Spectral Theory of the Fermi Polaron

TL;DR

This paper develops a rigorous mathematical framework for the Fermi polaron, connecting physical models with spectral theory, and introduces a variational principle that bounds the system's ground state energy.

Contribution

It provides a general renormalization approach for contact interactions and establishes a new variational principle for analyzing the Fermi polaron in two dimensions.

Findings

Derivation of TMS Hamiltonians via renormalization.

A variational principle linking eigenvalues to zero-modes.

Polaron and molecule energies are upper bounds to the ground state energy.

Abstract

The Fermi polaron refers to a system of free fermions interacting with an impurity particle by means of two-body contact forces. Motivated by the physicists' approach to this system, the present article develops a general mathematical framework for defining many-body Hamiltonians with two-body contact interactions by means of a renormalization procedure. In the case of the Fermi polaron the well-known TMS Hamiltonians are shown to emerge. For the Fermi polaron in a two-dimensional box a novel variational principle, established within the general framework, links the low-lying eigenvalues of the system to the zero-modes of a Birman-Schwinger type operator. It allows us to show, e.g., that the \emph{polaron}- and \emph{molecule} energies, computed in the physical literature, are indeed upper bounds to the ground state energy of the system.