Quantum mechanics of the fractional-statistics gas: Random-phase approximation

Abstract

A description of the fractional-statistics gas based on the complete summation of Hartree, Fock, ladder and bubble diagrams is presented. The superfluid properties identified previously in the randomphase-approximation (RPA) calculation of Fetter, Hanna, and Laughlin [Phys. Rev. B 39, 9679 (1989)] are substantially confirmed. The discrepancy between the RPA sound speed and the Hartree-Fock bulk modulus is found to be eliminated. The unusual Hall-effect behavior is found to vanish for the Bose gas test case but not for the fractional-statistics gas, implying that it is physically correct. Excellent agreement is obtained with the collective-mode dispersion obtained numerically by Xie, He, and Das Sarma [Phys. Rev. Lett. 65, 649 (1990)]. ment of the kth particle resulting from external forces F, is given to linear order bywhere l%'(t) & is the many-body wave function in the Schrodinger representation, the bracket denotes commutator, and x~( t)e(iris)Jvtx e( -iIA)fact J J

On the Collective Mode Spectrum for Composite Fermions at 1/3 Filling Factor

“…Concretely the Bethe-Salpeter equation and its contribution to the linear response of the FQHE regimen is considered in order to discuss the elementary excitation spectrum predicted by the model. Technically, the discussion in this paper is very close the one in the work [5] in which the fractional statistics gas was investigated. Here, as the FQHE problem is examined the discussion includes the Coulomb interaction and the presence of the external magnetic field.…”

“…The general formulations developed in [5] were closely followed. From them, it becomes clear that the collective mode spectrum should appear as included in the spectrum of the four-point Green function.…”

“…As mentioned above, it is possible to simplify the integral equation (7) introducing magneto-exciton wave functions [5], [8] which have the form…”

“…The explicit formulae for these terms and the one for F o are given in Appendix A. obtained The Feynman diagram representation of these contributions can be found in [5].…”

“…The method employed was the numerical solution of the Bethe-Salpeter equation for the four-points Green's function taken in the first approximation for the interaction kernel. Then, as the electromagnetic response kernel is linearly given in terms of the four-point function at coinciding points, it follows [5] that the collective mode spectrum can also be obtained as a byproduct.…”