannoudji, Ecole Normale Supérieure, Paris, France

"Composite bosons consist of two fermions and hence do not obey the usual bosonic commutation relations. The authors here develop comprehensive techniques and diagrams to describe such systems, focusing on two important examples: particle-hole excitons in semiconductors and two-particle Cooper pairs in superconductors. Future work will doubtless illuminate the important and relatively new field of BCS-BEC crossover involving paired fermions in ultra-cold dilute atomic gases." --Alexander Fetter, Stanford University, USA

"...full of insight and clarity, combining deep theoretical insight with a demand for intuitive and comprehensible behaviours. This link between excitons and Cooper pairs brings together two fields where there is fertile ground for connection and new ideas." --Jeremy Baumberg, University of Cambridge, UK

"The book by the noted many-body theorist Monique Combescot and Shiue-Yuan Shiau presents a novel theory of interacting excitons based on the fermion character of their constituent electron and hole and a unique unification of the concepts of Wannier and Frenkel excitons and Cooper pairs." --L. J. Sham, University of California San Diego, USA

"The topic is timely and interesting - in particular the similarities and differences between exciton condensates and condensation of Cooper pairs. There are books on excitons and there are books on superconductivity - this combination is rather unique and worth reading." --Pawel Hawrylak, University of Ottawa, Canada