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arXiv:cond-mat/0406045

Major difference between true bosons and "proteons"

Monique Combescot, Odile Betbeder-Matibet

Published Jun 2, 2004• 0 citations
arXivHTMLPDF

Abstract

We call "proteons" -- from the ever-changing greek sea-god $Πρωτε\upsilonς$ -- composite particles made of two fermions. Among them, are the semiconductor excitons, but also various atoms and molecules, like the giant molecules made of two $^{40}$K or $^6$Li atoms which have recently Bose condensed. In addition to their indistinguishability, these composite particles are ``ever-changing'' in the sense that there is no way to know with which fermions they are precisely made. As direct consequences, (i) the proteons are not true bosons, (ii) the basis made with proteon states is \emph{overcomplete}. In spite of these difficulties, these proteons do have a nice closure relation, unexpected at first, \emph{different from the boson one} and which makes the bosonization procedures used up to now to treat many-body effects between composite bosons, rather questionable, due to possibly incorrect sum rules resulting from it. This closure relation in particular explains, in a neat way, the surprising factor 1/2 between the inverse lifetime and the sum of scattering rates which exists for exact excitons but not for boson excitons, as we have recently shown.

DOI
10.48550/arxiv.cond-mat/0406045

Related papers

Exciton-exciton scattering: Composite boson versus elementary boson

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This paper shows the necessity of introducing a quantum object, the ``coboson,'' to properly describe, through a fermion scheme, any composite particle, such as the exciton, which is made of two fermions. Although commonly dealt with as elementary bosons, these composite bosons---cobosons in short---differ from them due to their composite nature which makes the handling of their many-body effects quite different from the existing treatments valid for elementary bosons. As a direct consequence of this composite nature, there is no correct way to describe the interaction between cobosons as a potential $V$. This is rather dramatic because, with the Hamiltonian not written as $H={H}_{0}+V$, all the usual approaches to many-body effects fail. In particular, the standard form of the Fermi golden rule, written in terms of $V$, cannot be used to obtain the transition rates of two cobosons. To get them, we have had to construct an unconventional expression for this Fermi golden rule in which $H$ only appears. Making use of this expression, we give here a detailed calculation of the time evolution of two excitons. We compare the results of this exact approach with the ones obtained by using an effective bosonic Hamiltonian in which the excitons are considered as elementary bosons with effective scatterings between them, these scatterings resulting from an elaborate mapping between the two-fermion space and the ideal boson space. We show that the relation between the inverse lifetime and the sum of the transition rates for elementary bosons differs from the one of the composite bosons by a factor of $1∕2$, so that it is impossible to find effective scatterings between bosonic excitons giving these two physical quantities correctly, whatever the mapping from composite bosons to elementary bosons is. The present paper thus constitutes a strong mathematical proof that, in spite of a widely spread belief, we cannot forget the composite nature of these cobosons, even in the extremely low-density limit of just two excitons. This paper also shows the (unexpected) cancellation in the Born approximation of the two-exciton transition rate for a finite value of the momentum transfer.

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Molecules made of two fermionic atoms are ``cobosons''---a contraction for ``composite bosons.'' These molecules ``feel'' each other not only through interactions between atoms, but also through the Pauli exclusion between their fermionic components. In order to point out the importance of this Pauli exclusion in cold atom physics, we have calculated here the energy change of $N$ identical cobosonic molecules when a similar molecule, made of atoms having possibly different spin states, is added. Due to the difference in the number of fermion exchanges with a molecule having zero, one, or two atoms identical to the ones already present in the system, we may think that this energy change can take three different values, even for spin-independent interatomic interactions. We actually find that the energy change, in the Born approximation, is exactly the same at all orders in density, when the added molecule has two or just one atom identical to the ones already there. In other words, the scattering length is predicted to be the same if the added molecule can have exchanges with one set of fermionic components or with two sets. This unexpected equality, which of course holds for spin-independent interactions only, results from a subtle balance in the fermion exchanges. To prove it, we make use of a recent extension of the composite-exciton many-body theory we have constructed, to any type of cobosons, the physical understanding of the various exchange processes being enlightened by the Shiva diagram representation (so named because of the multiarm structure reminiscent of the Hindu god Shiva) of this many-body theory.

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Physical Review Letters•May 17, 2011•19 citations
We call "proteons" -- from the ever-changing greek sea-god ΠρωτευςΠρωτε\upsilonςΠρωτευς -- composite particles made of two fermions. Among them, are the semiconductor excitons, but also various atoms and molecules, like the giant molecules made of two 40^{40}40K or 6^66Li atoms which have recently Bose condensed. In addition to their indistinguishability, these composite particles are ``ever-changing'' in the sense that there is no way to know with which fermions they are precisely made. As direct consequences, (i) the proteons are not true bosons, (ii) the basis made with proteon states is \emph{overcomplete}. In spite of these difficulties, these proteons do have a nice closure relation, unexpected at first, \emph{different from the boson one} and which makes the bosonization procedures used up to now to treat many-body effects between composite bosons, rather questionable, due to possibly incorrect sum rules resulting from it. This closure relation in particular explains, in a neat way, the surprising factor 1/2 between the inverse lifetime and the sum of scattering rates which exists for exact excitons but not for boson excitons, as we have recently shown.
This paper shows the necessity of introducing a quantum object, the ``coboson,'' to properly describe, through a fermion scheme, any composite particle, such as the exciton, which is made of two fermions. Although commonly dealt with as elementary bosons, these composite bosons---cobosons in short---differ from them due to their composite nature which makes the handling of their many-body effects quite different from the existing treatments valid for elementary bosons. As a direct consequence of this composite nature, there is no correct way to describe the interaction between cobosons as a potential VVV. This is rather dramatic because, with the Hamiltonian not written as H=H0+VH={H}_{0}+VH=H0​+V, all the usual approaches to many-body effects fail. In particular, the standard form of the Fermi golden rule, written in terms of VVV, cannot be used to obtain the transition rates of two cobosons. To get them, we have had to construct an unconventional expression for this Fermi golden rule in which HHH only appears. Making use of this expression, we give here a detailed calculation of the time evolution of two excitons. We compare the results of this exact approach with the ones obtained by using an effective bosonic Hamiltonian in which the excitons are considered as elementary bosons with effective scatterings between them, these scatterings resulting from an elaborate mapping between the two-fermion space and the ideal boson space. We show that the relation between the inverse lifetime and the sum of the transition rates for elementary bosons differs from the one of the composite bosons by a factor of 1∕21∕21∕2, so that it is impossible to find effective scatterings between bosonic excitons giving these two physical quantities correctly, whatever the mapping from composite bosons to elementary bosons is. The present paper thus constitutes a strong mathematical proof that, in spite of a widely spread belief, we cannot forget the composite nature of these cobosons, even in the extremely low-density limit of just two excitons. This paper also shows the (unexpected) cancellation in the Born approximation of the two-exciton transition rate for a finite value of the momentum transfer.
Molecules made of two fermionic atoms are ``cobosons''---a contraction for ``composite bosons.'' These molecules ``feel'' each other not only through interactions between atoms, but also through the Pauli exclusion between their fermionic components. In order to point out the importance of this Pauli exclusion in cold atom physics, we have calculated here the energy change of NNN identical cobosonic molecules when a similar molecule, made of atoms having possibly different spin states, is added. Due to the difference in the number of fermion exchanges with a molecule having zero, one, or two atoms identical to the ones already present in the system, we may think that this energy change can take three different values, even for spin-independent interatomic interactions. We actually find that the energy change, in the Born approximation, is exactly the same at all orders in density, when the added molecule has two or just one atom identical to the ones already there. In other words, the scattering length is predicted to be the same if the added molecule can have exchanges with one set of fermionic components or with two sets. This unexpected equality, which of course holds for spin-independent interactions only, results from a subtle balance in the fermion exchanges. To prove it, we make use of a recent extension of the composite-exciton many-body theory we have constructed, to any type of cobosons, the physical understanding of the various exchange processes being enlightened by the Shiva diagram representation (so named because of the multiarm structure reminiscent of the Hindu god Shiva) of this many-body theory.
This book bridges a gap between two major communities of Condensed Matter Physics, Semiconductors and Superconductors, that have thrived independently. Through an original perspective that their key particles, 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 the 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 start from 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 - a novel mechanism not visualized by standard Feynman diagrams. Advanced undergraduate or graduate students in physics with no prior background will benefit from this book. The developed concepts and methodology should also be useful to present researches on ultracold atomic gases, exciton-polaritons, and quantum information.
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 16 introduces the last part that addresses the bosonic condensation of composite bosons. To bridge the gap between elementary bosons and composite bosons, elementary fermion systems like semiconductor electrons and holes are also studied. Whereas the previous chapters provided a fairly precise knowledge of bosonic particles relevant for condensed matter physics, namely, Wannier excitons, Frenkel excitons, Cooper pairs, biexcitons, and polaritons, this chapter addresses an important question of current interest: what is the ground state of N composite bosons from the perspective of Bose-Einstein condensation? Or rather, how does the genuine Bose-Einstein condensation for elementary bosons persist in composite boson systems? This chapter lists important results with respect to condensate momentum fragmentation or spin fragmentation, correlation energy, and the dark nature of exciton condensate, to emphasize the connection between elementary bosons, elementary fermions, elementary fermion pairs, and composite bosons, from the composite boson perspective of their condensation.
This Letter provides the missing part of the newly constructed many-body formalism for composite quantum particles: the introduction of a finite temperature. The finite T formalism we propose deeply relies on the existence of a compact closure relation for the (overcomplete) set of N-composite-particle states. As a first application, we here calculate the energy mean value of the exciton gas outside the condensation regime. We show that carrier exchanges increase its temperature dependence compared to elementary bosons, a signature of the degree-of-freedom increase resulting from the particle composite nature.