Itinerant ferromagnetism in an interacting Fermi gas with mass imbalance

We study the emergence of itinerant ferromagnetism in an ultracold atomic gas with a variable mass ratio between the up- and down-spin species. Mass imbalance breaks the SU(2) spin symmetry, leading to a modified Stoner criterion. We first elucidate the phase behavior in both the grand canonical and canonical ensembles. Second, we apply the formalism to a harmonic trap to demonstrate how a mass imbalance delivers unique experimental signatures of ferromagnetism. These could help future experiments to better identify the putative ferromagnetic state. Furthermore, we highlight how a mass imbalance suppresses the three-body loss processes that handicap the formation of a ferromagnetic state. Finally, we study the time-dependent formation of the ferromagnetic phase following a quench in the interaction strength.

Figure 1

The phase behavior in the grand canonical ensemble with chemical potential imbalance $\Delta \mu /\mu$ and interaction strength $k_{\text{F}}a$ for three different values of mass imbalance. The solid red lines denote the onset of full polarization, into the phases shaded light and dark gray for up (${\varphi }_{\text{z}}/\rho =1$) and down (${\varphi }_{\text{z}}/\rho =-1$) spin, respectively. The dashed purple boundary separates systems that become polarized in the up-spin (above the dashed line, light blue shading) or down-spin (below the dashed line, dark blue shading) directions in the strongly interacting limit. Along the dashed purple line itself the polarization remains constant until reaching the black dot, past which polarization in either direction is equally favorable.

Figure 2

The phase behavior in the canonical ensemble with interaction strength $k_{\text{F}}a$ and net population imbalance $(N_{\uparrow }-N_{\downarrow })/(N_{\uparrow }+N_{\downarrow })$ for three different values of mass imbalance $r$. The light blue regions (Single Ph.) correspond to a paramagnetic gas, white (Ph. Sep) to a two-phase coexistence between two partially polarized gases, and dark gray (Ph. Sep, $\left|\frac{{\varphi }_{\text{z}}}{\rho }\right|=1$) to a two-phase coexistence between two fully polarized up and down spin gases.

Figure 3

Trap profiles for a gas at various interaction strengths with a mass imbalance $m_{\uparrow }/m_{\downarrow }=3/2$. For a cloud with $N_{\uparrow }/N_{\downarrow }=1$, (a)–(e) show the density of the heavier particles (dotted) and the density of lighter particles (solid) against radius. The axes are normalized by $\mathrm{ñ}$, the density of the heavier particles at the center of the trap when $k_{\text{F}}a=0$, and $\mathrm{R̃}$, the cloud size when $k_{\text{F}}a=0$. (f)–(j) show the density profiles for a cloud with $(N_{\uparrow }-N_{\downarrow })/(N_{\uparrow }+N_{\downarrow })=0.9$. At the center a grand canonical phase diagram reproduced from Fig. 1 shows dotted trajectories in parameter space corresponding to some of the trap profiles and (k) is referenced in the text.

Figure 4

A table of graphs with rows showing the cloud size ($R_{\text{tot}}$), kinetic energy ($E_{\text{K}}$) per particle of light and heavy species, and loss rate ($\Gamma$) as a function of interaction strength. The columns refer to population imbalances of $N_{\uparrow }/N_{\downarrow }=\{1/3,1,3,19\}$. Each graph shows three different mass imbalances $m_{\uparrow }/m_{\downarrow }=1$ by the solid red line, $m_{\uparrow }/m_{\downarrow }=3/2$ by the green dashed line, and the case of ${}^{40}\mathrm{K}$ and ${}^6\mathrm{Li}$, $m_{\uparrow }/m_{\downarrow }=20/3$, is the blue dotted line. The axes are normalized by the cloud size ($R_0$), kinetic energy of a particle at $R=0$ ($E_{\text{K}0}$), and peak loss rate (${\Gamma }_0$) for a mass and population balanced non-interacting cloud with the same total particle number.

Figure 5

(a) shows the growth rate ${\omega }_{\mathbf{q}}$ as a function of wave vector $q$ for collective modes at values of the dimensionless interaction strength $k_{\text{F}}a/k_{\text{F}}{a}_{\text{c}}-1=\{0.02,0.06,0.10\}$. Here $k_{\text{F}}{a}_{\text{c}}$ is the critical interaction strength of the Stoner transition, given in the general mass- and population-imbalanced case by the boundary in Fig. 1. The solid red curves correspond to the mass-balanced gas, and the dashed blue lines to a gas of ${}^6\mathrm{Li}$ and ${}^{40}\mathrm{K}$, where the chemical potentials have been tuned to give population balance at $k_{\text{F}}a=0$. (b) Summarizes the behavior of mode wave vector (primary $y$ axis) and maximum growth rate (secondary $y$ axis) in (a). The axes are normalized by the Fermi wave vector and chemical potential of the ${}^6\mathrm{Li}$ atoms.