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Low Temperature Physics

Quasilinear theory of quantum Fermi liquid

Nodar L. Tsintsadze, D. Alkhanishvili

Published Dec 1, 2016• 1 citation
Publisher

Abstract

Quasilinear theory of a weakly turbulent quantum Fermi liquid is presented. Landau's linear theory of Fermi liquids is generalized by consideration of weak nonlinear regime. A newly derived kinetic equation of the Fermi particles is used to derive a slowly varying distribution function f0, which satisfies the diffusion equation. It is shown that the magnitude of the diffusion coefficient D depends on weak interactions between atoms and the de Broglie waves diffraction.

DOI
10.1063/1.4973003

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Quasilinear theory of a weakly turbulent quantum Fermi liquid is presented. Landau's linear theory of Fermi liquids is generalized by consideration of weak nonlinear regime. A newly derived kinetic equation of the Fermi particles is used to derive a slowly varying distribution function f0, which satisfies the diffusion equation. It is shown that the magnitude of the diffusion coefficient D depends on weak interactions between atoms and the de Broglie waves diffraction.
With neutron star applications in mind, we developed a theory of diffusion in mixtures of superfluid, strongly interacting Fermi liquids. By employing the Landau theory of Fermi liquids, we determined matrices that relate the currents of different particle species, their momentum densities, and the partial entropy currents to each other. Using these results, and applying the quasiclassical kinetic equation for the Bogoliubov excitations, we derived general expressions for the diffusion coefficients, which properly incorporate all the Fermi-liquid effects and depend on the momentum transfer rates between different particle species. The developed framework can be used as a starting point for systematic calculations of the diffusion coefficients (as well as other kinetic coefficients) in superfluid Fermi mixtures, particularly, in superfluid neutron stars.
We consider a quantum Langevin kinetic equation for a system of fermions. We first derive the Langevin force noise correlation functions in Landau's Fermi-liquid kinetic theory from general considerations. We then use the resulting equation to calculate the equilibrium dynamic structure factor in the collisionless regime at low temperatures. The result is in agreement with the conventional many-body result. We then use the theory to derive both the fluctuating Navier-Stokes equations for a quantum fluid and the fluctuating hydrodynamic equations for fermions in the presence of quenched disorder. We also discuss the modifications needed for the fluctuating hydrodynamic equations to describe an electron fluid with long-ranged interactions, and we prove an HHH theorem for the nonlinear Landau kinetic equation.
Linear equations for the transverse spin dynamics in a weakly polarized degenerate Fermi liquid are derived from the Landau-Silin phenomenological kinetic equation with a general two-particle collision integral. Unlike a previous treatment where the Fermi velocity and density of states were taken to be constant independent of polarization we make no such assumption. The equations found describe the spin dynamics in a paramagnetic Fermi liquid with finite polarization as well in an itinerant ferromagnet. The results are confirmed by field theoretical calculations based on the integral equation for the vertex function. The transverse spin wave frequency in a polarized paramagnetic Fermi liquid is found to be proportional to k2{k}^{2}k2 with a complex diffusion coefficient such that the damping has a finite value proportional to the quasiparticles scattering rate at T=0T=0T=0. This behavior of a polarized Fermi liquid contrasts with the behavior of a Heisenberg ferromagnet in the hydrodynamic regime where the transverse spin-wave attenuation appears in terms proportional to k4{k}^{4}k4. The reactive part of the diffusion coefficient in a paramagnetic state at T=0T=0T=0 proves to be inversely proportional to the magnetization whereas in a ferromagnetic it is directly proportional to the magnetization. The dissipative part of the diffusion coefficient at T=0T=0T=0 in the paramagnetic state is polarization independent, whereas in the ferromagnetic state it is proportional to the square of the magnetization. Moreover, the spin wave spectrum in a ferromagnetic Fermi liquid proves to be unstable demonstrating the difficulty of applying a Fermi liquid description to itinerant ferromagnetism.
On the basis of the Kadanoff-Baym (KB) version of the time-dependent Green's function method, an Ansatz for the approximation of a spectral function is offered. The Ansatz possesses all the advantages of quasiparticle and extended quasiparticle approximations and satisfies the KB equation for a spectral function in the case of slightly nonequilibrium system when disturbances in space and time are taken into consideration in the gradient approximation. This feature opens opportunities for the microscopic derivation of the Landau kinetic equation for the quasiparticle distribution function of the normal Fermi liquid and provides the widening of these equations' temperature range of validity.