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Emptiness formation in the Lieb-Liniger gas: hydrodynamic instantons and a conjectured rate function

This paper proposes a parameter-free integral equation for the rate function governing the emptiness formation probability in the ground state of the repulsive Lieb-Liniger Bose gas, deriving it from a dual-field Fredholm-determinant representation and validating it against known limits and extensive numerical hydrodynamic simulations.

Original authors: Boris A. Khanikati, Alexander G. Abanov

Published 2026-08-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Boris A. Khanikati, Alexander G. Abanov

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 authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet, ultra-cold world of quantum gases, atoms behave less like tiny billiard balls and more like a single, unified wave. When scientists trap these atoms in a narrow, one-dimensional channel, they create a system where the rules of the quantum world are laid bare. One of the most intriguing questions in this field concerns the likelihood of a rare, spontaneous event: the sudden formation of a completely empty space within a crowd of particles. Imagine a dense fog where, for a fleeting moment, a perfect sphere of clear air appears out of nowhere. In the quantum realm, this is known as the "emptiness formation probability." It is a measure of how likely it is for a specific region to become entirely void of particles, a fluctuation that defies the usual tendency of matter to fill available space. Understanding this probability is not just an abstract exercise; it reveals the deep statistical laws that govern how quantum systems fluctuate and how they might behave under extreme conditions, offering a window into the fundamental nature of matter itself.

For decades, physicists have struggled to predict exactly how often this emptiness occurs in a specific type of quantum gas called the Lieb–Liniger gas, which consists of particles that repel each other. While scientists could calculate this probability for extreme cases—where particles barely interact or where they push against each other with immense force—a general formula for the middle ground remained elusive. Previous attempts to solve this relied on complex mathematical guesses that worked well in the extremes but failed to agree with computer simulations in the middle range. A team of researchers at Stony Brook University has now proposed a new, unified way to calculate this probability that works across the entire spectrum of interaction strengths, from the weakest push to the strongest repulsion.

The researchers approached the problem by looking at the mathematical structure that describes the quantum gas. They started with a precise, but incredibly complicated, description of the system involving a "Fredholm determinant," a type of mathematical object used to calculate probabilities in quantum mechanics. Instead of trying to solve this object directly, they treated the problem as a search for the most likely path the system takes to create an empty region. In the language of physics, this path is called an "instanton." They realized that to find the correct answer, they needed to consider two things happening at once: the shape of the empty region and the fluctuating fields that describe the particles. By analyzing how these two elements scale together as the empty region grows larger, they derived a new, self-contained equation. This equation does not require any adjustable parameters or guesses; it is determined entirely by the fundamental properties of the gas, specifically the strength of the repulsion between particles and their density.

The result of their work is a single, elegant integral equation that predicts the rate at which the probability of finding an empty space drops as the size of that space increases. To test their idea, the team performed massive computer simulations, essentially running the laws of quantum hydrodynamics to see what happens when they force a gap to form in the gas. They watched the system evolve in "imaginary time," a mathematical tool that allows physicists to study the most probable path of a rare event. The simulations revealed that the boundary of this empty region takes on a distinct, star-like shape, resembling a four-pointed star known as an astroid. This shape is consistent with what was seen in simpler systems, but here it appears in a complex, interacting quantum fluid.

When the researchers compared their new equation to the results of these computer simulations, they found a remarkable agreement. Across more than four orders of magnitude in interaction strength—covering everything from a very weakly interacting gas to one that behaves almost like a solid—their formula matched the simulation data to within a few percent. This level of precision is significant because it bridges the gap between the two known extremes. In the limit where particles interact very strongly, their formula correctly reproduces the behavior of free fermions, a well-understood state of matter. In the limit of weak interaction, it matches the predictions for a different type of fluid. Crucially, it also captures the subtle, first-order corrections in both limits, details that previous theories had missed or gotten wrong.

The paper also addresses a lingering question about the accuracy of their method. The small difference of three to five percent between their formula and the computer simulations is likely due to the limitations of the numerical grid used in the calculations, rather than a flaw in the theory itself. When the researchers refined their grid and ran the simulations longer, the numbers moved closer to their prediction, suggesting that the formula is indeed the correct answer. This work provides a powerful new tool for understanding rare fluctuations in quantum systems. It shows that even in a system as complex as a repulsive quantum gas, the probability of extreme events can be described by a clean, parameter-free mathematical law. The researchers have not just offered a guess; they have provided a rigorous framework that connects the microscopic scattering of particles to the macroscopic behavior of the entire gas, offering a unified view of how emptiness forms in the quantum world.

Technical Summary: Emptiness formation in the Lieb–Liniger gas: hydrodynamic instantons and a conjectured rate function

Problem Statement
The paper investigates the Emptiness Formation Probability (EFP) in the ground state of the repulsive one-dimensional Lieb–Liniger Bose gas. The EFP, denoted $P(R)$ , is the probability of finding no particles within a spatial interval of length $2R$ . For macroscopic intervals, the leading asymptotic behavior is governed by a rate function $f(\gamma_0)$ , defined via $-\log P(R) \sim (\rho_0 R)^2 f(\gamma_0)$ , where $\rho_0$ is the mean density and $\gamma_0 = c/\rho_0$ is the dimensionless interaction strength. While exact results exist for free fermions and specific limits (Tonks–Girardeau and weak coupling), a general analytical expression for $f(\gamma_0)$ at arbitrary repulsive coupling has remained an open challenge. Previous work by Korepin, Its, and Waldron (KIW) provided an exact dual-field Fredholm-determinant representation but proposed a conjecture for the large-distance asymptotics that recent hydrodynamic studies suggested might be parametrically incorrect in the weak-coupling limit.

Methodology
The authors employ a dual approach combining integrable operator theory and hydrodynamic variational methods:

Integrable Operator Analysis: Starting from the exact KIW dual-field Fredholm-determinant representation, the authors analyze the large-distance asymptotics. They identify a non-uniformity in the previous KIW analysis: terms subleading in the determinant expansion for a fixed dual field become leading order ( $O(R^2)$ ) after performing the dual-field vacuum average, because the relevant dual fields scale as $O(R)$ . By treating this as a uniform double-scaling problem (rescaling the field $\phi \to x\Phi$ ), they derive a joint saddle-point condition. This leads to a new conjecture for the rate function expressed as the solution to a parameter-free linear integral equation.
Hydrodynamic Variational Approach: The EFP is formulated as a variational problem for the Euclidean hydrodynamic action. The authors numerically minimize this action subject to the boundary condition of an empty interval at $\tau=0$ . The minimizer, known as the "emptiness instanton," describes the optimal rare fluctuation path. The numerical implementation utilizes a boundary-fitted coordinate system to resolve the astroid-like shape of the empty region's boundary and employs advanced optimization algorithms (Adam and L-BFGS) to solve for the density and velocity fields across a wide range of coupling strengths.

Key Contributions

A New Conjecture for the Rate Function: The paper proposes a specific integral equation for the function $g(t)$ that determines the rate function $f(\gamma_0)$ :
$\int_{-1}^{1} g(s) \log\left(1 + \frac{r^2}{(t-s)^2}\right) ds = 4t$
where $r = c/q$ is determined by the Lieb equation. The rate function is then given by $f(\gamma_0) = \frac{1}{2}(\gamma_0/r)^2 \int_{-1}^{1} t g(t) dt$ . This formulation is parameter-free and directly encodes the two-body scattering data of the Lieb–Liniger model.
Resolution of the KIW Discrepancy: The authors demonstrate that the previous KIW conjecture fails to capture the correct weak-coupling behavior because it neglects the quadratic contribution of the dual field in the vacuum average. The new conjecture corrects this by including the necessary quadratic term in the joint saddle-point analysis.
Numerical Validation: The paper presents extensive numerical minimization of the Lieb–Liniger hydrodynamic action over more than four orders of magnitude in coupling strength ( $\gamma_0$ ).

Results

Agreement with Limits: The conjectured rate function exactly reproduces the known leading-order results in both the weak-coupling ( $\gamma_0 \to 0$ ) and strong-coupling ( $\gamma_0 \to \infty$ ) limits. Furthermore, it correctly reproduces the first non-trivial subleading corrections in both regimes, which were previously derived independently via hydrodynamic perturbation theory.
Numerical Comparison: The numerically minimized hydrodynamic rate function $f_{\text{num}}(\gamma_0)$ follows the conjectured curve $f_{\text{conj}}(\gamma_0)$ closely across the entire interaction crossover. The systematic relative difference is approximately 3%–5%, with the numerical results lying slightly above the conjecture.
Convergence Analysis: Convergence tests involving grid refinement (increasing from $200\times200$ to $300\times300$ ) and extended optimization cycles show that the discrepancy decreases as numerical resolution improves. This suggests that the residual 3–5% difference is largely attributable to finite grid resolution and incomplete optimization convergence rather than a fundamental flaw in the conjecture.
Instanton Geometry: The numerical minimizers reveal emptiness instantons with astroid-like vacuum regions, consistent with findings for free fermions and polytropic fluids. The boundary shape evolves smoothly between weak and strong coupling limits.

Significance and Claims
The paper claims to provide a unified, parameter-free description of the leading EFP exponent for the repulsive Lieb–Liniger gas across all interaction strengths. The significance lies in:

Formal Derivation: Offering a formal derivation path from the exact Fredholm-determinant representation to the conjectured integral equation, identifying the specific asymptotic step (uniform double-scaling) that requires rigorous proof.
Empirical Robustness: Supporting the conjecture through independent analytical checks in limiting cases and high-precision numerical hydrodynamics.
Microscopic Connection: Highlighting that the logarithmic kernel of the integral equation is the integral of the Lieb–Liniger scattering kernel, suggesting a direct encoding of microscopic scattering data into the macroscopic large-deviation rate.

The authors remain modest regarding the status of the conjecture, noting that while it is strongly supported by numerical and asymptotic evidence, a complete rigorous derivation of the uniform double-scaling asymptotics remains an open mathematical problem. They also note that the structure of subleading terms (linear and logarithmic corrections to the $R^2$ exponent) remains an open question.

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