[Paper Review] Three comments on the Fermi gas at unitarity in a harmonic trap

D. T. Son|ArXiv.org|Jul 12, 2007

Cold Atom Physics and Bose-Einstein Condensates5 references3 citations

TL;DR

This paper provides a simple Hellmann-Feynman theorem-based proof of the virial theorem for a unitary Fermi gas in a harmonic trap, showing that the total energy is twice the average potential energy. It further derives that the odd-even energy splitting scales as $N^{1/9}\hbar\omega$ at large $N$, and identifies a $N^{-1/3}\hbar\omega$ energy level splitting for odd-$N$ systems due to surface quasiparticles, with implications for excitation spectra in trapped Fermi gases.

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}\hbarω, with an undetermined numerical constant. We also show that for odd N the lowest excitation energies are of order N^{-1/3}\hbarω.

Motivation & Objective

• To establish a simple, general proof of the virial theorem for unitary Fermi gases in harmonic traps using the Hellmann-Feynman theorem, independent of the local density approximation.
• To analyze the odd-even energy splitting in a unitary Fermi gas at large particle numbers, motivated by experimental observations of pairing effects.
• To determine the scaling of the lowest excitation energies in odd-$N$ systems, particularly the role of surface quasiparticles in determining the energy level spacing.
• To extend the analysis to anisotropic traps and explore the impact of trap geometry on the odd-even splitting and quasiparticle localization.

Proposed method

• Applies the Hellmann-Feynman theorem to the Hamiltonian dependence on trap frequency $\omega$, showing $\langle H\rangle = 2\langle V\rangle$ via energy scaling $E_N \propto \hbar\omega$.
• Uses dimensional analysis to derive the $N^{1/9}\hbar\omega$ scaling of the odd-even splitting by modeling the extra particle as a quasiparticle localized at the edge of the Fermi cloud.
• Constructs a model of a quasiparticle in a step-like potential barrier at the edge of the cloud, with energy $E_{\text{extra}} \propto (\hbar{\cal E})^{2/3}/m^{1/3}$, where ${\cal E}$ is the effective electric field.
• Relates the cloud radius $R \propto N^{1/6}$ to the particle number and combines with quasiparticle energy to derive the $N^{1/9}$ scaling of $\Delta_N$.
• Analyzes the orbital excitation spectrum of the surface quasiparticle using $E_\ell \propto \ell(\ell+1)/(m^* R^2)$, leading to a $N^{-1/3}\hbar\omega$ energy level spacing.
• Extends results to anisotropic traps by considering the minimal trap frequency $\omega_{\text{min}}$ and geometric mean $\bar{\omega}$, yielding modified scaling in Eq. (20).

Experimental results

Research questions

• RQ1 Does the virial theorem $\langle H\rangle = 2\langle V\rangle$ hold for unitary Fermi gases in harmonic traps, and can it be proven without the local density approximation?
• RQ2 How does the odd-even energy splitting $\Delta_N = E_N - \frac{1}{2}(E_{N-1} + E_{N+1})$ scale with particle number $N$ in the large-$N$ limit?
• RQ3 What is the origin and scaling of the lowest excitation energy in an odd-$N$ unitary Fermi gas, and how does it differ from even-$N$ systems?
• RQ4 How does the quasiparticle excitation spectrum change in anisotropic traps, particularly in prolate vs. oblate geometries?

Key findings

• The virial theorem $\langle H\rangle = 2\langle V\rangle$ holds for unitary Fermi gases in harmonic traps at zero or finite temperature, proven via the Hellmann-Feynman theorem without relying on the local density approximation.
• The odd-even energy splitting scales as $\Delta_N \sim N^{1/9}\hbar\omega$ for large odd $N$, derived from dimensional analysis of a quasiparticle localized at the edge of the Fermi cloud.
• The lowest excitation energies in odd-$N$ systems scale as $N^{-1/3}\hbar\omega$, arising from orbital quantization of surface quasiparticles on a spherical Fermi surface.
• In anisotropic traps, the odd-even splitting scales as $\Delta_N \sim \hbar\bar{\omega}^{1/3}\omega_{\text{min}}^{2/3} N^{1/9}$, with $\bar{\omega}$ the geometric mean frequency and $\omega_{\text{min}}$ the smallest trap frequency.
• In prolate traps with a unique minimum frequency, the ground state of the odd-$N$ system exhibits near-degeneracy due to quantum tunneling between two edge-localized states, leading to parity doubling.
• The scaling of the odd-even splitting is universal and independent of the pairing mechanism, depending only on the scale-free nature of the unitary Fermi gas and the quasiparticle localization at the edge.