Quantum Mechanical Limitations to Spin Diffusion in the Unitary Fermi Gas

We compute spin transport in the unitary Fermi gas using the strong-coupling Luttinger-Ward theory. In the quantum degenerate regime the spin diffusivity attains a minimum value of $D_s\simeq 1.3\hslash /m$ approaching the quantum limit of diffusion for a particle of mass $m$. Conversely, the spin drag rate reaches a maximum value of ${\Gamma }_{\mathrm{sd}}\simeq 1.2k_B{T}_F/\hslash$ in terms of the Fermi temperature $T_F$. The frequency-dependent spin conductivity ${\sigma }_s(\omega )$ exhibits a broad Drude peak, with spectral weight transferred to a universal high-frequency tail ${\sigma }_s(\omega \to \infty )={\hslash }^{1/2}C/3\pi (m\omega {)}^{3/2}$ proportional to the Tan contact density $C$. For the spin susceptibility ${\chi }_s(T)$ we find no downturn in the normal phase.

Figure 1

Spin diffusivity $D_s$ vs reduced temperature $T/T_F$ (solid red line) in the normal phase, $T>T_c\simeq 0.16T_F$. The experimental data [4] (blue squares) for the trapped gas are rescaled down by a factor of 4.7 to compensate for the effect of the trapping potential. The dashed black line is the result from kinetic theory, $D_s=1.1(T/T_F{)}^{3/2}\hslash /m$.

Figure 2

Spin conductivity ${\sigma }_s(\omega )$ (in units of $\hslash n/m{E}_F$) vs frequency (red circles) at $T=0.5T_F$. The Drude model (solid black line) has the same total spectral weight as ${\sigma }_s(\omega )$ given by the spin $f$-sum rule. Part of the spectral weight is transferred from lower frequencies into a universal high-frequency tail (dotted blue line) ${\sigma }_s(\omega \to \infty )=C/3\pi (m\omega {)}^{3/2}$ with Tan contact density $C=0.086k_F^4$ [9].

Figure 3

Spin drag rate ${\Gamma }_{\mathrm{sd}}$ (in units of $E_F/\hslash$) vs reduced temperature $T/T_F$ (solid red line). The experimental data [4] (blue squares) for a trapped gas are rescaled up by a factor of 5.3 to compensate for the effect of the trapping potential. The dashed black line is the result from kinetic theory, ${\Gamma }_{\mathrm{sd}}=0.9(T/T_F{)}^{-1/2}{E}_F/\hslash$.

Figure 4

Compressibility ${\chi }_n$ (circles) and spin susceptibility ${\chi }_s$ (squares) vs reduced temperature $T/T_F$. The experimental data [4] (full symbols) are compared to our Luttinger-Ward calculation (open symbols). The dashed black line is the susceptibility of the free Fermi gas.