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 as an exciton interacting with a carrier

The X$^-$ trion is essentially an electron bound to an exciton. However, due to the composite nature of the exciton, there is no way to write an exciton-electron interaction potential. We can overcome this difficulty by using a commutation technique similar to the one we introduced for excitons interacting with excitons, which allows to take exactly into account the close-to-boson character of the excitons. From it, we can obtain the X$^-$ trion creation operator in terms of exciton and electron. We can also derive the X$^-$ trion ladder diagram between an exciton and an electron. These are the basic tools for future works on many-body effects involving trions

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

How composite bosons really interact

The aim of this paper is to clarify the conceptual difference which exists between the interactions of composite bosons and the interactions of elementary bosons. A special focus is made on the physical processes which are missed when composite bosons are replaced by elementary bosons. Although what is here said directly applies to excitons, it is also valid for bosons in other fields than semiconductor physics. We in particular explain how the two basic scatterings -- Coulomb and Pauli -- of our many-body theory for composite excitons can be extended to a pair of fermions which is not an Hamiltonian eigenstate -- as for example a pair of trapped electrons, of current interest in quantum information.Comment: 39 pages, 12 figure

Commutation technique for interacting close-to-boson excitons

The correct treatment of the close-to-boson character of excitons is known to be a major problem. In a previous work, we have proposed a ``commutation technique'' to include this close-to-boson character in their interactions. We here extend this technique to excitons with spin degrees of freedom as they are of crucial importance for many physical effects. Although the exciton total angular momentum may appear rather appealing at first, we show that the electron and hole angular momenta are much more appropriate when dealing with scattering processes. As an application of this commutation technique to a specific problem, we reconsider a previous calculation of the exciton-exciton scattering rate and show that the proposed quantity is intrinsically incorrect for fundamental reasons linked to the fermionic nature of the excitons

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.

Faraday rotation in photoexcited semiconductors: an excitonic many-body effect

This letter assigns the Faraday rotation in photoexcited semiconductors to ``Pauli interactions'', \emph{i}. \emph{e}., carrier exchanges, between the real excitons present in the sample and the virtual excitons coupled to the $\sigma_{\pm}$ parts of a linearly polarized light. While \emph{direct Coulomb} interactions scatter bright excitons into bright excitons, whatever their spins are, \emph{Pauli} interactions do it for bright excitons \emph{with same spin only}. This makes these Pauli interactions entirely responsible for the refractive index difference, which comes from processes in which the virtual exciton which is created and the one which recombines are formed with different carriers. To write this difference in terms of photon detuning and exciton density, we use our new many-body theory for interacting excitons. Its multiarm ``Shiva'' diagrams for $N$-body exchanges make transparent the physics involved in the various terms. This work also shows the interesting link which exists between Faraday rotation and the exciton optical Stark effect