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 two cobosons, they in fact correspond to fermion exchanges between any number P of quantum particles, with 2 ≤P≤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 the 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 exact — and transparent — way.     

The exciton many-body theory extended to any kind of composite bosons

We have recently constructed a many-body theory for composite excitons, in which the possible carrier exchanges between N excitons can be treated exactly through a set of dimensionless “Pauli scatterings” between two excitons. Many-body effects with free excitons turn out to be rather simple because these excitons are the exact one-pair eigenstates of the semiconductor Hamiltonian, in the absence of localized traps. They consequently form a complete orthogonal basis for one-pair states. As essentially all quantum particles known as bosons are composite bosons, it is highly desirable to extend this free exciton many-body theory to other kinds of “cobosons” — a contraction for composite bosons — the physically relevant ones being possibly not the exact one-pair eigenstates of the system Hamiltonian. The purpose of this paper is to derive the “Pauli scatterings” and the “interaction scatterings” of these cobosons in terms of their wave functions and the interactions which exist between the fermions from which they are constructed. It is also explained how to calculate many-body effects in such a very general composite boson system.     

The Girardeau's fermion-boson procedure in the light of the composite-boson many-body theory

We reconsider the procedure developed for atoms a few decades ago by Girardeau, in the light of the composite-boson many-body theory we recently proposed. The Girardeau's procedure makes use of a so called “unitary Fock-Tani operator” which in an exact way transforms one composite bound atom into one bosonic “ideal” atom. When used to transform the Hamiltonian of interacting atoms, this operator generates an extremely complex set of effective scatterings between ideal bosonic atoms and free fermions which makes the transformed Hamiltonian impossible to write explicitly, in this way forcing to some truncation. The scatterings restricted to the ideal-atom subspace are shown to read rather simply in terms of the two elementary scatterings of the composite-boson many-body theory, namely, the energy-like direct interaction scatterings — which describe fermion interactions without fermion exchange — and the dimensionless Pauli scatterings — which describe fermion exchanges without fermion interaction. We here show that, due to a fundamental difference in the scalar products of elementary and composite bosons, the Hamiltonian expectation value for N ground state atoms obtained by staying in the ideal-atom subspace and working with boson operators only, differ from the exact ones even for N = 2 and a mapping to the ideal-atom subspace performed, as advocated, from the fully antisymmetrical atomic state, i.e., the state which obeys the so-called “subsidiary condition”. This shows that, within this Girardeau's procedure too, we cannot completely forget the underlying fermionic components of the particles if we want to correctly describe their interactions.     

Composite boson many-body theory for Frenkel excitons

We present a many-body theory for Frenkel excitons which takes into account their composite nature exactly. Our approach is based on four commutators similar to the ones we previously proposed for Wannier excitons. They allow us to calculate any physical quantity dealing with N excitons in terms of “Pauli scatterings” for carrier exchange in the absence of carrier interaction and “interaction scatterings” for carrier interactions in the absence of carrier exchange. We show that Frenkel excitons have a novel “transfer assisted exchange scattering”, specific to these excitons. It comes from indirect Coulomb processes between localized atomic states. These indirect processes, commonly called “electron-hole exchange” in the case of Wannier excitons and most often neglected, are crucial for Frenkel excitons, as they are the only ones responsible for the excitation transfer. We also show that in spite of the fact that Frenkel excitons are made of electrons and holes on the same atomic site, so that we could naively see them as elementary particles, they definitely are composite objects, their composite nature appearing through various properties, not always easy to guess. The present many-body theory for Frenkel excitons is thus going to appear as highly valuable to securely tackle their many-body physics, as in the case of nonlinear optical effects in organic semiconductors.     

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 composite bosons in other fields than semiconductor physics. We, in particular, explain how the two elementary 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.     

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/)ⁿ with n ≥ 2. An exact procedure to get these density dependent terms is given. Received 11 February 2002 / Received in final form 30 May 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: combescot@gps.jussieu.fr     

On the N-exciton normalization factor

The N-ground-state-exciton normalization factor, namely 〈v| B ₀ ^N B ₀ ^dagN| v〉 = N!F _N, with B ₀ ^d ag the exact ground state exciton creation operator, differs from N! because the excitons are not perfect bosons. The quantity F_N turns out to be crucial for problems dealing with interacting excitons. Indeed, the excitons feel each other not only through the Coulomb interaction but also through Pauli exclusion between their components. A quite novel purely Pauli contribution exists in their many-body effects, which relies directly on F_N. Following procedures used in the commutation technique we recently introduced to treat interacting close-to-bosons, and in the BCS theory of superconductivity, we rederive important relations verified by the F_N's. We also give new explicit expressions of F_N valid for η = Na _x ³/ small but N ² a _x ³/ large, as F_N does not read in terms of η but Nη, the exciton number N being possibly huge in macroscopic samples. Due to this superextensivity, F_N does not appear alone in physical quantities, but through ratios like F _{N + p}/F _N. We end this work by giving the η expansion of these ratios, useful for all purely Pauli many-body effects. Received 30 May 2002 / Received in final form 12 October 2002 Published online 27 January 2003 RID="a" ID="a"e-mail: combescot@gps.jussieu.fr     

Quasibosons

The similarity of the commutation relations for bosons and quasibosons (fermion pairs) suggests the possibility that all integral spin particles presently considered to be bosons could be quasibosons. The boson commutation relations for integral spin particles could be just an approximation to the quasiboson commutation relations that contain an extra term. Although the commutation relations for quasibosons are slightly more complex, it is a simpler picture of matter in that only fermions and composite particles formed of fermions exist. Mesons are usually referred to as bosons, but they must be quasibosons since their internal structure is fermion (quark) pairs. The photon is usually considered to be an elementary boson, but as shown here, existing experiments do not rule out the possibility that it is also a quasiboson. We consider how the quasiboson, composite nature of such a photon might manifest itself.     

Excitons dressed by a sea of excitons

We here consider an exciton i embedded in a sea of N identical excitons 0. If the excitons are taken as true bosons, a bosonic enhancement factor N is found for i=0. If the exciton composite nature is kept, this enhancement not only exists for i=0, but also for any exciton having a center of mass momentum equal to the sea exciton momentum. This physically comes from the fact that an exciton with such a momentum can be transformed into a sea exciton by Pauli scattering, i.e., carrier exchange with the sea, making this exciton i not so much different from a sea exciton. This possible scattering, directly linked to the composite nature of the excitons, is irretrievably lost when the excitons are bosonized. The underlying interest of this work is in fact the calculation of the scalar products of N-exciton states, which turns out to be quite tricky, due to possible carrier exchanges between excitons. This work actually constitutes a crucial piece of our many-body theory for interacting composite bosons, because all physical effects involving composite bosons ultimately end by the calculation of such scalar products. The skeleton diagrams we here introduce to represent them, allow to visualize many-body effects linked to carrier exchanges in an easy way. They are conceptually different from Feynman diagrams, because of the special feature of the Pauli scatterings which originate from boson statistics departure.     

Large N expansion for strongly-coupled boson–fermion mixtures

We study a many-body mixture of an equal number of bosons and two-component fermions with a strong contact attraction. In this system bosons and fermions can be paired into composite fermions. We construct a large N extension where both bosons and fermions have the extra large N degrees of freedom and the boson–fermion interaction is extended to a four-point contact interaction which is invariant under the O(N) group transformation, so that the composite fermions become singlet in terms of the O(N) group. It is shown that such O(N) singlet fields have controllable quantum fluctuations suppressed by 1/N factors and yield a systematic 1/N-expansion in terms of composite fermions. We derive an effective action described by composite fermions up to the next-to-leading-order terms in the large N expansion, and show that there can be the BCS superfluidity of composite fermions at sufficiently low temperatures.     

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 excitons and electrons. 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.     

The inconstant world of composite excitons

The composite nature of the exciton gives rise to very many carrier exchanges which cannot be accounted correctly through an effective Hamiltonian for bosonized particles. This forced us to construct a new many-body procedure [For a short review, see M. Combescot, O. Betbeder-Matibet, Solid State Commun. 134 (2005) 11]. Here we report a few important aspects of this procedure, using Shiva diagrams [M. Combescot, O. Betbeder-Matibet, Eur. Phys. J. B 55 (2007) 63] which greatly help not only to visualize, but also to calculate the subtle carrier exchanges which take place between composite excitons.     

Finite temperature formalism for composite quantum particles

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.     

On the excited quarks and leptons

Using a duality-like finite energy sum rule, we discuss the assumption of having excited fermions at the W scale in a supersymmetric(SUSY) and non-supersymmetric hypercolour theory where quarks and leptons are bound states of fermion and scalar preon constituents. We conclude that a SUSY-like composite model cannot have excited fermions having a mass smaller than 0.5 TeV. A non-SUSY composite model having composite fermions but elementary W bosons can produce an excited fermion mass of the order of M_W provided that the scalar vacuum condensate is of the order of the (TeV)² scale of compositeness.     

Standard Model without elementary scalars and high energy Lorentz violation

If Lorentz symmetry is violated at high energies, interactions that are usually non-renormalizable can become renormalizable by weighted power counting. Recently, a CPT invariant, Lorentz violating extension of the Standard Model containing two scalar-two fermion interactions (which can explain neutrino masses) and four fermion interactions (which can explain proton decay) was proposed. In this paper we consider a variant of this model, obtained suppressing the elementary scalar fields, and argue that it can reproduce the known low-energy physics. In the Nambu–Jona-Lasinio spirit, we show, using a large N _c expansion, that a dynamical symmetry breaking takes place. The effective potential has a Lorentz invariant minimum and the Lorentz violation does not reverberate down to low energies. The mechanism generates fermion masses, gauge-boson masses and scalar bound states, to be identified with composite Higgs bosons. Our approach is not plagued by the ambiguities of approaches based on non-renormalizable vertices. The low-energy effective action is uniquely determined and predicts relations among parameters of the Standard Model.     

X-ray edge spectra — a sea-boson perspective

The well-studied X-ray-edge problem is revisited using the sea-boson method. This approach is contrasted with the well-known theories of Mahan, Nozières and De Dominicis (MND). The present approach does not use the sudden approximation and the holes carry a momentum label unlike in the MND theory. We focus on the case of doped semiconductors rather than metals. The problem of electrons in a partially filled conduction band and holes in the initially hole-depleted valence band is recast in the sea-boson language. The resulting hamiltonian is shown to be equivalent to the electron-phonon hamiltonian with the excitons taking on the role of electrons and intra-conduction band particle-hole excitations known as ‘conductrons’ taking on the role of phonons. It is shown that the excitonic pole in the computed absorption spectra is replaced by a branch cut with a simple radical leading to a broadening of the exicton line due to these many-body effects. A critical comparison is made with the MND theory as well as with relevant experiments.     

Superfluidity near phase separation in Bose-Fermi mixtures

We study the transition to fermion pair superfluidity in a mixture of interacting bosonic and fermionic atoms. The fermion interaction induced by the bosons and the dynamical screening of the condensate phonons due to fermions are included using the nonperturbative Hamiltonian flow equations. We determine the bosonic spectrum near the transition towards phase separation and find that the superfluid transition temperature may be increased substantially due to phonon damping.     

Phase transitions of indirect excitons in coupled quantum wells: The role of disorder

The theory of what happens to a superfluid in a random field, known as the “dirty boson” problem, directly relates to a real experimental system presently under study by several groups, namely excitons in coupled semiconductor quantum wells. We consider the case of bosons in two dimensions in a random field, when the random field can be large compared to the repulsive exciton–exciton interaction energy, but is small compared to the exciton binding energy. The interaction between excitons is taken into account in the ladder approximation. The coherent potential approximation (CPA) allows us to derive the exciton Green's function for a wide range of the random field strength, and in the weak-scattering limit CPA results in the second-order Born approximation. For quasi-two-dimensional excitonic systems, the density of the superfluid component and the Kosterlitz–Thouless temperature of the superfluid phase transition are obtained, and are found to decrease as the random field increases.     

Theory of spin precession monitored by laser pulse

We first predict the splitting of a spin degenerate impurity level when this impurity is irradiated by a circularly polarized laser beam tuned in the transparency region of a semiconductor. This splitting, which comes from different exchange processes between the impurity electron and the virtual pairs coupled to the pump beam, induces a spin precession around the laser beam axis, which lasts as long as the pump pulse. It can thus be used for ultrafast spin manipulation. This effect, which has similarities with the exciton optical Stark effect we studied long ago, is here derived using the concepts we developed very recently to treat many-body interactions between composite excitons and which make the physics of this type of effects quite transparent. They, in particular, allow to easily extend this work to other experimental situations in which a spin rotates under laser irradiation.