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arXiv:1510.03134 · Physical Review A

Coboson many-body formalism for cold-atom dimers with attraction between different fermion species only

Monique Combescot, Shiue-Yuan Shiau, Yia‐Chung Chang

Published Jan 25, 2016• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.93.013624

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Journal of the Physical Society of Japan•Jun 29, 2012•1 citation
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 NNN-coboson normalization factor and the NNN-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.
We derive the scattering length of composite bosons (cobosons) within the framework of the composite-boson many-body formalism that uses correlated-pair states as a basis instead of free-fermion states. The integral equation constructed from this physically relevant basis makes transparent the role of fermion exchange in the coboson-coboson effective scattering. Three potentials used for Cooper pairs, fermionic-atom dimers, and semiconductor excitons are considered. While the sss-wave scattering length for the BCS-like potential is just equal to its Born value, the other two are substantially smaller. For fermionic-atom dimers and semiconductor excitons, our results, calculated within a restricted correlated-pair basis, are in good agreement with those obtained from procedures numerically more demanding. We also propose model coboson-coboson scatterings that are separable and thus easily workable and that produce scattering lengths which match quantitatively well with the numerically obtained values for all fermion mass ratios. These separable model scatterings can facilitate future works on many-body effects in coboson gases.
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.
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.
We present an exact many-body theory of ultracold fermionic gases for the Bose-Einstein condensation (BEC) regime of the BEC-BCS crossover. This is a purely fermionic approach which treats explicitly and systematically the dimers formed in the BEC regime as made of two fermions. We consider specifically the zero temperature case and calculate the first terms of the expansion of the chemical potential in powers of the density nnn. We derive first the mean-field contribution, which has the expected standard expression when it is written in terms of the dimer-dimer scattering length aM{a}_{M}aM​. We go next in the expansion to the Lee-Huang-Yang order, proportional to n3∕2{n}^{3∕2}n3∕2. We find the far less obvious result that it retains also the same expression in terms of aM{a}_{M}aM​ as for elementary bosons. The composite nature of the dimers appears only in the next term proportional to n2{n}^{2}n2.
We present a theoretical study of the BEC-BCS crossover in the Bose-Einstein-condensate (BEC) regime in the case of an unequal number of fermions of two species. We take full account of the composite nature of the dimers made of fermions. In the limit of low densities, we calculate the ground-state energy of the system, or equivalently the chemical potentials of each species, as well as the one-particle gap and the energy of an ``impurity'' immersed in a Fermi sea. For the chemical potentials we go up to order (density)4/3(\mathrm{density}){}^{4/3}(density)4/3. The results found involve the exact atom-dimer aAD{a}_{\mathrm{AD}}aAD​ and dimer-dimer aDD{a}_{\mathrm{DD}}aDD​ scattering lengths and therefore include the three- and four-body problems in the many-body problem. We briefly comment on the importance of the different mean-field corrections for recent experiments.
In this research, we develop a many-body Green's function approach for interacting composite bosons in the Bose–Einstein condensed phase. The resummation scheme dealing with the carrier–carrier exchange between many-body composite bosons in cold condensation is manifested as a diagrammatic technique through the introduction of a set of Dyson equations for composite bosons. We diagrammatically calculate the specific heat capacity of neutral composite bosons, made of mass-imbalanced fermions with opposite charges, up to the first orders of the Pauli exchange and Coulomb interaction. Condensate phenomena due to the effect of the carrier–carrier exchange are discussed. In comparison with the approximation that composite bosons behave like interacting fundamental bosons, our result shows that the energy reduction from the carrier–carrier exchange stabilizes composite-boson condensates.