Precise Momentum Map Confirms Theory for Ultra-Thin Fermionic Gases

Scientists investigate the behaviour of dilute Fermi gases to better understand many-body physics and phenomena such as superconductivity and neutron matter. Niels Benedikter and Emanuela L. Giacomelli, both from Universit`a degli Studi di Milano, alongside Asbjørn Bækgaard Lauritsen from CNRS & CEREMADE, Universit e Paris, Dauphine, PSL University, and Sascha Lill from the Department of Mathematical Sciences, Universitetsparken 5, 2100 Copenhagen, Denmark, present a rigorous derivation of the momentum distribution for such a gas, building upon previous perturbative arguments. Their collaborative work provides a precise description of the ground state energy, achieving accuracy up to the Huang–Yang formula, and confirms the formal result originally proposed by Belyakov, offering valuable insight into the fundamental properties of interacting fermionic systems.

Scientists have long sought to fully understand the behaviour of interacting quantum gases, systems crucial to advancements in condensed matter physics and quantum simulation. Recent work rigorously derived the momentum distribution of a dilute gas of interacting fermions, a significant step towards characterising these complex systems and confirming a previously established result proposed by Belyakov decades ago. The momentum distribution, a map of particle velocities in different directions, is fundamental to describing the properties of quantum gases created in ultracold atom experiments and builds upon decades of theoretical work surrounding the Huang-Yang formula, which describes the ground state energy density of such systems.

The team’s approach involved analysing a specific trial state, an approximation of the true ground state, and calculating its momentum distribution with high precision, overcoming significant mathematical challenges in dealing with complex particle interactions. The resulting formula provides a detailed picture of momentum distribution among the fermions, offering insights into the system’s collective behaviour. A key result demonstrates that even weak interactions can dramatically alter the expected “step-like” momentum profile of a non-interacting Fermi gas, highlighting the importance of accurately accounting for interactions when modelling these systems. Furthermore, the findings support the Landau conjecture, which posits that a well-defined Fermi surface remains a universal feature even in the presence of interactions.

This research delves into the complex behaviour of interacting fermionic systems, providing rigorous mathematical bounds crucial for understanding phenomena in condensed matter physics. The core of the work lies in establishing precise estimates for various operators and functions that describe these systems, particularly focusing on low-density regimes. Specifically, the study meticulously defines and bounds functions and operators essential for describing the system, including χ, related to low- and high-momentum components, with ∥χ ∥L2(Λ) ≤ Cρ^(-1/9). Bounds are also established for vσ and uσ, the Fourier transforms of functions modifying interactions, where ∥vσ∥2 ≤ Cρ^(1/2). Quadratic operators bσ(h, f, g), representing interactions between particles, are bounded by ∥bσ(h, f, g)∥≤∥h∥1∥fσ∥2∥gσ∥2, providing crucial control over interaction strength. The research also establishes bounds on more complex, multi-particle interactions, with Lemma 4.5 providing a key estimate for quartic terms. A significant portion of the work focuses on bounding fermionic creation and annihilation operators (aσ, a∗σ), demonstrating control through manipulation of their properties and introduction of cutoff functions, as detailed in Lemma 4.4. The study extends beyond individual operators, providing estimates for combinations of them, crucial for expressing physical quantities as sums or products. The introduction of low-momentum cutoffs simplifies calculations and allows for more precise bounds, with bounds on these cutoff operators further refining the analysis. A precise determination of momentum distributions underpins this work, beginning with the implementation of a trial state resolving the ground state energy to the precision of the Huang-Yang formula, expressed as 3/5(3π2)^(2/3) ρ^(5/3) + 2πaρ^2 + 4/35(11 −2 log 2)(9π)^(2/3) a^2ρ^(7/3). This formula details the ground state energy density and demonstrates the ability to characterise system behaviour as density (ρ) approaches zero. The chosen trial state, resolving the ground state energy to this level of detail, was crucial for accurately mapping the momentum distribution, prioritizing a rigorous analytical framework over purely numerical methods. Researchers constructed a trial state, Ψ, belonging to a Hilbert space h(N↑, N↓) and employed three explicit unitary transformations to analyse the averaged momentum distribution. To refine the analysis, the study introduced a “smeared” version of the excitation density operator, denoted as nexc q,α, involving a convolution with a periodic function, χq,α,σ. The function’s Fourier transform, F(χq,α,σ)(k), was defined to be one within a ball of radius ρ/3 + ασ centred at q, and zero otherwise, where α is a small constant. This smoothing procedure is essential for handling the thermodynamic limit, ensuring meaningful results as the system size increases. The researchers then compared their calculated excitation density with Belyakov’s prediction to validate their approach.

Scientists have, for the first time, rigorously confirmed a decades-old prediction regarding the momentum distribution of interacting fermions, a cornerstone in understanding the behaviour of quantum gases. This confirmation validates the underlying assumptions used to model systems ranging from ultracold atoms to the complex interactions within materials and provides a solid foundation for approximations used by researchers for years. The detailed expansion of the ground state energy density, encapsulated in the Huang-Yang formula, demonstrates an unprecedented level of precision in characterising these interactions as density approaches zero. While previous attempts often relied on perturbative methods lacking rigorous mathematical proof, this new result bypasses those limitations, offering a definitive answer for a specific regime. This confirmation has implications for quantum simulation, where researchers recreate and study complex quantum systems using controllable platforms like ultracold atoms, as accurate momentum distribution models are essential for interpreting experimental results and validating simulations. However, the work is limited to a simplified model and doesn’t address the complexities of real materials. 👉 More information 🗞 Momentum Distribution of the Dilute Fermi Gas 🧠 ArXiv: https://arxiv.org/abs/2602.12067 Tags: