Self-energy of an impurity in an ideal Fermi gas to second order in the interaction strength

We study in three dimensions the problem of a spatially homogeneous zero-temperature ideal Fermi gas of spin-polarized particles of mass $m$ perturbed by the presence of a single distinguishable impurity of mass $M$. The interaction between the impurity and the fermions involves only the partial $s$ wave through the scattering length $a$ and has negligible range $b$ compared to the inverse Fermi wave number $1/k_{\mathrm{F}}$ of the gas. Through the interactions with the Fermi gas the impurity gives birth to a quasiparticle, which will be here a Fermi polaron (or more precisely a monomeron). We consider the general case of an impurity moving with wave vector $\mathbf{K}\ne \mathbf{0}$: Then the quasiparticle acquires a finite lifetime in its initial momentum channel because it can radiate particle-hole pairs in the Fermi sea. A description of the system using a variational approach, based on a finite number of particle-hole excitations of the Fermi sea, then becomes inappropriate around $\mathbf{K}=\mathbf{0}$. We rely thus upon perturbation theory, where the small and negative parameter $k_{\mathrm{F}}a\to 0^{-}$ excludes any branches other than the monomeronic one in the ground state (as, e.g., the dimeronic one), and allows us a systematic study of the system. We calculate the impurity self-energy ${\Sigma }^{\left(2\right)}(\mathbf{K},\omega )$ up to second order included in $a$. Remarkably, we obtain an analytical explicit expression for ${\Sigma }^{\left(2\right)}(\mathbf{K},\omega )$, allowing us to study its derivatives in the plane $(K,\omega )$. These present interesting singularities, which in general appear in the third-order derivatives ${\partial }^3{\Sigma }^{\left(2\right)}(\mathbf{K},\omega )$. In the special case of equal masses, $M=m$, singularities appear already in the physically more accessible second-order derivatives ${\partial }^2{\Sigma }^{\left(2\right)}(\mathbf{K},\omega )$; using a self-consistent heuristic approach based on ${\Sigma }^{\left(2\right)}$ we then regularize the divergence of the second-order derivative ${\partial }_K^2\Delta E\left(\mathbf{K}\right)$ of the complex energy of the quasiparticle found in Trefzger and Castin [Europhys. Lett. 104, 50005 (2013)] at $K=k_{\mathrm{F}}$, and we predict an interesting scaling law in the neighborhood of $K=k_{\mathrm{F}}$. As a by product of our theory we have access to all moments of the momentum of the particle-hole pair emitted by the impurity while damping its motion in the Fermi sea at the level of Fermi's golden rule.

Figure 1

The singularity curves of the derivatives with respect to $(\overline{K},\varepsilon )\in {\mathbb{R}}^{+}\times \mathbb{R}$ of the dimensionless self-energy of order two ${\overline{\Sigma }}^{\left(2\right)}(\overline{K},\varepsilon )$. (i) On the parabolas ${\Delta }_{\sigma }^{\eta }=0$, where ${\Delta }_{\sigma }^{\eta }$ is the discriminant of the polynomial $P_{\sigma }^{\eta }\left(\lambda \right)$, $\sigma \in \{\alpha ,\beta ,\gamma ,\delta \}$ and $\eta \in \{+,-\}$, the roots of the polynomials are nondifferentiable functions of $(\overline{K},\varepsilon )$; this actually leads to a singularity of the derivatives of ${\overline{\Sigma }}^{\left(2\right)}(\overline{K},\varepsilon )$ on the solid-line portions, not on the dashed-line portions. (ii) ${\overline{\Sigma }}^{\left(2\right)}(\overline{K},\varepsilon )$ is not a smooth function of the roots on the horizontal half-line, where at least one (but also each) polynomial $P_{\sigma }^{\pm }\left(\lambda \right)$, $\sigma \in \{\alpha ,\beta ,\gamma ,\delta \}$, has a root ${\lambda }_0=0$, and on the oblique half-line (downwards for $\eta =+$, upwards for $\eta =-$), where at least one (but also each) polynomial $P_{\sigma }^{\eta }\left(\lambda \right)$, $\sigma \in \{\beta ,\gamma ,\delta \}$, has one root ${\lambda }_0=2$. Vertical dotted lines mark the abscissa of the tangent points of the parabolas with the singularity lines; the values of the abscissas are given explicitly as functions of the impurity-to-fermion mass ratio $r=M/m$, in a color code and obliquity code identifying, respectively, the parabola and the line that are tangent. The vertical dashed line marks the abscissa $\overline{K}=1$ of the crossing point of the horizontal and the oblique upwards singularity lines. In the figure, $r=3/2$.

Figure 2

Second-order derivative of the complex energy of the impurity [(a) real part shifted by $C=(9/4)ln\left[{(\rho g/{\epsilon }_{\mathrm{F}})}^2\right]$; (b) imaginary part], obtained by numerical resolution of the self-consistent equation (133), for a mass ratio $r=M/m=1$ and for values of $\rho g/{\epsilon }_{\mathrm{F}}$ equal to $-0.15$ (red stars), $-0.1$ (green plus signs), and $-0.01$ (blue circles), the last value having practically reached the limit $g\to 0^{-}$. In the chosen unit system, we can see emerge a scaling law of the form (143). Black solid line, analytic prediction in the limit $g\to 0^{-}$; the location of the corresponding discontinuities in the second-order derivative are identified by the vertical dotted lines of abscissas $x_{\mathrm{jump}}^{\prime }$ and $x_{\mathrm{jump}}$ from left to right [see Eq. (155)]. Black thin dashed line, analytic prediction corresponding to another possible writing of the function ${\partial }_{\overline{K}}^2{J}_{\beta }^{-}\left(2\right)$ before its analytic continuation; it differs from the black solid line only in between the two vertical dotted lines. Black thick dashed line, perturbation theory (129) deduced from Ref. [36], limited to its validity domain, that is, the tails.

Figure 3

Second-order derivative of the nonzero-temperature complex energy of the impurity of order two obtained by numerical integration of Eq. (158), for a mass ratio $r=M/m=1$, and for values of $T/T_{\mathrm{F}}$ equal to $0.1$ (red stars), $0.05$ (green plus signs), $0.025$ (blue circles). Black thick dashed line, zero-temperature result [36]. At $0<T\ll T_{\mathrm{F}}$, the typical scale on which the divergence of the zero-temperature result is interrupted is given by the difference of the abscissas of the crossing points between the nonzero (symbols) and zero-temperature (dashed line) results. It scales linearly in $T/T_{\mathrm{F}}$.