Superfluidity at the BEC-BCS crossover in two-dimensional Fermi gases with population and mass imbalance

We explore the zero-temperature phase behavior of a two-dimensional two-component atomic Fermi gas with population and mass imbalance in the regime of the BEC-BCS crossover. Working in the mean-field approximation, we show that the normal and homogeneous balanced superfluid phases are separated by an inhomogeneous superfluid phase of Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) type. We obtain an analytical expression for the line of continuous transitions separating the normal and inhomogeneous FFLO phases. We further show that the transition from the FFLO phase to the homogeneous balanced superfluid is discontinuous leading to phase separation. If the species have different masses, the superfluid phase is favored when the lighter species is in excess. We explore the implications of these findings for the properties of the two-component Fermi gas in the atomic trap geometry. Finally, we compare and contrast our findings with the predicted phase behavior of the electron-hole bilayer system.

Figure 1

The phase diagram shown as a function of $\left(h/E_b,\mu /E_b\right)$ for three different values of the mass ratio $r$. The diagrams were constructed assuming a uniform order parameter, neglecting the potential for inhomogeneous phase formation. The solid lines represent continuous phase boundaries, while the dashed lines denote first-order transitions into the balanced SF phase.

Figure 2

The phase diagram as a function of $m/n$ and interaction strength $E_b/E_F$ of the two-dimensional system with fixed majority and minority particle densities for three different mass ratios $r$. The SF phase (dotted line) is the line of zero population imbalance. Between the balanced SF and PP/FP phase (dashed line) lies a region of phase separation (PS).

Figure 3

Variation of the order parameter $\left|{\Delta }_{\mathbf{Q}}\right|/E_b$ with chemical potential difference $h/E_b$ and chemical potential $\mu /E_b$. The upper panel is at a mass ratio $r=2$, and the lower panel at equal masses, $r=1$. The thin black lines trace out the $\left|{\Delta }_{\mathbf{Q}}\right|/E_b$ variation, found by minimizing the thermodynamic potential (1), for several different chemical potentials. The thick black line is the locus of the second-order transition predicted by Ginzburg-Landau theory.

Figure 4

The left-hand column shows the phase diagram in $\left(\mu /E_b,h/E_b\right)$ at three different mass ratios $r$. The solid lines represent second-order phase boundaries, the dashed line denotes first-order phase transitions into the balanced SF phase. The FFLO phase is highlighted in gray. The trajectories followed in the sample traps in Sec. 5 are shown by thin dashed lines. The central column of diagrams focus more closely on the topology of the phase diagram close to the tricritical point region for $h/E_b<0$. Thermodynamic potential surfaces are shown in the right-hand column; the darker the more negative and so more favorable; label (a) is a PP normal state, (b) a FFLO state, (c) a SF state, and (d) a FP state. Different plots have different shading calibrations, (unequally spaced) contour lines are also shown.

Figure 5

The phase diagram in $\left(m/n,E_b/E_F\right)$ of the two-dimensional system with fixed majority and minority particle densities for three different mass ratios $r$. The SF phase (dotted line) is represented by the line of zero population imbalance $m/n=0$. The PP phase is separated from the FFLO phase (highlighted in gray) by a second-order phase boundary (solid line). In the phase separated (PS) region bordered by the dashed line, the system separates into a balanced superfluid (SF) and either a normal phase or, depending on the composition, an inhomogeneous FFLO phase. The right-hand column of graphs focus on the terminus of the PP-FFLO phase boundary on the side of negative polarization for the three featured mass ratios.

Figure 6

(Color online) Radial density profiles of four systems in identical harmonic traps with different values of population imbalance $h$ and mass ratio $r$. The solid line shows the radial density based on the primary $y$ axis. The dashed line shows the local population imbalance based on the secondary $y$ axis. The band shows where different phases exist with colors labeled on the right-hand side of the figure. With changing radial coordinate, the separate panels are associated with the trajectories highlighted in the $\left(h/E_b,\mu /E_b\right)$ phase diagram shown in Fig. 4.