Transport in the low-temperature limit of a Landau Fermi liquid

Abstract

The Boltzmann equation for Landau quasiparticles is solved for T ∼ 0 by a specialization of a method discussed by Sykes and Brooker. The quasiparticle distribution function is expanded in Legendre polynomials, assuming a boundary condition which imposes axial symmetry, and even-order terms are assumed to relax together with relaxation time τ _e , odd-order terms with relaxation time τ _o . By letting wavelength λ → ∞, with τ finite, one obtains a first-sound solution, and by lettingT → 0, and then λ → ∞, one obtains a zero-sound solution. When these solutions are used to calculate the pressure, it is found that the first-sound solution is consistent with hydrodynamics, exhibiting viscosity η=μ _s τ, while the zero-sound velocityc ₁=[ϱ⁻¹(B₁+4/3μ_s)]^1/2, so that phenomenologically zero-sound propagates like a longitudinal elastic wave in a glass. A higher zero-sound mode is also predicted, but is heavily damped. The heat flux is calculated and found to obey Vernotte's equation, which contains an intertial term, added to Fourier's law, that becomes significant asT → 0.