Itinerant chiral ferromagnetism in a trapped Rashba spin-orbit-coupled Fermi gas

We consider a repulsive two-component Fermi gas confined in a two-dimensional isotropic harmonic potential and subject to a large Rashba spin-orbit coupling. The single-particle dispersion can be tailored by the spin-orbit-coupling term, which provides an opportunity to study itinerant ferromagnetism in this system. We show that the interplay among spin-orbit coupling, correlation effect, and mean-field repulsion leads to a competition between ferromagnetic and nonmagnetic phases. The weakly correlated nonmagnetic and the ferromagnetic phases can be well described by the mean-field Hartree-Fock theory, while the transition between the ferromagnetic and a strongly correlated nonmagnetic phase is driven by beyond-mean-field quantum correlation effect. Furthermore, the ferromagnetic phase of this system possesses a chiral current density induced by the Rashba spin-orbit coupling, whose experimental signature is investigated.

Figure 1

Phase diagram in the $g\text{−}\tilde{\lambda }$ plane. Here we consider six repulsively interacting spin-1/2 fermions confined in a 2D harmoinc trap, subject to Rashba SO coupling. $g$ is the interaction strength, and is normalized to $g_{\mathrm{MF}}^0=2\pi {\hslash }^2/M$ which is the mean-field critical interaction strength for a 2D Fermi gas without SO coupling. $\tilde{\lambda }$ is the dimensionless SO coupling strength. We can see that the phase diagram contains three phases: the weakly correlated nonmagnetic phase, chiral magnetic phase, and strongly correlated nonmagnetic phase. The dashed line represents the mean-field results which contain only two regimes: a nonmagnetic phase below the dashed line and a ferromagnetic phase above the dashed line.

Figure 2

Left column: (a1)–(a3) show the ground state angular momentum per particle $J_z/N$, entanglement entropy (EE) and ground state spin fluctuations ${\left(\mathrm{\Delta }{S}_z\right)}^2$ as a function of interaction strength for $\tilde{\lambda }=7,N=6$. The non-zero $J_z/N$ indicates a magnetic ground state. Right column: (b1)–(b3) show three representative single-particle occupation number $n_m$ of state $|m\rangle$ for interaction strengthes marked by the yellow triangles in (a1).

Figure 3

(a) Spin texture of the ferromagnetic state and (b) total number density (color map) and the chiral current density (arrows) of the ferromagnetic state.

Figure 4

Density profiles of each spin species for different interaction strengths with $N=6$ and $\tilde{\lambda }=7$, by both ED (red solid lines: thick lines for spin up and thin lines for spin down) and HF (black dashed lines: thick lines for spin up and thin lines for spin down) methods.

Figure 5

(a) ED results for the ground-state energy $E_G$, the kinetic energy $E_{\mathrm{kin}}$, and the interaction energy $E_{\mathrm{int}}$ as functions of interaction strength. (b) The ratio of the interaction energy from the ED calculation and that from the HF calculation. The two vertical lines separate the parameter space into three phases according to the ED calculation: from left to right, we have the weakly correlated nonmagnetic phase, the ferromagnetic phase, and the strongly correlated nonmagnetic phase. Here $N=6$ and $\tilde{\lambda }=7$.

Figure 6

(a) Critical interaction strength $g_c$ for $\tilde{\lambda }=20$, above which the system is ferromagnetic. The two red triangles indicate the position where a new Landau level starts to be populated. (b) The single-particle energy bands (solid curved lines) and the Fermi level (dashed horizontal lines) for $N=58$ (upper panel) and $N=138$ (lower panel).

Figure 7

(a) Critical interaction strength $g_c$ obtained by ED calculation as a function of fermion number $N$ with SO coupling strength fixed at $\tilde{\lambda }=3$. (b) The occupation number in each band (red circles: lowest Landau level; green squares: second Landau level) shown for different particle numbers with $g=0.2g_{\mathrm{MF}}^0$.

Figure 8

(a) Ferromagnetic regime (shaded regions bounded between two lines) obtained by ED calculation for different atom number $N=4$ (dash dotted lines), 6 (dotted lines), and 8 (solid lines). (b) The EE as a function of interaction strength for atom number $N=4,6,8$ and SO coupling strength $\tilde{\lambda }=9$ (indicated by the red dashed line in (a)). The shaded regime between two vertical lines indicate the ferromagnetic regimes for different $N$ with the same corresponding line styles as in (a).

Figure 9

Time evolution of the atomic cloud after a sudden trap deformation. At $t=0$, the trapping frequency along the $y$ axis is suddenly changed from $\omega$ to $3.16\omega$, while that along the $x$ axis remains at $\omega$. The upper (lower) panel shows the dynamics of a nonmagnetic (ferromagnetic) state. Here $N=6,\tilde{\lambda }=7$, and ${\tau }_0=1/\omega$.

Figure 10

Average angular momentum per particle $L_z/N$ as a function of $N$. Note that the spin population oscillates in space such that the average spin per particle $S_z/N$ is very small. Hence $L_z/N\approx J_z/N$. Here we fix the interaction strength $g=0.1g_{\mathrm{MF}}^0$, the trap frequency is scaled as $\omega ={\omega }_0/\sqrt{N}$, and the dimensionless SO coupling strength is scaled accordingly as $\tilde{\lambda }={\tilde{\lambda }}_0{N}^{1/4}$. In the calculation, we take ${\tilde{\lambda }}_0=6$.

Figure 11

The schematic plot of the spin polarization (black arrow) and density current (red arrow) for (a) Rashba SO coupled system and (b) Dresselhaus SO coupled system, respectively.