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arXiv:cond-mat/0507675 · Physical Review B

Theory of transverse spin dynamics in a polarized Fermi liquid and an itinerant ferromagnet

V. P. Mineev

Published Oct 20, 2005• 23 citations
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Abstract

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 ${k}^{2}$ with a complex diffusion coefficient such that the damping has a finite value proportional to the quasiparticles scattering rate at $T=0$. 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 ${k}^{4}$. The reactive part of the diffusion coefficient in a paramagnetic state at $T=0$ 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 $T=0$ 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.

DOI
10.1103/physrevb.72.144418

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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.
The linear equations for transverse spin dynamics in a weakly polarized degenerate Fermi liquid with arbitrary relationship between temperature T and polarization $\ensuremath{\gamma}H$ are derived from Landau-Silin phenomenological kinetic equation with general form of two-particle collision integral. The temperature and polarization dependence of the spin current relaxation time is established. It is found in particular that at finite polarization transverse spin wave damping has a finite value at T=0.T=0.T=0. The analogy between temperature dependences of spin wave attenuation and ultrasound absorption in degenerate Fermi liquid at arbitrary temperature is presented. We also discuss spin-polarized Fermi-liquid in the general context of the Fermi-liquid theory and compare it with ``Fermi liquid'' with spontaneous magnetization.
We calculate the effective spin diffusion coefficient of weak ferromagnet and quasiequilibrium paramagnetic systems using Landau Fermi liquid theory. We find that the behavior of the diffusion coefficient of the quasiequilibrium system is determined primarily by the internal magnetic field, which, in turn, depends upon the nonequilibrium magnetization and the antisymmetric Landau parameters. We also show that this is qualitatively similar to the diffusion coefficient of the weak ferromagnetic system. We discuss its implication for the spin-polarized state created in cold atom systems.
We solve a recently derived kinetic equation for a dilute quantum system at arbitrary degeneracy and polarization, M. From this equation we develop a spin hydrodynamic equation that allows the generalization of the treatment of spin waves and the Leggett-Rice effect to polarized degenerate Fermi gases. We find a transverse spin-diffusion collision time ${\ensuremath{\tau}}_{\ensuremath{\perp}}$ that is often shorter than the corresponding longitudinal collision time ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\parallel}}}$ that is often shorter than the corresponding longitudinal collision time ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$. As T\ensuremath{\rightarrow}0, ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\parallel}}}$ approaches T-independent value, in constrast to ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\perp}}}$\ensuremath{\sim}1/T2{\mathit{T}}^{2}T2. Spin waves would thus continue to be damped even to T=0 K.
We derive the linearized spin dynamics for paramagnetic quantum liquids. In these systems, the existence of quantum-mechanical exchange in the presence of (an externally induced) broken symmetry in spin space gives rise to a coherent precession of the spins damped by diffusion. Such collective oscillations occur as a result of the precession of the spin current about a macroscopic local exchange field generated either by an external polarizing magnetic field (as in dilute nondegenerate spin-polarized gases) or by the long-lived nonequilibrium polarization (as in degenerate Fermi liquids). We use a unified approach based on the Kadanoff-Baym formulation of the kinetic equations to describe these phenomena. The connection between strongly and weakly interacting paramagnetic systems is made obvious by considering two special cases: (1) a weakly interacting spin-polarized quantum gas and (2) a parametrization of strongly correlated degenerate Fermi liquids in terms a weakly interacting gas of ``quasiparticles.'' Transport coefficients are calculated with a novel method which may be generalizable to systems with no translational invariance.
A microscopic theory is proposed for transverse dynamics and zero-temperature attenuation in polarized Fermi liquids. The transport equations are a set of two coupled equations in two ``partial transverse densities,'' which do not reduce to a single equation in a mixed component of a single-particle distribution. The effective interaction is linked to an irreducible vertex by an integral equation, and cannot be given as a limit of a full vertex. A framework for a generalized nonlocal Landau theory is established. The spectrum of attenuating spin waves is calculated at arbitrary polarizations and densities.