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Oxford University Press eBooks

Richardson-Gaudin Exact Solution

Monique Combescot, Shiue-Yuan Shiau

Published Dec 1, 2015• 0 citations
Publisher

Abstract

Chapter 10 deals with the exact solution to the BCS problem, as obtained by analytically solving Richardson-Gaudin equations. The chapter first provides a simple derivation of these equations and shows how to solve them for arbitrary N. The explicit N dependence of the N–Cooper pair energy obtained from this resolution provides invaluable insights into the role of the Pauli exclusion principle in paired fermion systems because the reduced BCS potential does not directly act between Cooper pairs. Revelation of this subtle point makes it possible to understand (i) why Cooper pairs can strongly overlap without breaking, whereas, at large density, an exciton system dissociates into an electron-hole plasma, and (ii) why the gap energy is far larger than the average pair energy. This understanding is further supported by the Cooper pair many-body parameter, which has the same form as the ones for Wannier or Frenkel excitons do.

DOI
10.1093/acprof:oso/9780198753735.003.0010

Related papers

Energy of N Cooper pair by analytically solving Richardson-Gaudin equations

Michel Crouzeix, Monique Combescot

This Letter provides the solution to a yet unsolved basic problem of Solid State Physics: the ground state energy of an arbitrary number of Cooper pairs interacting via the Bardeen-Cooper-Schrieffer potential. We here break a 50 year old math problem by analytically solving Richardson-Gaudin equations which give the exact energy of these $N$ pairs via $N$ parameters coupled through $N$ non-linear equations. Our result fully supports the standard BCS result obtained for a pair number equal to half the number of states feeling the potential. More importantly, it shows that the interaction part of the $N$-pair energy depends on $N$ as $N(N-1)$ only from N=1 to the dense regime, a result which evidences that Cooper pairs interact via Pauli blocking only.

arXiv:1111.5801•Nov 24, 2011•0 citations

Energy of N Cooper Pairs by Analytically Solving the Richardson-Gaudin Equations for Conventional Superconductors

Michel Crouzeix, Monique Combescot

This Letter provides the solution to a yet unsolved basic problem of solid state physics: the ground state energy of an arbitrary number of Cooper pairs interacting via the Bardeen-Cooper-Schrieffer potential. We here break a 50 yr old math problem by analytically solving Richardson-Gaudin equations which give the exact energy of these N pairs via N parameters coupled through N nonlinear equations. Our result fully supports the standard BCS result obtained for a pair number equal to half the number of states feeling the potential. More importantly, it shows that the interaction part of the N-pair energy depends on N as N(N-1) only from N=1 to the dense regime, a result which evidences that Cooper pairs interact via Pauli blocking only.

Physical Review Letters•Dec 22, 2011•16 citations

Two-Cooper-pair problem and the Pauli exclusion principle

W. V. Pogosov, Monique Combescot, Michel Crouzeix

While the one-Cooper-pair problem is now a textbook exercise, the energy of two pairs of electrons with opposite spins and zero total momentum has not been derived yet, the exact handling of Pauli blocking between bound pairs being not that easy for $N=2$ already. The two-Cooper-pair problem however is quite enlightening to understand the very peculiar role played by the Pauli exclusion principle in superconductivity. Pauli blocking is known to drive the change from 1 to $N$ pairs but no precise description of this continuous change has been given so far. Using Richardson's procedure, we here prove that Pauli blocking increases the free part of the two-pair ground-state energy but decreases the binding part when compared to two isolated pairs---the excitation gap to break a pair however increasing from one to two pairs. When extrapolated to the dense BCS regime, the decrease in the pair binding while the gap increases strongly indicates that at odd with common belief, the average pair binding energy cannot be on the order of the gap.

arXiv:0911.1688•Physical Review B•May 13, 2010•22 citations

Colloquium : Exactly solvable Richardson-Gaudin models for many-body quantum systems

J. Dukelsky, Stuart Pittel, Germán Sierra

The use of exactly solvable Richardson-Gaudin models to describe the physics of systems with strong pair correlations is reviewed. The article begins with a brief discussion of Richardson's early work, which demonstrated the exact solvability of the pure pairing model, and then shows how that work has evolved recently into a much richer class of exactly solvable models. The Richardson solution leads naturally to an exact analogy between these quantum models and classical electrostatic problems in two dimensions. This analogy is then used to demonstrate formally how BCS theory emerges as the large-$N$ limit of the pure pairing Hamiltonian. Several applications to problems of relevance to condensed-matter physics, nuclear physics, and the physics of confined systems are considered. Some of the interesting effects that are discussed in the context of these exactly solvable models include (i) the crossover from superconductivity to a fluctuation-dominated regime in small metallic grains; (ii) the role of the nucleon Pauli principle in suppressing the effects of high-spin bosons in interacting boson models of nuclei, and (iii) the possibility of fragmentation in confined boson systems. Interesting insight is also provided into the origin of the superconducting phase transition both in two-dimensional electronic systems and in atomic nuclei, based on the electrostatic image of the corresponding exactly solvable quantum pairing models.

arXiv:nucl-th/0405011•Reviews of Modern Physics•Aug 6, 2004•407 citations

Excitons and Cooper Pairs

Monique Combescot, Shiue-Yuan Shiau

Abstract This book bridges the gap between two major fields in condensed matter physics: semiconductors and superconductors. Through an original perspective that their key particles, excitons and Cooper pairs, are composite bosons (cobosons), the book raises fundamental questions: how does the Pauli exclusion principle affect the fermionic components of bosonic particles at the microscopic level, and how does this appear in the coboson many-body physics at the macroscopic level? What insights would the study of Wannier excitons, Frenkel excitons, and Cooper pairs provide about the “bosonic condensation” of composite bosons and how it differs from the Bose-Einstein condensation of elementary bosons? The book starts from a solid mathematical and physical foundation to derive excitons and Cooper pairs and introduces Shiva diagrams as a graphic support to grasp the many-body physics induced by fermion exchange in the absence of fermion-fermion interaction–a mechanism not visualized in standard Feynman diagrams. The book also covers composite bosons related to excitons: trions, biexcitons, and polaritons. The last part of the book is devoted to composite boson condensation, from the prototypical Bose-Einstein condensation of free elementary bosons and the consequence of interaction, to interacting electrons in the dense regime, to electron-hole pairs in the dense and dilute regimes. Advanced undergraduate and graduate students in physics, with no prior background, will benefit from this book, as numerous appendices provide the materials required to follow the various chapters. The concepts and formalism presented should also prove useful in research on ultracold atomic gases, polariton condensation, and quantum information.

Oxford University Press eBooks•Dec 1, 2015•81 citations

Composite Bosons

Monique Combescot, Shiue-Yuan Shiau

Chapter 19 addresses bosonic condensation in composite bosons. It first considers the coboson ground-state energy, which is analytically known by now in the case of N Cooper pairs interacting via the reduced BCS potential but not yet known for N Wannier or Frenkel excitons, owing to the complexity of the Coulomb potential. Still, the chapter provides hints on how to derive the density linear term of the N-exciton ground-state energy in the case of Wannier excitons. It also discusses the role of the exciton composite nature in condensate momentum fragmentation, spin fragmentation, and condensate brightness. Indeed, the excitons that have the lowest energy and thus are the ones expected to condense are not coupled to light; so, the exciton condensate must be dark. However, a bright exciton condensate can appear, coherent with the dark exciton condensate, above a density threshold, owing to coupling induced by carrier exchange between dark excitons and bright excitons.

Oxford University Press eBooks•Dec 1, 2015•0 citations
Chapter 10 deals with the exact solution to the BCS problem, as obtained by analytically solving Richardson-Gaudin equations. The chapter first provides a simple derivation of these equations and shows how to solve them for arbitrary N. The explicit N dependence of the N–Cooper pair energy obtained from this resolution provides invaluable insights into the role of the Pauli exclusion principle in paired fermion systems because the reduced BCS potential does not directly act between Cooper pairs. Revelation of this subtle point makes it possible to understand (i) why Cooper pairs can strongly overlap without breaking, whereas, at large density, an exciton system dissociates into an electron-hole plasma, and (ii) why the gap energy is far larger than the average pair energy. This understanding is further supported by the Cooper pair many-body parameter, which has the same form as the ones for Wannier or Frenkel excitons do.
This Letter provides the solution to a yet unsolved basic problem of Solid State Physics: the ground state energy of an arbitrary number of Cooper pairs interacting via the Bardeen-Cooper-Schrieffer potential. We here break a 50 year old math problem by analytically solving Richardson-Gaudin equations which give the exact energy of these NNN pairs via NNN parameters coupled through NNN non-linear equations. Our result fully supports the standard BCS result obtained for a pair number equal to half the number of states feeling the potential. More importantly, it shows that the interaction part of the NNN-pair energy depends on NNN as N(N−1)N(N-1)N(N−1) only from N=1 to the dense regime, a result which evidences that Cooper pairs interact via Pauli blocking only.
This Letter provides the solution to a yet unsolved basic problem of solid state physics: the ground state energy of an arbitrary number of Cooper pairs interacting via the Bardeen-Cooper-Schrieffer potential. We here break a 50 yr old math problem by analytically solving Richardson-Gaudin equations which give the exact energy of these N pairs via N parameters coupled through N nonlinear equations. Our result fully supports the standard BCS result obtained for a pair number equal to half the number of states feeling the potential. More importantly, it shows that the interaction part of the N-pair energy depends on N as N(N-1) only from N=1 to the dense regime, a result which evidences that Cooper pairs interact via Pauli blocking only.
While the one-Cooper-pair problem is now a textbook exercise, the energy of two pairs of electrons with opposite spins and zero total momentum has not been derived yet, the exact handling of Pauli blocking between bound pairs being not that easy for N=2N=2N=2 already. The two-Cooper-pair problem however is quite enlightening to understand the very peculiar role played by the Pauli exclusion principle in superconductivity. Pauli blocking is known to drive the change from 1 to NNN pairs but no precise description of this continuous change has been given so far. Using Richardson's procedure, we here prove that Pauli blocking increases the free part of the two-pair ground-state energy but decreases the binding part when compared to two isolated pairs---the excitation gap to break a pair however increasing from one to two pairs. When extrapolated to the dense BCS regime, the decrease in the pair binding while the gap increases strongly indicates that at odd with common belief, the average pair binding energy cannot be on the order of the gap.
The use of exactly solvable Richardson-Gaudin models to describe the physics of systems with strong pair correlations is reviewed. The article begins with a brief discussion of Richardson's early work, which demonstrated the exact solvability of the pure pairing model, and then shows how that work has evolved recently into a much richer class of exactly solvable models. The Richardson solution leads naturally to an exact analogy between these quantum models and classical electrostatic problems in two dimensions. This analogy is then used to demonstrate formally how BCS theory emerges as the large-NNN limit of the pure pairing Hamiltonian. Several applications to problems of relevance to condensed-matter physics, nuclear physics, and the physics of confined systems are considered. Some of the interesting effects that are discussed in the context of these exactly solvable models include (i) the crossover from superconductivity to a fluctuation-dominated regime in small metallic grains; (ii) the role of the nucleon Pauli principle in suppressing the effects of high-spin bosons in interacting boson models of nuclei, and (iii) the possibility of fragmentation in confined boson systems. Interesting insight is also provided into the origin of the superconducting phase transition both in two-dimensional electronic systems and in atomic nuclei, based on the electrostatic image of the corresponding exactly solvable quantum pairing models.
Abstract This book bridges the gap between two major fields in condensed matter physics: semiconductors and superconductors. Through an original perspective that their key particles, excitons and Cooper pairs, are composite bosons (cobosons), the book raises fundamental questions: how does the Pauli exclusion principle affect the fermionic components of bosonic particles at the microscopic level, and how does this appear in the coboson many-body physics at the macroscopic level? What insights would the study of Wannier excitons, Frenkel excitons, and Cooper pairs provide about the “bosonic condensation” of composite bosons and how it differs from the Bose-Einstein condensation of elementary bosons? The book starts from a solid mathematical and physical foundation to derive excitons and Cooper pairs and introduces Shiva diagrams as a graphic support to grasp the many-body physics induced by fermion exchange in the absence of fermion-fermion interaction–a mechanism not visualized in standard Feynman diagrams. The book also covers composite bosons related to excitons: trions, biexcitons, and polaritons. The last part of the book is devoted to composite boson condensation, from the prototypical Bose-Einstein condensation of free elementary bosons and the consequence of interaction, to interacting electrons in the dense regime, to electron-hole pairs in the dense and dilute regimes. Advanced undergraduate and graduate students in physics, with no prior background, will benefit from this book, as numerous appendices provide the materials required to follow the various chapters. The concepts and formalism presented should also prove useful in research on ultracold atomic gases, polariton condensation, and quantum information.
Chapter 19 addresses bosonic condensation in composite bosons. It first considers the coboson ground-state energy, which is analytically known by now in the case of N Cooper pairs interacting via the reduced BCS potential but not yet known for N Wannier or Frenkel excitons, owing to the complexity of the Coulomb potential. Still, the chapter provides hints on how to derive the density linear term of the N-exciton ground-state energy in the case of Wannier excitons. It also discusses the role of the exciton composite nature in condensate momentum fragmentation, spin fragmentation, and condensate brightness. Indeed, the excitons that have the lowest energy and thus are the ones expected to condense are not coupled to light; so, the exciton condensate must be dark. However, a bright exciton condensate can appear, coherent with the dark exciton condensate, above a density threshold, owing to coupling induced by carrier exchange between dark excitons and bright excitons.