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Physical Review Letters

Transverse spin diffusion in polarized Fermi gases

J. W. Jeon, William J. Mullin

Published Jun 5, 1989• 90 citations
Publisher

Abstract

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/${\mathit{T}}^{2}$. Spin waves would thus continue to be damped even to T=0 K.

DOI
10.1103/physrevlett.62.2691

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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 \ensuremathτ\ensuremath⊥{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremathτ\ensuremath⊥​ that is often shorter than the corresponding longitudinal collision time \ensuremathτ\ensuremath∥{\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\parallel}}}\ensuremathτ\ensuremath∥​ that is often shorter than the corresponding longitudinal collision time t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​. As T\ensuremath{\rightarrow}0, \ensuremathτ\ensuremath∥{\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\parallel}}}\ensuremathτ\ensuremath∥​ approaches T-independent value, in constrast to \ensuremathτ\ensuremath⊥{\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\perp}}}\ensuremathτ\ensuremath⊥​\ensuremath{\sim}1/T2{\mathit{T}}^{2}T2. Spin waves would thus continue to be damped even to T=0 K.
We report the solution of a recently derived kinetic equation for a dilute quantum system at arbitrary degeneracy and polarization. From this equation we have developed 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 τ⊥ that is often shorter than the corresponding longitudinal collision time, τ∥. As T→0, τ⊥ approaches a T‐independent value, in contrast to τ∥∼1/T2. Spin waves should thus continue to be damped even to T=0 K.
We compute spin diffusion in a dilute Fermi gas at arbitrary temperature, polarization, and strong interaction in the normal phase using kinetic theory. While the longitudinal spin diffusivity ${D}_{\ensuremath{\parallel}}$ depends weakly on polarization and diverges for small temperatures, the transverse spin diffusivity ${D}_{\ensuremath{\perp}}$ has a strong polarization dependence and approaches a finite value for $T\ensuremath{\rightarrow}0$ in the Fermi liquid phase. For a 3D unitary Fermi gas at infinite scattering length, the diffusivities reach a minimum near the quantum limit of diffusion $\ensuremath{\hbar}/m$ in the quantum degenerate regime and are strongly suppressed by medium scattering, and we discuss the importance of the spin-rotation effect. In two dimensions, ${D}_{\ensuremath{\perp}}$ attains a minimum at strong coupling $\ensuremath{-}1\ensuremath{\lesssim}\mathrm{ln}({k}_{F}{a}_{2D})\ensuremath{\lesssim}1$ and reaches ${D}_{\ensuremath{\perp}}\ensuremath{\sim}0.2...\phantom{\rule{0.16em}{0ex}}0.3\phantom{\rule{0.16em}{0ex}}\ensuremath{\hbar}/m$ at large polarization. These values are consistent with recent measurements of two-dimensional ultracold atomic gases in the strong coupling regime.
Recent experiments in a mixture of two hyperfine states of trapped Bose gases show behavior analogous to a spin-1/2 system, including transverse spin waves and other familiar Leggett-Rice-type effects. We have derived the kinetic equations applicable to these systems, including the spin dependence of interparticle interactions in the collision integral, and have solved for spin-wave frequencies and longitudinal and transverse diffusion constants in the Boltzmann limit. We find that, while the transverse and longitudinal collision times for trapped Fermi gases are identical, the Bose gas shows diffusion anisotropy. Moreover, the lack of spin isotropy in the interactions leads to the non-conservation of transverse spin, which in turn has novel effects on the hydrodynamic modes. PACS numbers: 03.75.Mn,05.30Jp,05.60.Gg,51.10.+y,67.20.+k. In recent JILA experiments,1,2 a mixture of two hyperfine states was found to segregate by species. The theoretical explanation3−8 for this behav-ior is based on the two states playing the role of a pseudo-spin-1/2 system,
A kinetic equation, which includes the effects of degeneracy, is derived for dilute, polarized systems by the Green's function method of Kadanoff and Baym. When the Born approximation is used for the self-energy, the equation reduces to a result due to Silin. In the Boltzmann limit our result is equivalent to the equation of Lhuillier and Laloë, with the addition of a mean-field drift term analogous to that appearing in the Landau-Silin equation. Our kinetic equation is used to derive an expression for the transverse spin-diffusion relaxation time, τ>, for a Fermi system. In the Boltzmann and low-polarization limits τ> reduces to τ ∥, the longitudinal relaxation time. However, in a highly polarized degenerate system τ> can be very much shorter than τ ∥.
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.
Recent experiments in a mixture of two hyperfine states of trapped Bose gases show behavior analogous to a spin-12\frac{1}{2}21​ system, including transverse spin waves and other familiar Leggett-Rice-type effects. We have derived the kinetic equations applicable to these systems, including the spin dependence of interparticle interactions in the collision integral, and have solved for spin-wave frequencies and longitudinal and transverse diffusion constants in the Boltzmann limit. We find that, while the transverse and longitudinal collision times for trapped Fermi gases are identical, the Bose gas shows unusual diffusion anisotropy in both dipole and quadrupole modes. Moreover, the lack of spin isotropy in the interactions leads to the nonconservation of transverse spin, which in turn has significant effects on the hydrodynamic modes.