The complete trans-series for conserved charges in the Lieb-Liniger model

This paper presents the complete trans-series solution for the rapidity density moments in the Lieb-Liniger model by explicitly constructing it from perturbative expansions and Volin's method, thereby satisfying resurgence relations, matching numerical data, and providing the full analytic trans-series for a coaxial circular plate capacitor.

The Gist

Imagine you are trying to predict the weather. You have a very complex system with wind, temperature, and humidity all interacting. If you try to solve the equations exactly, it's impossible. So, scientists usually use a "perturbative" approach: they start with a simple, calm day and add small corrections for the wind, then the rain, then the clouds.

In the world of quantum physics, the Lieb–Liniger model is like that complex weather system, but for a one-dimensional line of bosons (a type of particle, like atoms in a cold gas). Scientists have known how to calculate the "calm day" (the basic behavior) and the first few small corrections for decades.

However, there's a catch. Just like a weather forecast that gets weirdly wrong after a few days, these simple calculations eventually break down. They miss something huge: invisible, exponential corrections. These are like "ghost" effects that are so small they seem to vanish, but they actually hold the key to the true answer.

This paper by Zoltán Bajnok and his team is like a master key that unlocks the complete picture. They didn't just add a few more small corrections; they found the "Trans-Series." Think of a Trans-Series as a double-layered recipe:

1. Layer 1: The standard, step-by-step instructions (the perturbative part).
2. Layer 2: The secret, hidden ingredients (the non-perturbative, exponential parts) that you only add when the first layer starts to fail.

Here is how they did it, using some creative analogies:

1. The "Ghost" Corrections (The Trans-Series)

Imagine you are trying to measure the length of a table with a ruler. You get a number, say 1 meter. But you know there are tiny vibrations in the room making the measurement slightly off.

• Old Method: You just add a tiny "plus or minus 1 millimeter" to your answer.
• This Paper's Method: They realized the vibrations aren't just random noise; they are structured "ghosts" that appear in specific patterns. They found a mathematical formula that describes both the ruler measurement and the ghost vibrations simultaneously. This is the Trans-Series. It's the complete, perfect description of the table's length, including the ghosts.

2. The "Running Coupling" (The Magic Zoom Lens)

The math in this paper is incredibly messy because of "logarithms" (mathematical terms that grow slowly but get in the way).

• The Analogy: Imagine trying to read a map where the scale keeps changing as you zoom in and out. It's frustrating.
• The Solution: The authors introduced a "running coupling" (a variable called $v$). Think of this as a magic zoom lens. When they look at the problem through this lens, the messy logarithms disappear, and the map becomes clear and straight. This allowed them to see the pattern of the "ghost" corrections clearly.

3. The "Differential Detective Work"

To find the missing pieces of the puzzle, the team used a set of differential equations.

• The Analogy: Imagine you have a jigsaw puzzle, but you only have the picture of the sky (the first piece). You need to figure out what the mountains and trees look like.
• The Method: They found a rule (a differential equation) that says, "If you know the shape of the sky, you can mathematically deduce the shape of the mountains." By solving these rules, they could generate the higher-level "moments" (which represent conserved charges or energy in the system) without having to calculate them from scratch every time. It's like having a recipe that lets you bake a cake just by knowing how to make the frosting.

4. The "Capacitor Connection" (The Unexpected Side Quest)

One of the most fun parts of this paper is a side discovery. The math they used for the quantum particles is identical to the math used to calculate the capacitance of two circular metal plates (a capacitor) in a vacuum.

• The Analogy: It's like discovering that the recipe for a perfect chocolate cake is exactly the same as the blueprint for building a suspension bridge.
• The Result: Because they solved the "quantum cake" so perfectly, they also provided the first complete, high-precision solution for the "bridge" (the capacitor) for a long time. This shows how deep physics connections run: quantum particles and old-school electricity are speaking the same mathematical language.

5. The "Numerical Proof" (The Reality Check)

Finally, they didn't just trust their math. They ran massive computer simulations (using a method called the Thermodynamic Bethe Ansatz) to get a "ground truth" number.

• The Analogy: They built a super-accurate digital twin of the system.
• The Result: When they compared their "Trans-Series" prediction to the digital twin, they matched up to 96 decimal places. That is like measuring the distance from the Earth to the Sun and being off by less than the width of a human hair. This proves their "ghost" corrections are real and necessary.

Summary

In short, this paper is a tour de force in mathematical physics. The authors:

1. Found the complete recipe (Trans-Series) for a difficult quantum model, including the invisible "ghost" effects.
2. Used a magic lens (running coupling) to simplify the messy math.
3. Used rules of deduction (differential equations) to generate complex answers from simple ones.
4. Discovered that this quantum recipe is also the blueprint for a capacitor.
5. Proved it works with super-precision computer tests.

They turned a problem that was thought to be only solvable in parts into a complete, unified solution that works for both the quantum world and classical electricity.

Technical Summary

1. Problem Statement

The paper addresses the challenge of obtaining a complete analytical solution for the moments of the rapidity density in the non-relativistic Lieb–Liniger model (a 1D system of bosons with repulsive delta-function interactions).

• Context: The model is defined by a linear integral equation for the rapidity density $f(x)$. While the ground-state energy and density can be solved numerically or via low-density expansions, the large density (strong coupling) regime is notoriously difficult.
• The Gap: Previous work established an asymptotic perturbative expansion in the coupling constant. However, asymptotic series are incomplete; they miss non-perturbative corrections (exponentially suppressed terms like $e^{-1/g}$). A complete physical description requires a trans-series, which combines the perturbative series with these non-perturbative sectors.
• Goal: To derive the full trans-series solution for the moments $\phi_\ell$ (which correspond to conserved charges, with $\phi_0$ being density and $\phi_1$ being energy density) and verify its consistency with numerical solutions and resurgence theory.

2. Methodology

The authors employ a multi-step approach combining integrability techniques, differential equations, and resurgent analysis:

A. Perturbative Foundation (Volin's Method)

• They utilize Volin's method to solve the integral equation recursively.
• The integral equation is transformed into algebraic difference equations for the resolvent function $R(x)$.
• By analyzing the "bulk" and "edge" regimes of the resolvent, they extract the perturbative coefficients of the moments $\phi_\ell$ in terms of a running coupling variable $v$. This variable is chosen to eliminate logarithmic terms ($\log B$) that appear in the standard expansion, simplifying the series structure.

B. Differential Relations and Generalized Moments

• The authors introduce generalized moments $O_{\alpha, \beta}$ and boundary values $\chi_\alpha$ derived from a generalized integral equation with a source term $\cosh(\alpha x)$.
• They derive a system of Ordinary Differential Equations (ODEs) relating these quantities (e.g., $\dot{O}_{\alpha,\beta} = \frac{1}{\pi}\chi_\alpha \chi_\beta$).
• Crucially, they establish a second-order ODE linking the physical moments $\phi_\ell$ to $\phi_{\ell-1}$. This allows the recursive calculation of higher moments once the integration constants are fixed.

C. Wiener–Hopf and Trans-Series Construction

• Using the Wiener–Hopf technique, they construct the non-perturbative solution. The solution is expressed in terms of a perturbative basis $A_{\alpha, \beta}$ and boundary functions $a_\alpha$.
• The full trans-series is built by summing over poles of the Wiener–Hopf kernel, introducing exponential corrections of the form $e^{-2\kappa_r b}$ (where $\kappa_r = 2r$).
• The structure of the trans-series is:
  $$\phi_\ell = \phi_\ell^{(0)} + \sum_{n=1}^\infty e^{-n/g} \phi_\ell^{(n)}$$
  where each sector $\phi_\ell^{(n)}$ is itself an asymptotic series in the coupling.

D. Fixing Integration Constants

• The differential equations for the moments contain two integration constants. One is fixed by the perturbative boundary conditions.
• The second constant (the coefficient of the $1/v^2$ term) is determined by matching the perturbative expansion derived from Volin's method against the differential equation constraints.

E. Resurgence and Alien Derivatives

• The paper applies resurgence theory. They define alien derivatives ($\Delta_\kappa$) which relate the perturbative coefficients to the non-perturbative sectors.
• They propose a formula for the alien derivative of the perturbative basis: $\Delta_{\kappa_j} A_{\alpha, \beta} = 2S_{\kappa_j} A_{\alpha, -\kappa_j} A_{-\kappa_j, \beta}$.
• They demonstrate that the Stokes automorphism (median resummation) acting on the perturbative part reconstructs the full physical solution, confirming that the perturbative series contains all necessary information to recover the non-perturbative physics.

3. Key Contributions

1. Complete Trans-Series Solution: The paper provides the first explicit, complete trans-series for the moments of the Lieb–Liniger model, including arbitrary orders of non-perturbative corrections ($e^{-n/v}$).
2. Differential Recursion: They successfully generalize the differential relations known for relativistic models (like the O(3) sigma model) to the non-relativistic Lieb–Liniger case, allowing the generation of higher moments from lower ones.
3. Capacitance Application: As a by-product, they derive the complete analytic trans-series for the capacitance of a coaxial circular plate capacitor. This connects the Lieb–Liniger integral equation to a classical electrostatics problem (Maxwell-Kirchhoff problem), providing a full analytic expansion for small disk separations.
4. Physical Coupling Mapping: They provide the explicit transformation between the mathematical running coupling $v$ and the physical coupling $g$ (related to particle density), allowing the results to be expressed in terms of experimentally relevant variables.

4. Results

• Analytical Expressions: Explicit formulas for the first few moments ($\phi_0, \phi_1, \dots$) are provided up to high orders in $v$ and including multiple exponential sectors ($e^{-2/v}, e^{-4/v}, \dots$).
• Numerical Verification:
  • Resurgence Check: The authors numerically verified the alien derivative formula (Eq. 75) by analyzing the large-order asymptotics of the perturbative coefficients. The measured coefficients matched the theoretical predictions to high precision.
  • TBA Comparison: They compared the Borel-resummed trans-series against high-precision numerical solutions of the Thermodynamic Bethe Ansatz (TBA) integral equation.
  • Precision: At a specific coupling ($b=10$), the difference between the resummed trans-series (including 5 non-perturbative orders) and the numerical TBA solution was found to be of the order $10^{-96}$.
• Capacitance Formula: A new series expansion for the capacitance $C$ in terms of the disk separation ratio $\delta$ and logarithms was derived, extending previous results.

5. Significance

• Theoretical Physics: This work bridges the gap between perturbative expansions and non-perturbative physics in integrable 1D systems. It confirms that the resurgent structure observed in relativistic models also holds for non-relativistic models, suggesting a universal mathematical structure underlying integrable quantum field theories.
• Methodological Advance: It demonstrates the power of combining Volin's recursive algorithm with Wiener–Hopf factorization and resurgence theory to solve integral equations that are otherwise intractable analytically.
• Experimental Relevance: Since the Lieb–Liniger model is realized in cold atom experiments, having a complete analytical trans-series allows for precise predictions of conserved charges and correlation functions in regimes where standard perturbation theory fails.
• Cross-Disciplinary Impact: The connection to the capacitor problem provides a novel analytical tool for classical electrostatics, offering a complete solution to a problem that has been studied for over a century.

In conclusion, the paper establishes a rigorous framework for the non-perturbative analysis of the Lieb–Liniger model, validating the trans-series approach through both analytical consistency checks and extreme-precision numerical comparisons.