Momentum Distribution of the Dilute Fermi Gas

TL;DR

The paper establishes a rigorous connection between the momentum distribution of a low-energy trial state for a dilute three-dimensional spin-$\tfrac12$ Fermi gas and the perturbative Belyakov (and Huang–Yang) framework. By constructing a trial state via a particle–hole transformation and two quasi-bosonic Bogoliubov maps, and then performing a meticulous second-order Duhamel analysis, the authors extract the leading averaged excitation density as predicted by Belyakov, while tightly bounding all higher-order corrections. They prove energy-density accuracy at the Huang–Yang scale and provide explicit bounds for the averaged excitation density, including a near-Fermi-surface regime where Belyakov’s term is recovered. The results reinforce the expected Fermi-surface structure in the dilute regime and supply a solid mathematical bridge between rigorous many-body constructions and conventional perturbative physics. The work lays groundwork for a deeper understanding of momentum-space entanglement in interacting Fermi systems and validates the use of complex unitary conjugations to access low-energy observables.

Abstract

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)).

Figures (6)

• Figure 1: In $\mathrm{I}_{3;1}^{<\epsilon}$, fixing $(r,r')$, the integral runs over such $p \in \mathbb{R}^3$ that $p+r$ and $r'-p$ are outside the respective Fermi balls, but the energy $e_r$ still dominates. The set of such $p$ is an intersection of two annuli $A_r$ and $A_{r,r',x}$, defined in \ref{['eq:Ar_Arrx_Bx']}.
• Figure 2: Left: Example of a possible "dangerous area", in which the intersection volume of the two annuli $A_r$ and $A_{r,r',x}$ becomes large. Right: We represent the region $C_{r,x}^{(\mathrm{in})}$. If $r'$ is in this region, then $|A_r \cap A_{r,r',x}|$ becomes large.
• Figure 3: Left: The intersection angle between the two tangents at the intersection point of the annuli $A_r$ and $A_{r,r^\prime,x}$ is called $\beta$. Right: The angle $\tilde{\beta}$ quantifies the area of the dangerous region $C_{r,x}^{\mathrm{(in)}}$ at fixed $|r^\prime|$. This area is a spherical cap. Similarly, $C_{r,x}^{\mathrm{(out)}} \cap \partial B_s(0)$ is a spherical cap. The set $C^{(\mathrm{out})}_{r,x}$ is drawn on top for comparison.
• Figure 4: Left: Depiction of the first case $|p| = |q-r| \leq 2e_r x^{-1}$. The set $S^{(1)}_{q,r,x}$ is an intersection of two annuli. We simply bound the volume of $S^{(1)}_{q,r,x}$ by the volume of one annulus. Right: In the second case, $2 e_r x^{-1} \leq |q-r| \leq 2x - 2e_r x^{-1}$, we can use the tangent intersection angle $\beta$ to bound $|S^{(1)}_{q,r,x}|$.
• Figure 5: The third case $|q-r| \geq 2x - 2 e_r x^{-1}$. Here, the set $S^{(1)}_{q,r,x}$ has thicknesses $\delta$ and $\gamma$, which we conveniently bound using the angle $\alpha$.

Theorems & Definitions (41)

• Theorem 2.1: Main Result
• Remark 2.2: Choice of the state in Theorem \ref{['thm:main']}
• Remark 2.3: Choice of $\alpha$
• Remark 2.4: Averaging of the excitation density
• Remark 2.5: Validity of Belyakov's formula for $|q|\leq C\rho^{1/3}_\sigma$
• Remark 2.6: Other trial states
• Definition 3.1: Particle--Hole Transformation
• Definition 3.2: Scattering Solutions
• Definition 3.3: Quasi-Bosonic Transformations $T_1$ and $T_2$
• Proposition 3.4: Expansion of the Number of Excitations