Noncanonical-transformation approach to the problem of an itinerant particle interacting with a Fermi sea

We present a noncanonical-transformation solution to the problem of an itinerant particle interacting with a Fermi sea and apply the solution to the study of the excitonic effects on the optical spectra of a modulation-doped quantum wire. In the case of equal particle and Fermi electron mass, this provides the exact solution in one dimension. For unequal masses or higher dimensions, we provide the first term in the systematic expansion of the exact interacting eigenstates in terms of many-body correlations (${\mathit{e}}^{\mathit{S}}$ method). Our approximation conserves momentum, includes the three-body correlations between the itinerant particle and the electron and hole of one pair of Fermi sea excitations, and produces single Slater-determinant eigenstates. Numerical results were obtained for parameters relevant to modulation-doped quantum wires. They show strong dependence of the eigenstates on the total momentum of the system and reveal a new momentum scale of power-law behavior associated with the valence-subband width. For total momentum approaching the Fermi surface, this momentum scale diminishes to zero and large momentum excitations across the Fermi surface become important. In addition, for a total momentum approaching ${\mathit{k}}_{\mathit{F}}$ from above, a resonant structure in the wave function moving toward the Fermi surface is observed.