Phase structure of spin-imbalanced unitary Fermi gases

We investigate the phase structure of spin-imbalanced unitary Fermi gases beyond mean-field theory by means of the functional renormalization group. In this approach, quantum and thermal fluctuations are resolved in a systematic manner. The discretization of the effective potential on a grid allows us to accurately account for both first- and second-order phase transitions that are present on the mean-field level. We compute the full phase diagram in the plane of temperature and spin imbalance and discuss the existence of other conjectured phases such as the Sarma phase and a precondensation region. In addition, we explain on a qualitative level how we expect that in situ density images are affected by our findings and which experimental signatures may potentially be used to probe the phase structure.

Figure 1

Mean-field phase diagram of the spin-imbalanced UFG. We show the superfluid-to-normal transition and the Sarma crossover. The location of the phase boundaries can be obtained from the mean-field expression for the grand-canonical potential, Eq. (3), or from the FRG flow by omitting bosonic fluctuations. In the latter approach it is possible to additionally resolve the precondensation line (black dotted line), below which a minimum at ${\Delta }_{0,k}$ appears during the RG flow but vanishes for $k\to 0$, thereby leaving the system in the normal phase. We discuss the phenomenon of precondensation in Sec. 4 in detail.

Figure 2

Phase diagram of the spin-imbalanced UFG beyond the mean-field approximation. The phase boundaries are obtained from the FRG evolution of the effective potential including the feedback of bosonic fluctuations. The critical temperature of the balanced system is found to be $T_{\mathrm{c}}/\mu =0.40$. For small $\delta \mu /\mu$ we find a second-order phase transition with a reduced critical temperature. For low temperatures, spin imbalance results in a breakdown of superfluidity by means of a first-order phase transition. We extract $\delta {\mu }_{\mathrm{c}}/\mu =0.83$ for the critical imbalance at zero temperature. The second-order line terminates in a tricritical point (CP). We indicate the Sarma crossover by the green dash-dotted line. The region between the precondensation line (black dotted line) and the phase boundary gives an estimate for the pseudogap region, as explained in the main text.

Figure 3

Scale evolution of the minimum ${\Delta }_{0,k}$ of the effective average potential close to a first-order phase transition at $\delta {\mu }_c$. The evolution proceeds from the ultraviolet ($k=\Lambda$, $t=0$) to the infrared ($k\to 0$, $t\to -\infty$). The insets show the shape of the effective potential $U_k\left(\Delta \right)$ at several points along the scale evolution. The solid red lines correspond to a point in the broken phase ($\delta {\mu }_{\mathrm{SF}}=0.78\mu$), where the global minimum of the effective potential is nonzero in the infrared. The dashed blue line represents a point with $\delta {\mu }_{\mathrm{NF}}=0.79\mu$, where the global minimum in the infrared is located at ${\Delta }_{0,k=0}=0$. For all plots, $T/\mu =0.17$.

Figure 4

Schematic in situ density profile $n_{\sigma }\left(r\right)$ for a population-imbalanced ensemble with $N_1>N_2$ at low temperature. The blue (dark gray) and red (light gray) points correspond to atoms in hyperfine states $|1\rangle$ and $|2\rangle$, respectively. For $T<T_{\mathrm{CP}}$ the superfluid transition is of first order, such that the superfluid inner region is separated from the polarized normal gas by a jump in density at the critical radius $r_{\mathrm{c}}$.

Figure 5

Regularization scheme dependence of the phase boundary. We display the phase diagram obtained by applying the fermion regulator $R_{\psi \sigma }$ from Eqs. (13) (blue long-dashed line, “Reg. 1”) and (14) (red short-dashed line, “Reg. 2”).

Figure 6

Using a Taylor expansion of the effective average action $U_k\left(\rho \right)$ to order ${\rho }^2\sim {\varphi }^4$ (brown dotted line) in the flow equation, the location of the second-order line deviates quantitatively from the grid solution (blue dashed line) as we increase $\delta \mu /\mu$.