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arXiv:1012.1607 · New Journal of Physics

Spin diffusion in Fermi gases

Georg M. Bruun

Published Mar 16, 2011• 48 citations
arXivHTMLPDF

Abstract

We examine spin diffusion in a two-component homogeneous Fermi gas in the normal phase. Using a variational approach, analytical results are presented for the spin diffusion coefficient and the related spin relaxation time as a function of temperature and interaction strength. For low temperatures, strong correlation effects are included through the Landau parameters, which we extract from Monte Carlo results. We show that the spin diffusion coefficient has a minimum for a temperature somewhat below the Fermi temperature with a value that approaches the quantum limit ∼ℏ/ m in the unitarity regime, where m is the particle mass. Finally, we derive a value for the low-temperature shear viscosity in the normal phase from the Landau parameters.

DOI
10.1088/1367-2630/13/3/035005

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•Oct 28, 2015•1 citation
We examine spin diffusion in a two-component homogeneous Fermi gas in the normal phase. Using a variational approach, analytical results are presented for the spin diffusion coefficient and the related spin relaxation time as a function of temperature and interaction strength. For low temperatures, strong correlation effects are included through the Landau parameters, which we extract from Monte Carlo results. We show that the spin diffusion coefficient has a minimum for a temperature somewhat below the Fermi temperature with a value that approaches the quantum limit ∼ℏ/ m in the unitarity regime, where m is the particle mass. Finally, we derive a value for the low-temperature shear viscosity in the normal phase from the Landau parameters.
Using kinetic theory, we calculate the shear viscosity and the spin-diffusion coefficient as well as the associated relaxation times for a two-component Fermi gas in two dimensions, as a function of temperature, coupling strength, polarization, and mass ratio of the two components. It is demonstrated that the minimum value of the viscosity decreases with the mass ratio, since Fermi blocking becomes less efficient. We furthermore analyze recent experimental results for the quadrupole mode of a two-dimensional gas in terms of viscous damping, obtaining a qualitative agreement using no fitting parameters.
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.
We compute the shear viscosity, thermal conductivity, and spin diffusivity of a Fermi gas with short-range interactions in the Fermi liquid regime of the normal phase, that is, at temperatures T much lower than the Fermi temperature T_{F} and larger than the superfluid critical temperature T_{c}. In line with recent advances in the precision of cold atom experiments, we provide exact results up to the second order in the interaction strength. We extend the Landau-Salpeter equation to compute the collision amplitude beyond the forward-scattering limit, covering all collisions on the Fermi surface. We treat the collision kernel exactly, leading to significant corrections beyond relaxation-time or variational approximations. The transport coefficients, as functions of the s wave scattering length a and Fermi wave number k_{F}, follow (1+γk_{F}a)/a^{2} up to corrections of order O(a^{0}), with a positive coefficient γ for the viscosity and negative one for the thermal conductivity and spin diffusivity.
We investigate the temperature dependence of the shear viscosity and spin diffusion in a two-dimensional Fermi gas with contact interactions, as realized in ultracold atomic gases. We describe the transport coefficients in terms of a Boltzmann equation and present a full numerical solution for the degenerate gas. In contrast to previous works we take the medium effects due to finite density fully into account. This effect reduces the viscosity-to-entropy ratio $\ensuremath{\eta}/s$ by a factor of 3, and similarly for spin diffusion. The trap-averaged viscosity agrees well with recent measurements by Vogt et al. [Phys. Rev. Lett. 108, 070404 (2012)].
The scattering properties of spin-polarized Fermi gases are dominated by ppp-wave interactions. Besides their inherent angular dependence, these interactions differ from their sss-wave counterparts as they also require the presence of a finite effective range in order to understand the low-energy properties of the system. In this article we examine how the shear viscosity and thermal conductivity of a three-dimensional spin-polarized Fermi gas in the normal phase depend on the effective range and the scattering volume in both the weakly and the strongly interacting limits. We show that, although the shear viscosity and the thermal conductivity both explicitly depend on the effective range near resonance, the Prandtl number which parametrizes the ratio of momentum to thermal diffusivity does not have an explicit interaction dependence both at resonance and for weak interactions in the low-energy limit. In contrast to sss-wave systems, ppp-wave scattering exhibits an additional resonance at weak attraction from a quasi-bound state at positive energies, which leads to a pronounced dip in the shear viscosity at specific temperatures.
In the first part of the thesis I derive a full quantitative formula which describes the amplitude and frequency of magnetic oscillations in two-dimensional Dirac systems. The investigations are on the basis of graphene, but they generally also hold for other two-dimensional Dirac systems. Starting from the Luttinger-Ward functional I derive an expression for the oscillatory part of the grand potential of graphene in a magnetic field. The amplitude of this ex- pression is usually called the Lifshitz-Kosevich (LK) formula. I perform the computation for the clean and the disordered system, and I study the effect of electron-electron interactions on the oscillations. I discuss my results by comparing them to the analogue expressions for the two-dimensional Fermi gas (2DEG) which have been derived by Adamov et al.. I find that, unlike in the 2DEG, a finite temperature and impurity scattering also affects the oscillation frequency. Further I find that in graphene, compared to the 2DEG, additional interaction induced damping effects occur: to two-loop order electron-electron interactions do lead to an additional damping factor in the amplitude of the LK-formula. Moreover the renormalization effects cannot fully be accounted for by renormalizations of the Fermi velocity but they also have to be described by field renormalizations. In the second part of the thesis I investigate the temperature dependence of the shear viscosity and spin diffusion in a two-dimensional, two-component Fermi gas, as realized in ultracold atomic gases. I implement a contact interaction that only acts between fermions in different hyperfine states. The transport coefficients are obtained within a kinetic approach. I solve the linearized Boltzmann equation by using a variational principle and present a full numerical solution for the degenerate gas. In contrast to previous works I take the medium effect due to finite density fully into account. This effect reduces the viscosity to particle density ratio, η/n, by a factor of four for strong interactions; and similarly for spin diffusion. The lowest value I obtain for the viscosity to entropy ratio is η/s = 0.15h ̄/kB, and it occurs close to the phase transition to the superfluid phase. This value is about twice the conjectured lower bound of η/s = 1/(4π) ̄h/kB, computed using the AdS/CFT correspondence. I compare my result for the shear viscosity to the measurements by Vogt et al., who measured the damping rate of the quadrupole mode of a trapped Fermi gas confined to two dimensions. This damping rate is related to the shear viscosity of the gas and our numerical results agree well with the experiment.