Unconventional superfluidity of fermions in Bose-Fermi mixtures

We examine a two-dimensional mixture of single-component fermions and dipolar bosons. We calculate the self-enregies of the fermions in the normal state and the Cooper-pair channel by including first-order vertex correction to derive a modified Eliashberg equation. We predict the appearance of superfluids with various nonstandard pairing symmetries at experimentally feasible transition temperatures within the strong-coupling limit of the Eliashberg equation. Excitations in these superfluids are anyonic and follow non-Abelian statistics.

Figure 1

Schematic diagram of phonon propagator including the polarizing effect of the fermions. The thick dashed line corresponds to the dressed phonon propagator $D(i{\omega }_s,k_{\perp })$, while the thin dashed line corresponds to noninteracting phonon propagator $D_0(i{\omega }_s,k_{\perp })$. The solid line corresponds to the free fermion propagator.

Figure 2

The dressed excitation spectrum $\Omega ({\mathrm{k⃗}}_{\perp })$ (in units of $\hslash {\omega }_b$) as a function of ${\mathrm{k⃗}}_{\perp }$ for various values of the Bose-Fermi interaction: ${\mathcal{G}}_{\mathrm{bf}}=0$ (upper dash-dotted line), $0.51$ (middle dashed line), and $0.57$ (bottom solid line). The fixed parameters are $m_f=53$, $\frac{3g}{4\pi g_{\mathrm{dd}}}=0.6$, and $g_{3\mathrm{D}}=2.5.$

Figure 3

Effective fermion-fermion interaction strength ${\lambda }_L(0,0)$ in the angular momentum channels $L=0$ (dashed line), $L=1$ (solid line), $L=3$ (dotted line), and $L=5$ (dash-dotted line) as functions of the fermion dimensionality parameter $\eta$. We fixed $m_f/m_b=53/52$, $g_{3\mathrm{D}}=4$, $3g/8\pi g_{\mathrm{dd}}=0.6$, and ${\mathcal{G}}_{\mathrm{bf}}=0.5$.

Figure 4

Fermion self-energy in normal state including the first-order vertex correction. The solid line denotes the fermion propagator while the thick dashed line denotes the dressed phonon propagator of Eq. (6).

Figure 5

Plot of the vertex correction in $P_v^0$ as a function of $\frac{{\epsilon }_F}{\pi T}$ for different values of the dimensionality parameter $\eta =1.5$ (dotted line), $\eta =1$ (solid line), and $\eta =0.6$ (dashed line). We fixed $m_f/m_b=53/52$, $\frac{3g}{4\pi g_{\mathrm{dd}}}=0.6,{\mathcal{G}}_{}=0.5$, and $g_{3\mathrm{D}}=3.8$.

Figure 6

$\chi (T)$ as a function of $\frac{{\epsilon }_F}{\pi T}$ for different values of the dimensionality parameter $\eta =1.5$ (dotted line), $\eta =1$ (solid line), and $\eta =0.6$ (dashed line). We fixed $m_f/m_b=53/52$, $\frac{3g}{4\pi g_{\mathrm{dd}}}=0.6$, ${\mathcal{G}}_{}=0.5$, and $g_{3\mathrm{D}}=3.8$.

Figure 7

Schematic diagram of fermion self-energy in the Cooper-pair channel including the first-order vertex correction. The solid line denotes the fermion propagator while the thick dashed line denotes the dressed phonon propagator of Eq. (6). The first diagram after the equals sign denotes the direct interaction, the second and third diagrams denote the first-order vertex corrections, and the fourth diagram denotes the cross-term.

Figure 8

Plot of cross-term interaction $P_c^L(0,{\epsilon }_F)$ as a function of $\eta$ for $m_f/m_b=53/52$, $\pi T={\epsilon }_F$, $g_{3\mathrm{D}}=4.0$, ${\mathcal{G}}_{\mathrm{bf}}=0.5$, and $\frac{3g}{4\pi g_{\mathrm{dd}}}=0.6$. From the left-hand side, the upper curve (solid line), the middle curve (dashed line), and the bottom curve (dotted line) correspond to $p$-, $h$-, and $f$-wave angular momentum channels, respectively.

Figure 9

Effective interaction ${\lambda }_L^{\Delta }(\pi T={\epsilon }_F)$ as a function of $\eta$ for $m_f/m_b=53/52$, $\pi T={\epsilon }_F$, $g_{3\mathrm{D}}=4.0$, ${\mathcal{G}}_{\mathrm{bf}}=0.5$, and $\frac{3g}{4\pi g_{\mathrm{dd}}}=0.6$. From the left-hand side, the top curve (dotted line), the middle curve (dashed line), and the bottom curve (solid line) correspond to ${\lambda }_3^{\Delta }(T),{\lambda }_5^{\Delta }(T)$, and ${\lambda }_1^{\Delta }(T)$, respectively.

Figure 10

Critical temperature $T_c$, in units of Fermi energy ${\epsilon }_F$, as a function of $\eta$. The top curve (dash-dotted), the middle curve (dashed), and the bottom curve (solid) correspond to transition temperatures of $p$-wave, $h$-wave, and $f$-wave order parameters, respectively, with the following parameters: (a) $m_f/m_b=1.7$, dipole strength $g_{3\mathrm{D}}=2.5$; (b) $m_f/m_b=53/52$, dipole strength $g_{3\mathrm{D}}=4.0$.