⚛️ high-energy theory

From Divergent Series to Geometry: Resurgence of the Quantum Metric

This paper demonstrates that the divergent perturbative series of the quantum metric tensor in anharmonic oscillators can be accurately reconstructed using Borel–Padé resummation guided by resurgence theory, thereby extending non-perturbative techniques from energy eigenvalues to quantum geometry.

Original authors: Marcos J. Hernández, Bogar Díaz, J. David Vergara

Published 2026-08-05

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CC BY 4.0

Original authors: Marcos J. Hernández, Bogar Díaz, J. David Vergara

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

Imagine you are trying to predict the path of a rollercoaster, but your map is made of a series of tiny, shaky steps. In the world of quantum physics—the study of the tiniest building blocks of our universe—scientists often use a tool called "perturbation theory" to make these predictions. Think of it like trying to guess the shape of a complex sculpture by stacking one block on top of another. Usually, this works great for the first few blocks. But in many tricky quantum systems, if you keep stacking blocks forever, the tower doesn't get taller and more accurate; instead, it starts to wobble and eventually collapses into chaos. The math becomes a "divergent series," a runaway train of numbers that grows so fast it seems to break the laws of logic.

For decades, physicists have been stuck with this broken map. They knew the tower was shaky, but they also knew that if they stopped stacking at just the right moment, the shape they saw was surprisingly close to the truth. The big question was: How do we fix the map so we can see the whole picture without the tower falling apart? Enter "resurgence theory," a mathematical magic trick that suggests these broken, divergent series aren't actually broken at all. Instead, they are hiding secret clues about the universe's deeper, invisible rules. By using a special kind of "resummation" (a fancy word for re-organizing the numbers), scientists can decode the chaos and reveal the exact shape of the sculpture, even when the original map was a mess. This is crucial because understanding these hidden rules helps us predict how quantum systems behave, which is the foundation for future technologies like quantum computers and new materials.

Now, let's look at what this specific paper does. The authors, Marcos J. Hernández, Bogar Díaz, and J. David Vergara, decided to test this mathematical magic trick on a new, unexplored part of the quantum world: the "Quantum Metric Tensor" (QMT). If the energy levels of a quantum system are like the speed of the rollercoaster, the QMT is like a map of the track's curvature and how sensitive the ride is to tiny changes in the track's design. It tells us how "close" two different quantum states are to each other. The problem? When scientists tried to calculate this QMT using standard methods, they hit the same wall as before: the math exploded into a divergent series that grew too fast to handle.

The team tackled three different types of quantum "rollercoasters" to see if resurgence theory could fix the QMT map. First, they looked at a quartic oscillator (a system with a specific type of bumpy potential). They found that, just like with energy levels, the QMT numbers grew factorially (getting huge, fast). By applying a technique called Borel–Padé resummation—which is like taking the chaotic numbers, smoothing them out, and then re-stitching them together—they were able to reconstruct the exact QMT values. When they compared their "resurrected" math to a perfect computer simulation (exact diagonalization), the results matched beautifully, especially for the ground state (the system's resting position).

Next, they moved to a sextic oscillator, which is an even bumpier, more complex ride. Here, the numbers didn't just grow fast; they grew in a weird, double-factorial pattern that standard math tools couldn't handle. The authors had to invent a "generalized" version of their smoothing tool (called a generalized Borel–Leroy transform) to fit this specific shape. They found that by tweaking a few parameters, this new tool could also perfectly reconstruct the QMT, proving that the method is flexible enough to handle even the wildest quantum curves.

Finally, they took the experiment into higher dimensions, looking at d-dimensional oscillators (systems that exist in 3, 4, 5, or even 6 dimensions at once). They discovered that while the math got more complicated as the dimensions increased, the core idea still held up. The divergent series could still be tamed and resummed to give accurate results.

The paper suggests that these techniques are powerful and reliable, but it also notes a few important limits. The method works best for the ground state and gets slightly less accurate as the system gets more excited (like a rollercoaster going faster and faster). Also, while the energy calculations were slightly more precise than the QMT calculations, both improved significantly as the authors included more terms in their series. The authors explicitly state that they did not find any "Berry curvature" (a related quantum property) in these specific examples because it was zero, so they couldn't test the method on that part yet. However, their findings strongly suggest that resurgence theory is a robust way to turn broken, divergent quantum maps into accurate, usable guides, bridging the gap between messy approximations and the hidden, non-perturbative truths of the quantum universe.

Technical Summary: From Divergent Series to Geometry: Resurgence of the Quantum Metric

Problem Statement
Perturbation theory is a cornerstone of quantum mechanics, yet in physically relevant systems such as anharmonic oscillators, perturbative expansions are typically divergent asymptotic series rather than convergent ones. While these series encode accurate information when truncated optimally, their large-order behavior is intrinsically linked to non-perturbative physics (e.g., instantons, tunneling). Historically, resurgence theory and Borel resummation techniques have been successfully applied to extract exact results from these divergent energy eigenvalues. However, the perturbative expansions of the Quantum Metric Tensor (QMT)—a central object in quantum geometry that quantifies the distinguishability of nearby states and governs fidelity susceptibilities—have not been systematically analyzed using these resurgent methods. The core problem addressed is whether the divergent asymptotic series of the QMT, which inherit the divergence of the underlying wave functions, can be reliably resummed to yield accurate non-perturbative physical predictions.

Methodology
The authors apply resurgence theory and Borel–Padé resummation techniques to the perturbative expansions of the QMT in three representative anharmonic oscillator models:

The Quartic Oscillator: A standard benchmark where perturbative coefficients exhibit factorial growth ( $n!$ ).
The Sextic Oscillator: A higher-order system where coefficients follow a non-standard growth law ( $\Gamma(2n + \beta)$ ), requiring generalized Borel–Leroy transforms.
The $d$ -Dimensional Quartic Oscillator: A system with spherical symmetry to test the scalability of these methods in higher-dimensional configuration spaces.

The methodology involves:

High-Order Calculation: Using the BenderWu package (for 1D) and the LagrangeMesh package (for $d$ -dimensions) to compute perturbative coefficients for energy eigenvalues and QMT components up to high orders (e.g., $m=100$ or $m=200$ ).
Resurgence Analysis: Identifying the large-order behavior of the coefficients to determine the nature of singularities in the Borel plane.
Resummation:
- For the quartic case, applying standard Borel transforms followed by Padé approximants and Laplace integrals. For alternating series, the Principal Value (PV) prescription is employed to handle spurious poles on the positive real axis.
- For the sextic case, utilizing generalized $(\alpha, \beta)$ -Borel transforms with $\alpha=2$ to accommodate $\Gamma(2n+\beta)$ growth. A variational numerical criterion is used to optimize the parameter $\beta$ to minimize the error against exact values.
Validation: Comparing the resummed results against exact numerical diagonalization of the Hamiltonian and the Lagrange Mesh method.

Key Contributions

Extension to Quantum Geometry: This work represents the first application of resurgence theory and Borel resummation specifically to the Quantum Metric Tensor, extending techniques previously reserved for energy spectra.
Universal Non-Perturbative Scales: The analysis identifies that the divergent QMT series exhibit factorial growth consistent with resurgence theory. For the quartic oscillator, the large-order behavior is controlled by a singularity at $\lambda/k^{3/2} = -1/3$ , implying non-perturbative corrections of the form $\exp(1/3\lambda)$ .
Generalized Resummation for Higher-Order Potentials: The paper demonstrates that for the sextic oscillator, standard Borel resummation is insufficient due to $\Gamma(2n+\beta)$ growth. It successfully implements generalized Borel–Leroy transforms, showing that these methods can capture the underlying non-perturbative physics even in strongly anharmonic regimes.
Dimensional Robustness: The study confirms that resurgence techniques remain effective in $d$ -dimensional systems. While the dimensional dependence modifies the asymptotic laws of the coefficients (introducing factors of $d$ in the Gamma functions), the series remain Borel summable, and the resummation accurately reproduces exact results.

Results

Accuracy: Borel–Padé resummation yields results that converge systematically to the exact values obtained via numerical diagonalization as the truncation order $m$ increases. This holds true for both energy eigenvalues and QMT components across the quartic, sextic, and $d$ -dimensional models.
Ground vs. Excited States: The resummation is most accurate for the ground state. For excited states, the accuracy deteriorates as the quantum number $N$ increases, particularly for small values of the harmonic parameter $k$ . This is attributed to the growth of matrix elements and the increase in the exponents $\beta$ governing the factorial growth, which proportionally enlarges non-perturbative contributions.
Analytic Structure: The Padé approximants for the QMT components do not display the same regular pole accumulation patterns as the energy spectrum, suggesting a more intricate analytic structure for the metric tensor. However, the accumulation of poles around the expected singularity ( $u = -1/3$ ) in the Borel plane confirms the connection to instanton-like effects.
Comparison of Methods: In all examples, Borel resummation of the energy spectrum performs slightly better than that of the QMT components. The authors attribute this to the faster growth rate of the QMT coefficients compared to the energy coefficients.

Significance and Claims
The paper claims that its findings extend resurgent techniques from the domain of energy eigenvalues to the Quantum Metric Tensor, thereby highlighting the interplay between quantum geometry and non-perturbative physics. The work demonstrates that:

Divergent perturbative expansions of the QMT are not pathological but are resurgent objects containing hidden non-perturbative data.
Suitable resummation procedures (Borel–Padé and generalized Borel–Leroy) can reconstruct exact physical results from these divergent series with high precision, often requiring fewer computational resources than direct diagonalization for high-precision needs.
The framework is robust across different anharmonic potentials and spatial dimensions, preserving the connection between the large-order behavior of perturbative coefficients and non-perturbative scales.

The authors conclude that resurgence and Borel resummation are versatile tools capable of bridging perturbative and non-perturbative physics in quantum geometry, providing a rigorous method to extract meaningful physical information from asymptotic series that would otherwise be considered divergent.

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