Shiva diagrams for composite-boson many-body effects : How they work

The purpose of this paper is to show how the diagrammatic expansion in fermion exchanges of scalar products of $N$-composite-boson (``coboson'') states can be obtained in a practical way. The hard algebra on which this expansion is based, will be given in an independent publication. Due to the composite nature of the particles, the scalar products of $N$-coboson states do not reduce to a set of Kronecker symbols, as for elementary bosons, but contain subtle exchange terms between two or more cobosons. These terms originate from Pauli exclusion between the fermionic components of the particles. While our many-body theory for composite bosons leads to write these scalar products as complicated sums of products of ``Pauli scatterings'' between \emph{two} cobosons, they in fact correspond to fermion exchanges between any number P of quantum particles, with $2 \leq P\leq N$. These $P$-body exchanges are nicely represented by the so-called ``Shiva diagrams'', which are topologically different from Feynman diagrams, due to the intrinsic many-body nature of Pauli exclusion from which they originate. These Shiva diagrams in fact constitute the novel part of our composite-exciton many-body theory which was up to now missing to get its full diagrammatic representation. Using them, we can now ``see'' through diagrams the physics of any quantity in which enters $N$ interacting excitons -- or more generally $N$ composite bosons --, with fermion exchanges included in an \emph{exact} -- and transparent -- way.Comment: To be published in Eur. Phys. J.

The trion: two electrons plus one hole versus one electron plus one exciton

We first show that, for problems dealing with trions, it is totally hopeless to use the standard many-body description in terms of electrons and holes and its associated Feynman diagrams. We then show how, by using the description of a trion as an electron interacting with an exciton, we can obtain the trion absorption through far simpler diagrams, written with electrons and \emph{excitons}. These diagrams are quite novel because, for excitons being not exact bosons, we cannot use standard procedures designed to deal with interacting true fermions or true bosons. A new many-body formalism is necessary to establish the validity of these electron-exciton diagrams and to derive their specific rules. It relies on the ``commutation technique'' we recently developed to treat interacting close-to-bosons. This technique generates a scattering associated to direct Coulomb processes between electrons and excitons and a dimensionless ``scattering'' associated to electron exchange inside the electron-exciton pairs -- this ``scattering'' being the original part of our many-body theory. It turns out that, although exchange is crucial to differentiate singlet from triplet trions, this ``scattering'' enters the absorption explicitly when the photocreated electron and the initial electron have the same spin -- \emph{i}. \emph{e}., when triplet trions are the only ones created -- \emph{but not} when the two spins are different, although triplet trions are also created in this case. The physical reason for this rather surprising result will be given

Many-body effects between unbosonized excitons

We here give a brief survey of our new many-body theory for composite excitons, as well as some of the results we have already obtained using it. In view of them, we conclude that, in order to fully trust the results one finds, interacting excitons should not be bosonized: Indeed, all effective bosonic Hamiltonians (even the hermitian ones !) can miss terms as large as the ones they generate; they can even miss the dominant term, as in problems dealing with optical nonlinearities

Optical signatures of a fully dark exciton condensate

We propose optical means to reveal the presence of a dark exciton condensate that does not yield any photoluminescence at all. We show that (i) the dark exciton density can be obtained from the blueshift of the excitonic absorption line induced by dark excitons; (ii) the polarization of the dark condensate can be deduced from the blueshift dependence on probe photon polarization and also from Faraday effect, linearly polarized dark excitons leaving unaffected the polarization plane of an unabsorbed photon beam. These effects result from carrier exchanges between dark and bright excitons.Comment: 5 pages, 4 figure

Effective bosonic hamiltonian for excitons : a too naive concept

Excitons, being made of two fermions, may appear from far as bosons. Their close-to-boson character is however quite tricky to handle properly. Using our commutation technique especially designed to deal with interacting close-to-boson particles, we here calculate the exact expansion in Coulomb interaction of theexciton-exciton correlations, and show that a naive effective bosonic hamiltonian for excitons cannot produce these X-X correlations correctly

Effects of fermion exchanges on the polarization of exciton condensates

Exchange processes are responsible for the stability of elementary boson condensates with respect to their possible fragmentation. This remains true for composite bosons when single fermion exchanges are included but spin degrees of freedom are ignored. We here show that their inclusion can produce a "spin-fragmentation" of a condensate of dark excitons, i.e., an unpolarized condensate with equal amount of dark excitons with spins (+2) and (-2). Quite surprisingly, for spatially indirect excitons of semiconductor bilayers, we predict that the condensate polarization can switch from unpolarized to fully polarized, depending on the distance between the layers confining electrons and holes. Remarkably, the threshold distance associated to this switching lies in the regime where experiments are nowadays carried out.Comment: 5 pages, 1 figur

Density expansion of the energy of N close-to-boson excitons

Pauli exclusion between the carriers of $N$ excitons induces novel many-body effects, quite different from the ones generated by Coulomb interaction. Using our commutation technique for interacting close-to-boson particles, we here calculate the hamiltonian expectation value in the $N$-ground-state-exciton state.Coulomb interaction enters this quantity at first order only by construction ; nevertheless, due to Pauli exclusion, subtle many-body effects take place, which give rise to terms in $(Na_x^3/\mathcal{V})^n$ with $n\geq2$ >. An \emph{exact} procedure to get these density dependent terms is given

Commutation technique for an exciton photocreated close to a metal

Recently, we have derived the changes in the absorption spectrum of an exciton when this exciton is photocreated close to a metal. The resolution of this problem -- which has similarities with Fermi edge singularities -- has been made possible by the introduction of ``exciton diagrams''. The validity of this procedure relied on a dreadful calculation based on standard free electron and free hole diagrams, with the semiconductor-metal interaction included at second order only, and its intuitive extention to higher orders. Using the commutation technique we recently introduced to deal with interacting excitons, we are now able to \emph{prove} that this exciton diagram procedure is indeed valid at any order in the interaction.