Hartree-Fock treatment of Fermi polarons using the Lee-Low-Pine transformation

TL;DR

This paper develops a Hartree-Fock approach to the Fermi polaron problem using the Lee-Low-Pine transformation, achieving accurate results in one dimension and suggesting a pathway for higher-dimensional studies.

Contribution

It introduces a Hartree-Fock method combined with the Lee-Low-Pine transformation for Fermi polarons, aligning well with exact solutions in one dimension and enabling future extensions.

Findings

HF results agree with Bethe ansatz in 1D

Method effectively handles repulsive polarons

Potential for extension to higher dimensions

Abstract

We consider the Fermi polaron problem at zero temperature, where a single impurity interacts with non-interacting host fermions. We approach the problem starting with a Frohlich-like Hamiltonian where the impurity is described with canonical position and momentum operators. We apply the Lee-Low-Pine (LLP) transformation to change the fermionic Frohlich Hamiltonian into the fermionic LLP Hamiltonian which describes 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 in which a pair of host fermions with momenta $\mathbf{k}$ and $\mathbf{k}'$ interact with a potential proportional to $\mathbf{k}\cdot\mathbf{k}'$. We apply the HF theory, which has the advantage of not restricting the number of particle-hole pairs, to repulsive Fermi polarons in one dimension. When the impurity and host fermion masses are equal our variational ansatz, where HF orbitals are expanded in terms of free-particle states, produces results in excellent agreement with McGuire's exact analytical results based on the Bethe ansatz. This work raises the prospect of using the HF ansatz and its time-dependent generalization as building blocks for developing all-coupling theories for both equilibrium and nonequilibrium Fermi polarons in higher dimensions