[Paper Review] Epsilon expansion for a Fermi gas at infinite scattering length

Yusuke Nishida, D. Son|arXiv (Cornell University)|Apr 21, 2006

Physics of Superconductivity and Magnetism4 citations

TL;DR

This paper introduces an epsilon expansion around four spatial dimensions to systematically study the unitary Fermi gas, treating the dimensionality deficit ε = 4−d as a small parameter. It computes the chemical potential to Fermi energy ratio and the gap-to-chemical potential ratio to next-to-leading order, yielding results consistent with Monte Carlo simulations when extrapolated to d=3.

ABSTRACT

We show that there exists a systematic expansion around four spatial dimensions for Fermi gas in the unitarity regime. We perform the calculations to leading and next-to-leading orders in the expansion over epsilon=4-d, where d is the dimensionality of space. We find the ratio of chemical potential and Fermi energy to be mu/eF=1/2 epsilon^3/2 + 1/16 epsilon^5/2 ln epsilon -0.0246 epsilon^5/2 and the ratio of the gap in the fermion quasiparticle spectrum and the chemical potential to be Delta/mu=2/epsilon-0.691. The minimum of the fermion dispersion curve is located at |p|=(2m epsilon_0)^1/2 where epsilon_0/mu=2+O(epsilon). Extrapolation to d=3 gives results consistent with Monte Carlo simulations.

Motivation & Objective

• To develop a systematic analytical framework for the unitary Fermi gas, which lacks a small expansion parameter in three dimensions.
• To exploit the special role of four spatial dimensions, where the two-body wavefunction exhibits a 1/r² singularity, enabling a controlled expansion.
• To compute key thermodynamic and spectral properties—such as μ/εF and Δ/μ—to next-to-leading order in ε.
• To test the viability of the ε expansion for extrapolation to physical three dimensions, where no small parameter exists in standard approaches.

Proposed method

• Formulate the unitary Fermi gas using a local four-Fermi interaction Lagrangian, followed by a Hubbard-Stratonovich transformation to introduce an auxiliary bosonic field.
• Apply dimensional regularization in d = 4−ε dimensions, treating ε as a small expansion parameter.
• Expand the effective action and self-energy diagrams in powers of ε, computing one- and two-loop contributions to the fermion and boson propagators.
• Use the Nambu–Gor’kov formalism to describe the superfluid state, with the order parameter φ₀ determining the gap Δ.
• Compute the fermion quasiparticle dispersion relation by solving the Dyson equation with self-energy corrections from vertex and loop diagrams.
• Extrapolate results from ε → 0 to ε = 1 (d=3) using leading and next-to-leading order terms, assessing convergence and consistency with simulations.

Experimental results

Research questions

• RQ1 Can a systematic expansion in ε = 4−d be constructed for the unitary Fermi gas, despite the absence of a small parameter in d=3?
• RQ2 What are the leading and next-to-leading order corrections to μ/εF and Δ/μ in the ε expansion?
• RQ3 How does the minimum of the fermion quasiparticle dispersion shift from p=0, and what is its momentum dependence?
• RQ4 To what extent do the extrapolated results at ε=1 agree with Monte Carlo simulations and experimental data?
• RQ5 Is the ε expansion a viable analytical tool for non-perturbative systems like the unitary Fermi gas?

Key findings

• The ratio of chemical potential to Fermi energy is μ/εF = ½ε³/² + ¹⁄₁₆ε⁵/²lnε − 0.0246ε⁵/² + ⋯, with the leading correction arising from two-loop diagrams.
• The gap-to-chemical potential ratio is Δ/μ = 2ε⁻¹ − 0.691 + ⋯, indicating a strong enhancement of the gap at small ε.
• The minimum of the fermion quasiparticle dispersion occurs at |p| = (2mε₀)¹/², with ε₀/μ = 2 + O(ε), showing a significant shift from the mean-field value.
• Extrapolation to ε=1 (d=3) yields μ/εF ≈ 0.475, ε₀/μ ≈ 2, and Δ/μ ≈ 1.31, in good agreement with Monte Carlo simulations (Δ/μ ≈ 1.2) and experiments (ξ ≈ 0.46–0.51).
• The convergence of the ε expansion at ε=1 suggests that the unitary Fermi gas can be effectively described as a weakly interacting system of fermionic and bosonic quasiparticles in d=3.
• The next-to-leading order correction to Δ/μ arises from a cancellation between two-loop diagrams and subleading terms in one-loop diagrams, confirming consistency of the perturbative framework.