Polaron in a $p+ip$ Fermi topological superfluid

We study polaron excitations induced by an impurity interacting with a two-dimensional $p+ip$ Fermi superfluid. As the Fermi-Fermi pairing interaction is tuned, the background Fermi superfluid undergoes a topological phase transition. We show that such a transition is accompanied by a discontinuity in the second derivative of the polaron energy, regardless of the impurity-fermion interaction. We also identify a polaron to trimer crossover when the Fermi superfluid is in the strongly interacting, thus topologically trivial, regime. However, the trimer state is metastable against the molecular state where the impurity binds a Bogoliubov quasiparticle from the Fermi superfluid. By comparing the polaron to molecule transition in our system with that of an impurity in a noninteracting Fermi sea, we find that pairing interactions in the background Fermi superfluid effectively facilitate the impurity-fermion binding. Our results suggest the possibility of using the impurity as a probe for detecting topological phase transitions in the background; they also reveal interesting competitions between various many-body states in the system.

Figure 1

Chemical potential $\mu$ of the background $p$-wave superfluid as a function of the pairing order parameter $\mathrm{\Delta }$.

Figure 2

(a) The lowest branch of the polaron energy versus $E_b$. (b) The lowest branch of the polaron energy versus $\mu$. (c) The first derivative of the lowest branch of the polaron energy with respect to $\mu$. (d) Impurity residue. The parameters are $Q=0, m_b=m$, and momentum cutoff $k_c=20k_F$. In (a) and (d), the solid green line is for $\mu =0.5E_F$, the solid blue line is for $\mu =0.2E_F$, the dashed-dotted black line is for $\mu =0$, the dashed blue line is for $\mu =-0.2E_F$, and the dashed green line is for $\mu =-0.5E_F$. In (b) and (c), the solid green line is for $E_b=0.5E_F$, the dashed blue line is for $E_b=0.75E_F$, and the dashed-dotted black line is for $E_b=E_F$.

Figure 3

Momentum distribution of the trimer probability $|{\psi }_{{\mathbf{k}}_1,{\mathbf{k}}_2}{|}^2$ in the $(k_1,k_2)$ plane, where the polar angles ${\theta }_1$ and ${\theta }_2$ of ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ have been integrated for (a) $E_b=0.5E_F$ with $\mu =-0.5E_F$, (b) $E_b=10E_F$ with $\mu =-0.5E_F$, (c) $E_b=0.5E_F$ with $\mu =0.5E_F$, and (d) $E_b=10E_F$ with $\mu =0.5E_F$. The other parameters are $Q=0, m_b=m$, and momentum cutoff $k_c=20k_F$.

Figure 4

Molecular and polaron energies in the $Q=0$ sector with (a) $\mu =0.5E_F$; (b) $\mu =0.2E_F$; (c) $\mu =-0.2E_F$; (d) $\mu =-0.5E_F$. Here, we choose $m_b=m$ and $k_c=20k_F$. The red solid curve is the lowest branch of the polaron energy and the black dashed curve is the molecular energy.