Three comments on the Fermi gas at unitarity in a harmonic trap

TL;DR

This paper discusses key theoretical aspects of the unitary Fermi gas in a harmonic trap, including a virial theorem, odd-even energy splitting, and excitation energies, providing insights into its physical properties.

Contribution

It offers a concise proof of the virial theorem, analyzes the odd-even splitting behavior, and estimates excitation energies for the unitary Fermi gas in a trap.

Findings

Virial theorem states average energy is twice the potential energy.

Odd-even splitting scales as N^{1/9} with particle number.

Lowest excitation energies for odd N scale as N^{-1/3}.

Abstract

In this note we consider three issues related to the unitary Fermi gas in a harmonic trap. We present a short proof of a virial theorem, which states that the average energy of a particle system at unitarity in a harmonic trap is twice larger than the average potential energy. The theorem is valid for all systems with no intrinsic scale, at zero or finite temperature. We discuss the odd-even splitting in a unitarity Fermi gas in a harmonic trap. We show that at large number of particles N the odd-even splitting is proportional to N^{1/9}\hbar\omega, with an undetermined numerical constant. We also show that for odd N the lowest excitation energies are of order N^{-1/3}\hbar\omega.