Lee-Low-Pine Transformation Inspired Hartree-Fock Treatment of the Fermi Polaron Problem

We consider the Fermi polaron problem, where a single impurity interacts with non-interacting host fermions at zero temperature. We apply the Lee-Low-Pine (LLP) transformation to eliminate the impurity degrees of freedom, changing the Hamiltonian into a fermionic LLP Hamiltonian describing a many-body system containing host fermions only. We adapt the self-consistent Hartree-Fock (HF) approach, first proposed by Edwards, to the fermionic LLP Hamiltonian where a pair of host fermions interact with a potential very different from the usual two-body interaction. We apply our HF theory, which has the advantage of not imposing any restrictions on the number of particle-hole pairs, to repulsive Fermi polarons in one dimension. We find that for the case where the impurity has the same mass as the host fermion, the results calculated from our variational ansatz, where the HF orbitals are expanded as the superposition of the free-particle states, are in an excellent agreement with the exact ones of McQuire using Bethe ansatz. This work raises the prospect of using the HF ansatz and its time-dependent counterpart as building blocks for developing all-coupling theory for both equilibrium and nonequilibrium Fermi polarons at higher dimensions.