Spin transport in a unitary Fermi gas close to the BCS transition

We consider spin transport in a two-component ultracold Fermi gas with attractive interspecies interactions close to the BCS pairing transition. In particular, we consider the spin-transport relaxation rate and the spin-diffusion constant. Upon approaching the transition, the scattering amplitude is enhanced by pairing fluctuations. However, as the system approaches the transition, the spectral weight for excitations close to the Fermi level is decreased by the formation of a pseudogap. To study the consequence of these two competing effects, we determine the spin-transport relaxation rate and the spin-diffusion constant using both a Boltzmann approach and a diagrammatic approach. The former ignores pseudogap physics and finite lifetime effects. In the latter, we incorporate the full pseudogap physics and lifetime effects, but we ignore vertex corrections, so that we effectively calculate single-particle relaxation rates instead of transport relaxation rates. We find that there is qualitative agreement between these two approaches, although the results for the transport coefficients differ quantitatively.

Figure 1

The dimensionless spin-transport relaxation rate $\hslash /{\epsilon }_F{\tau }_D$ obtained from Eq. (7) as a function of the reduced temperature $T/T_F$. From top to bottom the curves correspond to interaction parameters $-1/k_{F}a=\{0,0.56,1.0,1.4\}$. Each curve terminates at the value of the transition temperature corresponding to its $k_{F}a$ value: in decreasing order these are $T/T_F\approx \{0.50,0.24,0.13,0.064\}$. The dashed line is the high-temperature asymptote $1/{\tau }_D\propto 1/\sqrt{T}$ predicted in Ref. [6].

Figure 2

The dimensionless spin-diffusion constant $D_{s}m/\hslash$ obtained from the result of Fig. 1 using the Einstein relation. From bottom to top the curves correspond to interaction parameters $-1/k_{F}a=\{0,0.56,1.0,1.4\}$. Each curve terminates at the value of the transition temperature corresponding to its $k_{F}a$ value. The dashed line is the high-temperature asymptote $D_s=(\hslash /m)(9{\pi }^{3/2}/16\sqrt{2}){(T/T_F)}^{3/2}$.

Figure 3

The self-energy $\Sigma$ is the many-body $T$ matrix closed with a single noninteracting Green's-function line. The external lines are indicated by the dashed lines.

Figure 4

The imaginary (solid line) and real (dashed line) parts of the self-energy at unitarity for $\hslash k=\sqrt{2m\mu }$ for (a) $k_{B}T/\mu =1$ and (b) $T=T_c$.

Figure 5

The spectral function $\rho$ at unitarity for $\hslash k=\sqrt{2m\mu }$ for $k_{B}T/\mu =1$ (dashed line) and $T=T_c$ (solid line).

Figure 6

The solution of the number equation giving $\mu$ as a function of temperature. The dashed line is the noninteracting result. The solid line gives the result at unitarity.

Figure 7

The density of states $\nu$ vs $\omega$. The dashed curve is the noninteracting result for $T=0$, while the solid curve corresponds to the unitary case for $T=T_c$.

Figure 8

The spin-transport relaxation rate $1/{\tau }_D$ as a function of temperature at unitarity. The dashed line is the Boltzmann result, and the solid line is the diagrammatic result obtained by evaluating Eq. (15).

Figure 9

The spin-diffusion constant $D_s$ as a function of temperature at unitarity. The dashed line is the Boltzmann result, and the solid line is the diagrammatic result obtained by evaluating the spin susceptibility in Eq. (16) and using the Einstein relation.

Figure 10

The spin susceptibility obtained by evaluating Eq. (16) scaled by the noninteracting zero-temperature value ${\chi }_0=3n/2{\epsilon }_F$ as the solid line. The dashed line corresponds to the noninteracting result. The solid curve terminates at the critical temperature $T_c=0.29T_F$.