Mass-gap description of heavy impurities in Fermi gases

Abstract

We present a unified theory that connects the quasiparticle picture of Fermi polarons for mobile impurities to the Anderson orthogonality catastrophe for static impurities. By operator reordering of the underlying many-body Hamiltonian, we obtain a modified fermionic dispersion relation that features a recoil-induced energy gap, which we call the `mass gap'. We show that the resulting mean-field Hamiltonian exhibits an in-gap state for finite impurity mass, which takes a key role in Fermi polaron and molecule formation. We identify the mass gap as the microscopic origin of the quasiparticle weight of Fermi polarons and derive a power-law scaling of the weight with the impurity-to-fermion mass ratio. The associated in-gap state is shown to give rise to the emergence of the polaron-to-molecule transition away from the limiting case of the Anderson orthogonality catastrophe in which the transition is absent.

Figures (11)

• Figure 1: Mass-gap model. a) Sketch of the modified dispersion relation $E_{\mathbf{k}}$ and opening of the mass gap $\Delta(M)$. The in-gap state and bound state are depicted with a red dashed line. A particle-hole excitation from the lower to the upper band is shown in blue. b) As the interaction strength increases, the single-particle energies (solid lines) decrease compared to the non-interacting levels (dashed lines), and the in-gap state energy $E_g$ moves from the edge of the upper band to the lower band edge. The polaron-to-molecule transition (depicted as yellow star) occurs at the critical interaction strength $1/(k_F a_c)$, when the in-gap state crosses the Fermi energy $E_F$. The attractive polaron is described by occupying the lowest energy states, including the bound state . The molecule is obtained by the additional occupation of the in-gap state . c) The repulsive polaron corresponds to the excited configuration occupying the in-gap state instead of the bound state.
• Figure 2: Exact energy spectrum. Energy spectrum of the gapped Hamiltonian $\hat{\cal H}_\text{quad}$ obtained from exact diagonalization at finite mass ratio $M/m=3$ as function of interaction strength $1/(k_Fa)$. Due to the gapped dispersion relation $E_{\mathbf{k}}$, the bound state appears already at a negative scattering length. The in-gap state crosses the Fermi level $E_F$ (solid line) at positive scattering length, signaling the polaron-to-molecule transition (yellow star).
• Figure 3: Polaron-to-molecule transition from the OC. a) Phase diagram of the impurity problem in 3D as function of the inverse mass ratio $m/M$ and interaction strength $1/(k_Fa)$ as predicted by our theory. b) Polaron and molecule energy as a function of $m/M$ for a fixed interaction strength $1/(k_Fa)=0.5$. The polaron-to-molecule transition is highlighted with a yellow star.
• Figure 4: Emergence of the quasiparticle weight. a) Quasiparticle weight $Z$ of the attractive polaron as function of the mass ratio $m/M$. Red symbols represent the mean-field prediction of the mass-gap model. The solid line corresponds to a power-law with exponent $\alpha = (\delta_F/\pi)^2$. Blue symbols show results from the Chevy ansatz Chevy2006, which fails to capture the OC. b) Numerically calculated exponent $\alpha$ of the power-law decay $Z(M)\sim (m/M)^{\alpha}$ (red symbols) as function of $1/(k_Fa)$ compared to the analytical expression $(\delta_F/\pi)^2$ (solid line).
• Figure S1: Modified scattering phase shift $\tilde{\delta}(E)$ for finite-mass particles with gapped dispersion relation $E_{\mathbf{k}}$ obtained from Eq. \ref{['eq:new-T-matrix']} and \ref{['eq:phase-shift']}.