We derive a bosonized Hamiltonian for a semiconductor quantum well with a light field by a new bosonization method taking exactly into account the composite-particle effects of excitons. Then we apply the bosonized Hamiltonian to the four-wave-mixing (FWM) processes. The obtained Hamiltonian describes two-boson states corresponding to bound and unbound two-exciton states and guarantees exclusion of unphysical states, which cannot exist in the fermionic space due to the Pauli exclusion principle. Calculated results of the FWM signals are in good agreement with the experimental ones, and therefore, the bosonized Hamiltonian turns out to be able to describe two-exciton correlations effectively.
We propose an exact bosonization scheme for two two-fermion composites with identical internal structures, that is, a mapping of the two-exciton (four-fermion) subspace to a two-boson subspace. We obtain analytical, exact expressions of the boson-boson interactions and the boson-photon ones taking into full account that the commutation relation of the composite bosons deviates from the ideal-boson commutation due to internal motions of the composite. We can distinguish the "composite-particle effects' from the Coulomb interactions among the fermions in the interactions. With this method, origins of optical nonlinearity in a system with two 1s excitons are studied in terms of the mutual excitonic correlations and exciton-photon interactions.
Excitons are bound electron-hole pairs in semiconductors whose interactions influence optical properties and determine their collective behavior and emergent correlated phases. Despite the theoretical and experimental significance of these phenomena, a microscopic theory of exciton-exciton interactions that takes into account the full composite nature of these quasiparticles in systems with an arbitrary single-particle Hamiltonian is lacking. Here, we present a general theory of exciton-exciton interactions that builds upon our previous publication [Physical Review B 113, 155440 (2026)], which applied only to conventional semiconductors with weak band hybridization. By contrast, the theory presented here is valid in virtually any crystalline semiconducting system and can be applied to the study of biexciton formation and correlated excitonic phases in systems where band topology, quantum geometry, and moiré effects play a significant role. In addition to the analytical theory, we also present results in simple systems with flat bands with a nontrivial quantum metric and show that the biexcitonic spectrum differs strongly from that in the case without quantum geometry. Our theory paves the way for accurate quantitative calculations of biexciton spectra and excitonic many-body phases in modern condensed-matter systems.
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.
We revisit the approach proposed by Mukamel and coworkers to describe interacting excitons through infinite series of composite-boson operators for both, the system Hamiltonian and the exciton commutator -- which, in this approach, is properly kept different from its elementary boson value. Instead of free electron-hole operators, as used by Mukamel's group, we here work with composite-exciton operators which are physically relevant operators for excited semiconductors. This allows us to get \emph{all} terms of these infinite series explicitly, the first terms of each series agreeing with the ones obtained by Mukamel's group when written with electron-hole pairs. All these terms nicely read in terms of Pauli and interaction scatterings of the composite-exciton many-body theory we have recently proposed. However, even if knowledge of these infinite series now allows to tackle
N-body problems, not just 2-body problems like third order nonlinear susceptibility
χ(3), the necessary handling of these two infinite series makes this approach far more complicated than the one we have developed and which barely relies on just four commutators.
We revisit the approach proposed by Mukamel and co-workers to describe interacting excitons through infinite series of composite-boson operators for both the system Hamiltonian and the exciton commutator---which, in this approach, is properly kept different from its elementary-boson value. Instead of free-electron-hole operators, as used by Mukamel's group, we here work with composite-exciton operators which are physically relevant operators for excited semiconductors. This allows us to get all terms of these infinite series explicitly, the first terms of each series agreeing with the ones obtained by Mukamel's group when written with electron-hole pairs. All these terms nicely read in terms of Pauli and interaction scatterings of the composite-exciton many-body theory we have recently proposed. However, even if knowledge of these infinite series now allows us to tackle
N-body problems, not just two-body problems like third-order nonlinear susceptibility ${\ensuremath{\chi}}^{(3)}$, the necessary handling of these two infinite series makes this approach far more complicated than the one we have developed and which barely relies on just four commutators.