Unlike the Coulomb potential that acts between all semiconductor carriers, the potential commonly used for BCS superconductors and cold-atom gases acts between different fermion species only, these species differing by their spin or hyperfine index. The coboson formalism we develop here evidences that such composite bosons interact through fermion exchange only, thus rendering the structure of the Shiva diagrams that visualize their many-body effects far simpler. A separable potential is used to obtain analytical results for the
N-coboson normalization factor and the
N-coboson ground-state energy within the Born approximation, in terms of the Born dimer-dimer scattering length. This formalism opens the route toward approaching complex many-body effects, such as Bose-Einstein condensation, through a different perspective.
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=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
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.
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.
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.
Particles made of two fermions can in many cases be treated as elementary bosons, but the conditions for this treatment to be valid are nontrivial. The so-called ``coboson formalism'' is a powerful tool to tackle compositeness effects relevant for instance for exciton physics and ultracold atomic dimers. A key element of this theory is an ansatz for the ground state of
N pairs, built from the single-pair ground state combined with the exclusion principle. We show that this ansatz can fail in one-dimensional systems which fulfill the conditions expected to make the ansatz valid. Nevertheless, we also explain how coboson theory can recover the correct ground state. Thus, our work highlights limitations and strengths of the formalism and leads to a better treatment of composite bosons.
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.