Eur. Phys. J. B 58, 291-296 (2007)
https://doi.org/10.1140/epjb/e2007-00235-y

Damping of zero sound in Luttinger liquids

Institut für Theoretische Physik, Universität Frankfurt, Max-von-Laue Strasse 1, 60438 Frankfurt, Germany

Corresponding author: ^a pk@itp.uni-frankfurt.de

Received: 20 June 2007
Revised: 30 July 2007
Published online: 8 September 2007

Abstract

We calculate the damping γ_q of collective density oscillations (zero sound) in a one-dimensional Fermi gas with dimensionless forward scattering interaction F and quadratic energy dispersion k² / 2 m at zero temperature. Using standard many-body perturbation theory, we obtain γ_q from the expansion of the inverse irreducible polarization to first order in the effective screened (RPA) interaction. For wave-vectors | q| /k_F ≪F (where k_F = m v_F is the Fermi wave-vector) we find to leading order γ_q ∝| q |³ /(v_F m²). On the other hand, for F ≪| q| /k_F most of the spectral weight is carried by the particle-hole continuum, which is distributed over a frequency interval of the order of q²/m. We also show that zero sound damping leads to a finite maximum proportional to |k - k_F | ^{-2 + 2 η} of the charge peak in the single-particle spectral function, where η is the anomalous dimension. Our prediction agrees with photoemission data for the blue bronze K_0.3MoO₃. We comment on other recent calculations of γ_q.