Theory of exciton spectra of incompressible quantum liquids

Energy and optical spectra of excitons against a background of incompressible quantum liquids (IQL's) are investigated by finite-size computations in a spherical geometry and by symmetry arguments based on the composite fermion theory. Properties of excitons are governed by the parameter h/l, where h is a separation between electron and hole confinement planes and l is a magnetic length. When h/l{}1, the energy spectrum comprises a single exciton branch L₀ and a quasicontinuum above it. With increasing h/l a multiple-branch exciton spectrum develops. Different branches Lₘ may be classified by the index m, which identifies the minimum angular momentum, Lₘ, of the Lₘ branch. There are two types of branches. The branches of the first type are symmetrically compatible with a model of an exciton as a neutral entity consisting of a valence hole and several fractionally charged quasiparticles. All these anyon branches have m values exceeding some critical value (m{≥}3 for the {ν}=1/3 IQL), and they are generically related to some specific states from the low-energy sector of the electron subsystem, and drop down below the original L₀ branch with increasing h/l. Comparative investigation of the number-of-particle dependencies of the electron and exciton spectra shows that these properties survive in the macroscopic limit and establishes a connection between anyon branches and the basic low-energy physics of IQL's.The branches of the second type remain above L₀ and cannot be treated in terms of low-energy electronic excitations. We argue that the sphere-onto-plane projection rule for neutral composite particles has a form L-Lₘ=kR, where k is the particle quasimomentum and R is the sphere radius. Since a plane rather than a sphere is the dynamical space of an exciton, this projection rule clarifies the physical meaning of the multiple-branch spectra and establishes the selection rules for optical transitions. In particular, it identifies Lₘ's as internal angular momenta of the excitons belonging to different branches. Momentum dependencies of the probabilities of magnetoroton-assisted transitions suggest that magnetorotons of an {ν}=1/3 IQL are L₂ quasiparticles. We also show that doublet emission spectra originating from zero- and single-magnetoroton transitions persist in a wide region of the parameter values.