Damping of phonons in Bose gas at low temperatures

We consider homogeneous Bose gas in a large cubic box with periodic boundary conditions interacting with a small potential with a positive Fourier transform. We compute the imaginary part of the phononic excitation spectrum in the lowest order of perturbation theory in thermodynamic limit at low temperatures and low momentum. Our analysis is based on perturbation theory of the standard Liouvillean. We use two approaches: the first, motivated by the standard representation of operator algebras, examines resonances near zero; the second analyzes the 2-point correlation function in the energy-momentum space.

💡 Research Summary

The paper investigates the damping of phonon excitations in a homogeneous Bose gas confined to a large cubic box with periodic boundary conditions. The authors consider a weak two‑body interaction potential v(x) whose Fourier transform is positive, and they introduce a coupling constant κ to control the strength of the interaction. By replacing the zero‑momentum mode with a c‑number √ν (ν being the chemical potential) they obtain the Bogoliubov Hamiltonian, whose quadratic part can be diagonalized by a Bogoliubov transformation. The resulting quasiparticle dispersion relation is

 ω_bg(k)=√{k⁴/4+ν v̂(k) v̂(0) k²},

which for small momenta reduces to the linear phonon form ω_bg(k)≈√ν |k|.

To study the finite‑temperature behavior the authors work with the Liouvillean L, i.e. the generator of the Heisenberg dynamics on the doubled Hilbert space. In this representation there are two kinds of quasiparticles: “left” excitations (above the thermal equilibrium) and “right” holes (below the equilibrium). The Liouvillean is split into a quadratic Bogoliubov part L_bg,ν and interaction parts √κ L_3,ν (three‑body term) and κ L_4,ν (four‑body term). Perturbation theory in κ shows that the first non‑trivial contribution to the quasiparticle energy appears at order κ and consists of a Feynman‑Hellmann term (real part) and a Fermi‑Golden‑Rule term (imaginary part). The imaginary part, which determines the damping rate, depends only on the three‑body Liouvillean L_3,ν.

Two complementary computational schemes are employed. The first is the “standard representation” approach, rooted in the modular theory of W*‑algebras. One constructs one‑quasiparticle vectors by acting with left and right creation operators on the KMS state, and studies how these vectors acquire a complex energy in the thermodynamic limit. The second is a Green‑function method: time‑ordered two‑point correlation functions G(ω,k) are analyzed in energy‑momentum space. The Bogoliubov Liouvillean produces a singular shell at ω=±ω_bg(k); the interaction broadens this shell, and the width of the broadened peak is precisely the damping rate. Both approaches lead to the same explicit formulas.

The damping rate is decomposed into two contributions:

 γ(k)=γ_B(k)+γ_L(k).

γ_B is the Beliaev damping, arising from the decay of a left quasiparticle into two left quasiparticles. γ_L is the Landau damping, arising from the decay of a left quasiparticle into one left and one right quasiparticle (hence absent at zero temperature). The amplitudes of the three‑quasiparticle vertices are encoded in the functions

 j(k;p,q)=ν v̂(0) v̂(k)(s_k−c_k)(c_p s_q+c_q s_p)+…

 κ(k;p,q)=ν v̂(0) v̂(k)(s_k−c_k)(c_p s_q+s_p c_q)+…

where s_k and c_k are the Bogoliubov coefficients. The exact damping rates are given by the integrals

 γ_B(k)=π∫ d³p (2π)⁻³ j(k;p,k−p)² δ