Hamiltonian Description of Composite Fermions
R. Shankar, Yale University
I begin with a description of the work done with G. Murthy in which we start with electrons, do the Chern-Simons phase transformation, and introduce collective coordinates which oscillate at the cycltron frequency. We decouple and freeze the oscillators in the infrared, thereby obtaining a low energy long wavelength theory corresponding to the LLL. The theory now has composite fermions of charge e^* = e/ (2ps+1), the correct dipole moment, and the correct magnetic moment e/2m. The hamiltonian is just the coulomb interaction written in terms of the CF variables and there are constraints that pay for the oscillators. I show how this hamiltonian can be used to compute gaps and scaling relations among them for Jain fractions and compare the results to numerical work of Park and Jain. Finally I discuss how the small q forms of our charge and constraint operators could be extended to all q on algebraic grounds. It is seen that the constraints describe the incompressiblity of the vortices (whose charge is -2p/(2ps+1)) and that our charge operator is the sum of the electron and vortex charge densities, i.e., the charge density of the composite fermion. In short, our approach allows one to work directly with the CF, which have simple wavefunctions, but couple nonminimally to external potentials, being composite objects.

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