Momentum Distribution of the Dilute Fermi Gas

We consider a dilute quantum gas of interacting spin-1/2 fermions in the thermodynamic limit. For a trial state that resolves the ground state energy up to the precision of the Huang–Yang formula, we rigorously derive its momentum distribution. Our result agrees with the formal perturbative argument of Belyakov (Sov. Phys. JETP 13: 850–851 (1961)).

💡 Research Summary

The paper presents a rigorous derivation of the momentum distribution for a dilute spin‑½ Fermi gas in three dimensions, focusing on the thermodynamic limit where the particle density ρ tends to zero while the scattering length a remains fixed (a³ρ≪1). Building on recent advances that proved the Huang–Yang formula for the ground‑state energy density up to the a²ρ^{7/3} term, the authors construct a specific trial state Ψ that reproduces the energy density with an error of order ρ^{7/3}+L^{-120}. The trial state is obtained by applying three unitary transformations to the vacuum: a particle‑hole transformation R that creates the non‑interacting Fermi sea, and two Bogoliubov‑type transformations T₁ and T₂ generated by operators B₁ and B₂. These transformations encode pair correlations and are expanded to second order; higher‑order contributions are controlled through a detailed many‑body analysis.

The central result, Theorem 2.1, states that for any momentum q in the reciprocal lattice and any smoothing parameter 0<α<1/27, the averaged excitation density n_exc^{q,α}=∑σ n_exc^{q,α,σ} computed in the state Ψ differs from the expression originally proposed by Belyakov (1961) by at most C ρ^{5/3+α}. The Belyakov formula involves double integrals over two Fermi balls (one for each spin component) and a kernel that depends on the smoothing function χ{q,α,σ}, which is a characteristic function of a small ball of radius ρ^{1/3}+ασ centered at q. Moreover, for momenta satisfying |q|≤C ρ^{1/3}_σ, the Belyakov prediction itself scales as ρ^{5/3+3α}_σ, bounded above and below by positive constants times this power.

The proof proceeds by (i) expressing the number operator in second‑quantized form, (ii) conjugating it with the three unitaries, and (iii) using a second‑order Duhamel expansion to isolate the leading contributions. The authors develop a suite of integral estimates for the kernels u and v (the projectors onto occupied and unoccupied momentum states) and for the pair‑creation amplitudes appearing in B₁ and B₂. They then bound the norms of commutators such as