Transport of spin and mass at normal-superfluid interfaces in the unitary Fermi gas

Transport in strongly interacting Fermi gases provides a window into the nonequilibrium behavior of strongly correlated fermions. In particular, the interface between a strongly polarized normal gas and a weakly polarized superfluid at finite temperature presents a model for understanding transport at normal-superfluid and normal-superconductor interfaces. An excess of polarization in the normal phase or a deficit of polarization in the superfluid brings the system out of equilibrium, leading to transport currents across the interface. We implement a phenomenological mean-field model of the unitary Fermi gas, and investigate the transport of spin and mass under nonequilibrium conditions. We consider independently prepared normal and superfluid regions brought into contact, and calculate the instantaneous spin and mass currents across the normal-superfluid (NS) interface. For an unpolarized superfluid, we find that spin current is suppressed below a threshold value in the driving chemical potential differences, while the threshold nearly vanishes for a critically polarized superfluid. The mass current can exhibit a threshold in cases where Andreev reflection vanishes, while in general Andreev reflection prevents the occurrence of a threshold in the mass current. Our results provide guidance to future experiments aiming to characterize spin and mass transport across NS interfaces.

Figure 1

Schematic of nonequilibrium and equilibrium states of a phase-separated spin-imbalanced Fermi gas at finite temperature. Blue: Majority (spin up), yellow: minority (spin down); lines represent Cooper pairing.

Figure 2

Phase diagram of the two-component Fermi mixture in the unitarity limit, consisting of the superfluid phase (SF), the forbidden region (FR), and the normal phase (N) in the coordinates of $p$ and $T/T_{F\uparrow }$. The dashed line denotes the approximate phase boundary beyond the tricritical temperature $T_{c3}$. Square (red): Quantum critical point from Ref. [52] at polarization $p_{c0}=0.39$. Circle (blue): The tricritical point from Ref. [52], at polarization $p_{c3}=0.24$ and temperature $T_{c3}=0.06T_{F\uparrow }$. Triangle (green): Critical temperature $T_c=0.167T_F$ at zero polarization from Ref. [5].

Figure 3

Thermodynamic states considered, for the case ${\mu }_N={\mu }_S$. For the subcase $h_s=0$ (unpolarized superfluid), the red dot indicates the state of the superfluid region while the red line indicates the allowed states of the normal region, given the temperature $T=0.05{\mu }_S$ and the condition ${\mu }_N={\mu }_S$. For the other subcase, $h_s=h_c$ (critically polarized superfluid), the blue dot indicates the state of the superfluid region while the blue dashed line indicates the allowed states of the normal region. The solid black and black dashed curves show the normal-superfluid phase boundary above and below the tricritical point, respectively. The maximum value of $h_N/{\mu }_N$ on the red solid and blue dotted lines corresponds to $p_N=0.99$, while the minimum $h_N/{\mu }_N$ corresponds to the critical polarization in the normal phase of $p_{Nc}=0.34$ at $T/{\mu }_N=T/{\mu }_s=0.05$.

Figure 4

Net, spin, spin-up, and spin-down current densities for the case ${\mu }_N={\mu }_S$ as functions of $h_N/{\mu }_S$, under the conditions (a) $h_S=0$ and (b) $h_S=h_c$. For both plots, the temperature is $T=0.05{\mu }_S$. The vertical dashed line in (a) denotes the $h_N$ value at which $\delta {\mu }_{\uparrow }=\delta {\mu }_{\downarrow }=E_{min}$, corresponding to normal-region polarization $p_N=0.44$. The horizontal dashed line in both plots is the $J_i=0$ line. The horizontal axes start at $h_N=h_c(T=0.05{\mu }_S)=0.91{\mu }_s$.

Figure 5

Chemical potentials and current densities across the interface, away from mechanical equilibrium, for parameters $p_N=99\%$, and $h_S=h_c, {\mu }_N={\mu }_S, T=0.05{\mu }_S$. The left (right) side of the figure corresponds to the normal (superfluid) region. The horizontal axis is qualitative, showing the directions and relative magnitudes of the currents. The vertical axis shows the chemical potentials quantitatively.

Figure 6

Thermodynamic states considered, for the case of mechanical equilibrium at temperature $T=0.05{\mu }_S$. The maximum value of $h_N/{\mu }_N$ on the red solid (blue dotted) line corresponds to $p_N=0.96$.

Figure 7

Chemical potentials and schematic current densities across the interface, at mechanical equilibrium, with $p_N=0.96$, and $h_S=h_c, T=0.05{\mu }_S$.

Figure 8

Net, spin, spin-up, and spin-down current densities as functions of $h_N$, under the conditions (a) $h_S=0$ and (b) $h_S=h_c$. For both plots, temperature is $T=0.05{\mu }_S$. The dashed vertical line denotes the $h_N$ value at which $\delta {\mu }_{\downarrow }=E_{min}$.

Figure 9

The normal and Andreev contributions to the net current as functions of $h_N$, under the conditions (a) $h_S=0$ and (b) $h_S=h_c$. For both plots, temperature is $T=0.05{\mu }_S$.

Figure 10

Normal (non-Andreev) contributions to the spin-up and spin-down currents on the $\alpha$ branch versus $h_N$. (a) $h_S=0$, (b) $h_S=h_c$. For both plots, temperature is $T=0.05{\mu }_S$. The dashed vertical line denotes the $h_N$ value at which $\delta {\mu }_{\downarrow }=E_{min}$, while the dotted vertical line indicates $\delta {\mu }_{\uparrow }=E_{min}$.

Figure 11

Dispersion curve of superfluid excitation energy for both branches at $T=0.05{\mu }_S$ and maximal superfluid polarization $h_S=h_c$. The solid line represents $E_{\alpha }$, and the dashed line represents $E_{\beta }$. The energy and the wave vector are normalized by ${\mu }_S$ and $k_s$, respectively, with $k_s=2m{\mu }_S/{\hslash }^2$ and ${\mu }_S$ the average chemical potential (7).

Figure 12

Alpha-branch scattering regimes on the excitation energy $E_{\alpha }$ vs transverse kinetic energy ${\xi }_{\perp }$ plane. The case shown has ${\mu }_N={\mu }_S$ and $p_N=0.8$. The other parameters, normalized by ${\mu }_S$, are $T=0.05, h_S=0, U_S=1.13, U_h=0, U_{N\uparrow }=0.16, U_{N\downarrow }=2.10$.

Figure 13

Spin current $J^{\text{spin}}/J_0$ plotted vs temperature $T/{\mu }_S$ under mechanical equilibrium with $p_N=99\%, h_S=0$.

Figure 14

Normalized chemical potential differences $\delta {\mu }_{\uparrow (\downarrow )}$ plotted vs $h_N$ under mechanical equilibrium, at $T=0.05{\mu }_S$, for (a) $h_s=0$ and (b) $h_s=h_c$. The dashed line shows the minimum of the superfluid excitation spectrum at the given value of $h_s$.

Figure 15

Zeeman field $h_N$ versus normal region polarization $p_N$ under the conditions (a) ${\mu }_N={\mu }_S$ and (b) mechanical equilibrium. In case (a), $h_N/{\mu }_S$ is independent of $h_S$ because ${\mu }_N/{\mu }_S$ is constant.