The Physical Significance of Singularities in the Chern–simons Fermi Liquid Description of a Partially Filled Landau Level

We analyze the linear response of a half filled Landau level to long wavelength and low frequency driving forces, using Fermi liquid theory for composite fermions. This response is determined by the composite fermions quasi–particle effective mass, m, and quasi–particle Landau interaction function f(θ − θ). Analyzing infra–red divergences of perturbation theory, we get an exact expression for m, and conjecture the form of the f(θ − θ). We then conclude that in the limit of infinite cyclotron frequency, and small q, ω, the composite fermion excitation spectrum is continuous for 0 < ω < γ e 2 ǫh q, with γ an unknown number. For fractional quantum Hall states near a half filled Landau level, we derive an exact expression for the energy gap. The fermionic Chern–Simons theory of the half filled Landau level maps the problem of interacting electrons in a strong magnetic field onto that of composite fermions at zero magnetic field, interacting with a Chern–Simons gauge field. This mapping opens the way to a study of low energy excitations of the system by an application of Fermi liquid theory. In this paper we discuss such an application. Due to space constraints, very few details are given. We refer the reader to Ref. [1] for further details. Fermi liquid theory outlines a formal method for a calculation of the exact small wave– vector q, low frequency ω and zero temperature limit of linear response functions. In practice, the exact calculation is usually intractable, and approximations are needed. In section (1) below, we remind the reader of the formal method. Then, in section (2), we attempt to use it to extract information about the linear response of electrons at, and near, a half filled Landau level. 1. Derivation of linear response function by Fermi liquid theory – a review The Fermi liquid theory prescription for the calculation of linear response functions can be put into the following steps: • Formulate the Hamiltonian of the system. • Calculate the effective mass of quasi–particles at the Fermi level m and the quasi– particle Landau interaction function f(k− k). This calculation is done in two stages. First, calculate the single particle self energy Σ(k, ω) for states near the Fermi level,