Impurity effects on BCS-BEC crossover in ultracold atomic Fermi gases

We present a systematic investigation of the effects of “nonmagnetic” impurities on the $s$-wave BCS-BEC crossover in atomic Fermi gases within a pairing fluctuation theory. Both pairing and impurity scattering $T$ matrices are treated self-consistently at the same time. While the system is less sensitive to impurity scattering in the Born limit, for strong impurity scatterers, both the frequency and the gap function are highly renormalized, leading to significant suppression of the superfluid $T_c$, the order parameter, and the superfluid density. We also find the formation of impurity bands and smearing of coherence peak in the fermion density of states, leading to a spectrum weight transfer and finite lifetime of Bogoliubov quasiparticles. In the BCS regime, the superfluidity may be readily destroyed by the impurity of high density, leading to a superfluid-insulator quantum phase transition at zero temperature. In comparison, the superfluidity in unitary and BEC regimes is relatively more robust.

Figure 1

Feynman diagrams for the regular impurity $T$ matrix $T_{\omega }$ and the anomalous impurity $T$ matrix $T_{{\mathrm{\Delta }}^{†}}$. The crosses denote the impurities and the dashed lines represent the impurity potential. The dressed thin solid and thick solid lines represent impurity dressed noninteracting and interacting fermion propagators, respectively. The shaded elliptical region denotes self-consistent impurity dressing of the double pairing vertex structure. See Ref. [4] for details.

Figure 2

The impurity renormalization functions ${\mathrm{\Sigma }}_{\omega }$ and ${\mathrm{\Gamma }}_0\left(\omega \right)$ at $T_c$, in a unitary Fermi gas at an intermediate impurity scattering level, with $n_i=0.05n, u/E_F^{}=20$. Re (Im) denotes the real (imaginary) part of the functions. There are sharp features near $\omega =\pm \mathrm{\Delta }$. The broad peak of ${\mathrm{\Sigma }}_{\omega }^{\prime \prime }$ at $\omega /\mathrm{\Delta }\approx -3$ (black curve) indicates the formation of an impurity band. The horizontal axes are rescaled by the gap $\mathrm{\Delta }$ of a clean system.

Figure 3

Effects of different impurity scattering strength $u$, calculated at $T_c$, from the Born limit $u/E_F^{}=1$ to the unitary limit $u/E_F^{}=100$ on the fermion DOS $N\left(\omega \right)$ in the BCS regime ($1/k_F^{}a=-1.5$), with impurity density $n_i^{}=0.05n$. The impurity band (IB) splits from the main band (blue curve) for an intermediate value of $u/E_F^{}=20$. The inset presents the details of the coherence peaks, sharing the same axis labels. Here the clean limit gap $\mathrm{\Delta }\approx 0.018E_F^{}$.

Figure 4

Effects of impurity scattering on the fermion DOS in the BCS regime ($1/k_F^{}a=-1$), with impurity scattering strength $u=10E_F^{}$ and impurity density varying from $n_i^{}=0.01n$ to $n_i^{}=0.1n$. Detailed structure of $N\left(\omega \right)$ near the gap edge, and the impurity band are shown in the left and right insets, respectively. Here $\mathrm{\Delta }\approx 0.085E_F^{}$ is the gap value in the clean limit, and all data were calculated at $T_c$.

Figure 5

Effects of impurity scattering on the fermion DOS at (a) $1/k_F^{}a=0$ and (b) 1, corresponding to unitary and BEC cases, respectively, with the impurity density $n_i^{}=0.05n$, for different impurity scattering strength $u$ varying from the Born limit $u/E_F^{}=2$ to the unitary limit $u/E_F^{}=100$. Calculated at $T_c$, here $(\mu /E_F^{},\mathrm{\Delta }/E_F^{})=(0.62,0.64)$ and $(-0.80,1.33)$, and the main band bottom is located at $\omega /\mathrm{\Delta }\approx -1.4$ and 1.17 in the clean limit for (a) and (b), respectively.

Figure 6

Behavior of $T_c$ throughout the BCS-BEC crossover in the clean limit (black dotted line) and in the presence of strong impurity scattering with density $n_i=0.05n$ and strength $u=100E_F^{}$ (red solid line). For comparison, also plotted is the mean field value of $T_c$ with the impurities (blue dashed line).

Figure 7

The ratio $2\mathrm{\Delta }\left(0\right)/T_c$ as a function of $1/k_{F}a$ in the BCS through unitary regimes with an impurity density $n_i/n=0.05$. Plotted in the inset is the zero $T$ DOS in the presence of strong impurity scattering with density $n_i=0.05n$ and strength $u=100E_F^{}$ (black solid line) at $1/k_{F}a=-2.2$, where $T_c$ already vanishes. For comparison, also plotted in the inset is the corresponding DOS in the clean limit (blue dashed line). Here the inset and the main figure share the same legends.

Figure 8

Superfluid transition temperature $T_c$ of a unitary Fermi gas as a function of impurity scattering strength $u$, from the Born limit through the unitary limit, with an impurity density $n_i=0.05n$.

Figure 9

(a) Superfluid gaps and order parameter ${\mathrm{\Delta }}_{sc}$ and (b) superfluid density $n_s/n$ in a unitary Fermi gas, as a function of $T/T_c^0$, for different densities of impurity, including the clean limit (black solid line) and $n_i/n=0.03$ (red dotted) and 0.05 (dashed lines), with scattering strength $u/E_F^{}=100$. For clarity, we show the total gap $\mathrm{\Delta }/E_F^{}$ and the pseudogap ${\mathrm{\Delta }}_{pg}/E_F^{}$ only the $n_i/n=0.05$ case. Shown in the inset is a continuous evolution of the corresponding ratio $2\mathrm{\Delta }\left(0\right)/T_c$ vs $n_i/n$. Here $T_c^0/T_F^{}=0.256$ is the $T_c$ in the clean limit.

Figure 10

Fractional superfluid density $n_s/n$ in a unitary Fermi gas at $T=0$ as a function of the relative impurity density $n_i/n$, for both unitary (blue solid) and Born (black dashed curves) scatterers.