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The Lee-Huang-Yang Formula in the Hard-Core Simplified Approach to the Bose gas

This paper establishes the existence and uniqueness of solutions to the hard-core variant of the Simplified Approach for the dilute Bose gas and proves that both the ground state energy and condensate fraction satisfy the Lee-Huang-Yang formula and Bogolyubov's prediction, respectively, by demonstrating the convergence of soft-core approximations to the hard-core limit.

Original author: Mohamed El Hedi Bahri, Ian Jauslin

Published 2026-11-04

📖 4 min read🧠 Deep dive

CC BY 4.0

Original author: Mohamed El Hedi Bahri, Ian Jauslin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a room filled with trillions of tiny, invisible particles, all moving in a chaotic dance. Under normal conditions, these particles behave like a gas, bouncing off one another and filling every corner of the space. But if you cool them down enough, something extraordinary happens. They lose their individuality and collapse into a single, unified state, moving in perfect unison. This phenomenon, known as Bose-Einstein condensation, was predicted nearly a century ago and has been observed in laboratories with atoms like rubidium and sodium. It is a state of matter where the rules of the quantum world become visible on a scale large enough to measure.

For decades, physicists have tried to write down the exact mathematical rules that govern this behavior, especially when the particles interact with one another. The challenge is that these interactions are incredibly complex. When particles get close, they repel each other with a force that can be modeled as a hard wall: they simply cannot occupy the same space. This "hard-core" behavior is a realistic description of how real atoms act, but it makes the mathematics of the system notoriously difficult to solve. While scientists have long had a good theoretical guess for how the energy of such a system should behave at very low densities, proving that this guess is correct for a gas of hard, impenetrable particles has remained a major open problem in physics.

In this new work, researchers at Rutgers University have finally bridged that gap. They focused on a specific mathematical tool called the "Simplified Approach," which was introduced in the 1960s to make these complex quantum problems more manageable. This approach reduces the behavior of the entire gas to a single equation that describes how the particles correlate with one another. Previously, this method had only been rigorously proven to work for "soft" potentials, where particles repel each other gently over a distance. The new study extends this method to the "hard-core" case, where the repulsion is infinite and instantaneous, effectively treating the particles as solid spheres that cannot overlap.

The team constructed a solution to this difficult equation using a step-by-step iterative process. They started with a simple guess and refined it repeatedly, proving that this process converges to a unique, stable answer. Once they had this solution, they calculated two key properties of the gas: its energy per particle and the fraction of particles that have joined the condensate. Their calculations confirmed that the energy follows a famous formula known as the Lee-Huang-Yang formula. This formula, which includes a specific correction term involving the square root of the density, had been conjectured to be true for hard-core gases but had never been rigorously derived from this specific simplified equation. The researchers proved that as the density of the gas approaches zero, the energy predicted by their hard-core equation matches this formula exactly, including the precise numerical coefficients.

They also looked at the condensate fraction, which measures how many particles are part of the unified quantum state versus how many are left out. Their results showed that the fraction of particles in the condensate decreases in a very specific way as the density increases, matching the predictions made by the foundational theory of Bogolyubov. This agreement is significant because it demonstrates that the Simplified Approach, despite being a reduction of the full many-body problem, captures the essential physics of the hard-core Bose gas with high accuracy.

To reach these conclusions, the authors did not simply solve the hard-core equation in isolation. Instead, they treated the hard-core potential as the limit of a series of "soft" potentials that become increasingly steep and tall. They showed that as these soft potentials grow infinitely high, the solutions to the equations for the soft potentials smoothly converge to the solution for the hard-core case. This allowed them to transfer known results from the soft-core scenario to the hard-core scenario, proving that the errors in their approximations remain controlled even as the potential becomes infinitely strong.

The findings provide a rigorous mathematical foundation for understanding the ground state of a dilute Bose gas with hard-core interactions. By confirming that the Simplified Approach yields the correct energy and condensate fraction in the low-density limit, the study validates a powerful theoretical framework that had previously been unproven for realistic, hard-sphere interactions. This work does not just offer a new calculation; it establishes that the simplified model is a reliable tool for describing the quantum behavior of atoms that cannot pass through one another, bringing the theoretical understanding of these systems into alignment with the physical reality of atomic gases.

Technical Summary: The Lee–Huang–Yang Formula in the Hard-Core Simplified Approach to the Bose Gas

Problem Statement
The paper addresses the mathematical description of the ground state of a dilute Bose gas with hard-core interactions. While the "Simplified Approach" (introduced by Lieb in 1963) has been rigorously established for integrable soft-core potentials, its application to hard-core potentials—physically relevant for realistic atomic Bose gases and mathematically challenging due to the infinite gradient of the potential—remained unproven. Specifically, the authors seek to determine if the Simplified Approach, when adapted to hard-core potentials, correctly predicts the ground-state energy per particle ( $e$ ) and the condensate fraction ( $\eta$ ) in the low-density limit ( $\rho \to 0$ ). The goal is to verify if these predictions align with the seminal Lee–Huang–Yang (LHY) formula for energy and Bogolyubov's theory for the condensate fraction.

Methodology
The authors employ a rigorous analytical framework centered on the "Hard-Core Simple Equation," an integro-differential equation defined on $\mathbb{R}^3$ with a boundary condition at the hard-core radius (set to 1). The core of their methodology involves a limit argument where the hard-core potential is approximated by a sequence of soft-core potentials $v_m(x) = m \mathbb{1}_{\{|x|\le 1\}}$ with increasing height $m$ .

Key methodological steps include:

Iterative Constructions: The authors construct solutions for both the hard-core and soft-core equations using monotone iteration schemes. For the hard-core case, they define a sequence $\tau_n$ and prove its convergence to a unique solution $\tau$ .
Intermediate Iterations: To overcome the difficulty of directly comparing soft-core and hard-core iterates (due to competing monotonicities in density and potential height), the authors introduce two auxiliary iteration sequences, $\gamma_n$ and $\omega_n$ . These serve as upper and lower bounds, creating an envelope that allows for the comparison of the soft-core solutions $\sigma_m$ against the hard-core solution $\tau$ .
Uniform Estimates: A critical technical achievement is establishing that the error terms in the low-density expansions for the soft-core equations are uniform with respect to the potential height $m$ . This ensures that the asymptotic behavior survives the limit $m \to \infty$ .
Fixed-Energy Framework: The proofs are conducted at a fixed energy $e$ , computing the density $\rho$ as a function of $e$ . The authors demonstrate that for small $e$ , the mapping $e \mapsto \rho$ is invertible and comparable, allowing the final results to be re-expressed in terms of the small density parameter $\rho$ .

Key Contributions and Results
The paper establishes the following main results:

Existence and Uniqueness: The authors prove the existence and uniqueness (in a restricted sense, $0 \le \tau \le 1$ ) of the solution to the Hard-Core Simple Equation.
Convergence of Soft to Hard Core: They demonstrate that as the soft-core potential height $m \to \infty$ , the solutions $\sigma_m$ and their densities $\bar{\rho}_m$ converge to the hard-core solution $\tau$ and density $\rho_{hc}$ in the $L^1$ norm. Crucially, they establish a convergence rate of $O(m^{-1/2})$ .
Lee–Huang–Yang Formula for Energy: The ground-state energy per particle $e$ for the hard-core Simple Equation is shown to satisfy the LHY formula:
$e = 2\pi\rho_{hc} \left( 1 + \frac{128}{15\sqrt{\pi}} \sqrt{\rho_{hc}} + o(\sqrt{\rho_{hc}}) \right)$
where the scattering length $a_0$ for the hard-sphere potential is 1. This confirms that the Simplified Approach recovers the correct asymptotic energy for the many-body hard-sphere Bose gas.
Condensate Fraction: The condensate fraction $\eta$ predicted by the hard-core equation is shown to satisfy:
$\eta = 1 - \frac{8}{3\sqrt{\pi}} \sqrt{\rho_{hc}} + o(\sqrt{\rho_{hc}})$
This result is in agreement with Bogolyubov's theory.

Significance
The paper extends the rigorous mathematical theory of the Simplified Approach from soft-core to hard-core potentials. By proving that the hard-core Simple Equation yields the correct Lee–Huang–Yang expansion for energy and the correct condensate fraction, the authors validate the Simplified Approach as an accurate effective theory for the ground state of realistic, interacting Bose gases with hard-core repulsion. The work bridges the gap between phenomenological theories and first-principles derivations for specific observables in the dilute limit, providing a mathematically sound foundation for the use of the Simple Equation in regimes where the potential is singular. The results generalize previous findings by Carlen, Jauslin, and Lieb to the physically critical hard-core case.

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