The Fractional Statistics Gas: Beyond the RPA

This work establishes and tests procedures which can determine the properties of the high-temperature superconductors using the t-J model with spinon and holon quasiparticles obeying fractional statistics. A simpler problem with similar physics, the fractional statistics gas, is studied. Processes augmenting the Random Phase Approximation analysis of the fractional statistics gas by Fetter, Hanna, and Laughlin (Phys. Rev. B 39, 9679 (1989)) are computed. The results allow a qualitative understanding of the corrections due to the analogous Feynman diagrams to the t-J model computations of Tikofsky, Laughlin, and Zou (Phys. Rev. Lett. 69, 3670 (1993)). First, the linear response of the fractional statistics gas is calculated, including Hartree, Fock, and ladder Feynman diagrams (Q. Dai, J. L. Levy, A. L. Fetter, C. B. Hanna, and R. B. Laughlin, Phys. Rev. B 46, 5642 (1992)). The superfluid properties found by Fetter, Hanna, and Laughlin remain. The computed sound speed agrees with that found from the Hartree-Fock bulk modulus. The Hall effect vanishes for the bosons but not the semions, suggesting that the semion Hall effect found previously is correct. The collective mode spectrum agrees well with the numerical results of Xie, He, and Das Sarma (Phys. Rev. Lett. 65, 649 (1990)). In the t-J model, similar processes would have a limited effect upon the optical conductivity, but could cancel the Hall effect. In addition, the spin susceptibility and gap of the fractional statistics gas was examined (J. L. Levy and R. B. Laughlin, Phys. Rev. B, in press). Interactions with the density oscillations of the system substantially decrease the spin gap to a value of (0.2 +/- 0.2) hbaromega_{c }, much less than the mean-field value of hbaromega_{c}. The lower few Landau levels remain visible, though broadened and shifted, in the spin susceptibility. As a check of the methods, the single-particle Green's function of the non-interacting Bose gas viewed in the fermionic representation, as computed by the same approximation scheme, agrees well with the exact results. The same mechanism would reduce the gap of the t-J model without eliminating it, and would also broaden the response functions.