A SOLUBLE MODEL FOR THE FORMATION OF EXCITON BANDS AND ELECTRON-HOLE DROPLETS IN ONE-DIMENSION

We consider an integrable model consisting of two one-dimensional para bolic bands of opposite mass (m = +/-1/2, respectively) separated by a gap, 2DELTA. The bands contain spinless fermions, one band correspond s to the conduction band and the other one to the valence band of a se miconductor or semimetal. The holes in the valence band and the spinle ss electrons in the conduction band are locally attracted via a delta- function potential. This model can be mapped onto the two-component Fe rmi gas with delta-function interaction, so that the two components la bel the bands: the chemical potential corresponds to a magnetic field, and DELTA to the chemical potential. We use Gaudin and Yang's exact s olution of the many-body problem to study the formation of exciton ban ds. The properties of the ground and metastable states, the excitation spectrum and the thermodynamics of the model are obtained. In the gro und or metastable states the particles (electrons in the conduction ba nd and holes in the valence band) are either paired in exciton bound s tates or unpaired. Their spectrum of elemental excitations is approxim ately parabolic. At finite T many-particle bound states (string soluti ons of the Bethe ansatz equations) can be populated; at low T these st ates are strongly delocalized and can be interpreted as electron-hole droplets. The low-T properties of the model are discussed.