⚛️ high-energy theory

Analytic Spread Complexity from Level Statistics: From Chaos to Integrability

This paper establishes an analytic framework linking spread complexity to local spectral statistics by demonstrating that the Krylov kernel's banded structure allows the complexity's finite-time peak and late-time behavior to be directly expressed through the Fourier transforms of nearest-neighbour spacing distributions, thereby unifying the description of both chaotic and integrable quantum systems.

Original authors: Pallab Basu, Suman das, Bigboy Madlala

Published 2026-08-11

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Pallab Basu, Suman das, Bigboy Madlala

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 the universe as a giant, invisible orchestra playing a song that never repeats. In the world of quantum physics, this "song" is the behavior of tiny particles, and the "notes" are the specific energy levels they can hold. For decades, scientists have tried to figure out how to tell if this orchestra is playing a chaotic, jazz-like improvisation or a rigid, predictable classical march. To do this, they use a special tool called "complexity." Think of complexity not as a difficult math problem, but as a measure of how much a quantum system has "scrambled" or spread out over time. If you drop a drop of ink into a glass of water, the ink spreads until it's everywhere; in quantum mechanics, information spreads in a similar way, and "spread complexity" tracks how far that information has traveled.

For a long time, scientists noticed something strange and exciting about this spreading in chaotic systems: the complexity doesn't just grow forever. Instead, it rises, hits a sharp peak (like a mountain top), and then settles down to a flat plateau. It's as if the ink spreads, hits a wall, and then slowly settles. This peak is a huge clue that the system is chaotic, but nobody knew exactly why the peak happened or what part of the music was causing it. Was it the distance between the notes? The rhythm? The paper you are about to read dives into this mystery, treating the energy levels of the system like a map and the complexity like a traveler moving across it, trying to find the secret recipe that creates that mountain peak.

The Map and the Traveler

This paper, titled "Analytic Spread Complexity from Level Statistics," is like a detective story where the detectives are trying to solve the mystery of that "mountain peak" in quantum complexity. The authors, Pallab Basu, Suman Das, and Bigboy Madlala, propose a new way to look at the problem. Instead of watching the complexity grow step-by-step in a confusing, abstract space, they decided to look at the "energy map" of the system.

They discovered that the "kernel" (a fancy word for the rulebook that tells the complexity how to move) has a very specific shape. Imagine a giant spreadsheet where the rows and columns represent different energy levels. The authors found that the most important numbers in this spreadsheet are clustered tightly around the main diagonal (where the row number equals the column number). As you move away from this center line, the numbers drop off very quickly, like a hill that flattens out into a valley. This means the complexity doesn't care about the relationship between energy levels that are miles apart; it only really cares about neighbors.

The "Universal" Shortcut

Here is where the story gets clever. The authors realized that because the kernel is so focused on neighbors, they could replace the messy, complicated kernel of a real chaotic system with a much simpler, "universal" one. They compared it to a uniform grid, like a perfectly spaced ladder. They call this their "approximate kernel-universality hypothesis."

Think of it like this: If you want to know how a crowd of people moves through a crowded room, you don't need to track every single person's unique personality. If the room is chaotic, the movement is often governed by simple, local rules—people bumping into their immediate neighbors. The authors suggest that for the purpose of calculating complexity, you can pretend the energy levels are just rungs on a perfect ladder. This "ladder" model is easy to solve with math, and surprisingly, it predicts the behavior of real, messy chaotic systems with amazing accuracy.

The Secret Ingredient: The Nearest Neighbor

By using this simple ladder model, the authors derived a master formula. This formula says that the height and shape of the complexity peak are controlled almost entirely by the "nearest-neighbor spacing distribution."

Let's break that down with an analogy. Imagine the energy levels are people standing in a line. The "spacing" is the distance between them.

In a chaotic system (like the Gaussian Orthogonal Ensemble or GOE, and Gaussian Unitary Ensemble or GUE), the people hate standing too close to each other. They push apart, creating a "repulsion." This repulsion creates a specific pattern of distances.
In an integrable system (like a Poisson distribution), the people don't care about each other. They stand randomly, like raindrops on a window.

The paper shows that the "mountain peak" in complexity only appears when there is this "repulsion" or "pushing apart" between neighbors. The authors found that the peak is essentially the Fourier transform (a mathematical way of looking at patterns) of the distribution of these nearest-neighbor distances.

They tested this by simulating the math. For chaotic systems, their simple formula predicted the peak height and position almost perfectly. For example, for the GUE ensemble, they predicted a peak at a specific time (around 2.764 in their "unfolded" time units), and the simulations matched this closely.

The Case of the Missing Peak

The most dramatic proof of their idea comes from what happens when the "pushing apart" is removed. The authors looked at Poisson statistics, which represents a system with no chaos (integrable). In this scenario, the energy levels are just random, independent numbers.

When they applied their formula to this random case, the "mountain peak" completely disappeared. Instead of rising and falling, the complexity just grew slowly and smoothly until it hit the plateau. There was no overshoot, no mountain. This confirms their theory: the peak isn't a random accident; it is a direct fingerprint of the short-range correlations (the "pushing apart") between energy levels. If the levels don't repel each other, the peak vanishes.

How Sure Are They?

The authors are very confident in their findings, but they are careful to distinguish between what they proved and what they simulated.

The Structure: They proved mathematically that the kernel is "banded" (focused on neighbors) and that it has a "semicircular" shape.
The Hypothesis: The idea that the complex kernel can be replaced by the simple "uniform lattice" kernel is an approximation. They tested this numerically with simulations of 1,000 energy levels, and the results matched the exact calculations incredibly well. The difference was so small it was almost invisible.
The Conclusion: They suggest that the peak is controlled by the nearest-neighbor spacing. They didn't just guess; they derived a formula that works for both chaotic and non-chaotic systems, and the math checks out in their simulations.

Why It Matters

This paper is a big deal because it connects two worlds that were previously separate: the "dynamics" (how things move and change) and the "statistics" (the pattern of the energy levels). Before this, scientists knew that chaos created a peak, but they didn't have a simple, analytic way to say why or how high it would be.

Now, they have a "recipe." If you know the pattern of the distances between energy levels (the spacing distribution), you can calculate the complexity peak without having to run a supercomputer simulation. It's like being able to predict the shape of a wave just by looking at the spacing of the pebbles on the beach, without needing to watch the ocean.

The authors also point out that this might help us understand black holes and the "holographic principle" (the idea that our 3D universe might be a projection of a 2D surface). In those theories, the growth of complexity is linked to the growth of a wormhole. If the peak in complexity is really just a reflection of how energy levels repel each other, it might give us a microscopic clue about how the geometry of space-time itself is built from quantum information.

In short, this paper takes a confusing, jagged mountain of data and shows us that it's actually just a smooth, predictable hill built from the simple, local rules of how energy levels keep their distance from one another. The "chaos" isn't a mystery; it's just the sound of neighbors politely pushing each other apart.

Technical Summary: Analytic Spread Complexity from Level Statistics

Problem Statement
Spread complexity (a specific form of Krylov complexity) has emerged as a diagnostic tool for quantum chaos, characterized by a distinct growth–peak–plateau time evolution in chaotic systems. While the dynamical origin of this peak has been partially understood through transition probabilities, its microscopic spectral origin remains incompletely resolved. Specifically, it is unclear which spectral correlations control the peak's height, why this height varies across different random-matrix symmetry classes (e.g., GOE vs. GUE), and how spread complexity relates quantitatively to standard spectral statistics. Furthermore, the behavior of spread complexity in integrable (Poisson) limits, where the peak is absent, requires a unified theoretical framework.

Methodology
The authors develop an analytic framework that reformulates spread complexity directly in the energy eigenbasis, establishing a bridge between Krylov dynamics and local spectral statistics. The methodology proceeds through the following steps:

Kernel Representation: Starting from the definition of spread complexity as the expectation value of the Krylov position operator, the authors derive an exact expression (Eq. 2.19) where complexity is a double sum over energy eigenstates. This expression separates into a structural "Krylov kernel," $K(i,j)$ , which depends on the Lanczos polynomials, and oscillatory phase factors determined by energy differences.
Diagonal Decomposition: The double sum is reorganized based on the separation $k$ between energy levels in the ordered spectrum. The authors demonstrate that the expansion converges rapidly, implying that the dynamics are dominated by nearby energy levels.
Analysis of Kernel Structure: Through numerical analysis of random-matrix ensembles, the authors identify that the Krylov kernel possesses an approximately banded structure in the energy basis. The magnitude of matrix elements decays rapidly away from the principal diagonal, following a $k^{-2}$ decay for the integrated off-diagonal weights. Additionally, the kernel exhibits a semicircular profile along the diagonal.
Approximate Kernel Universality Hypothesis: Motivated by the banded structure, the authors propose that after "unfolding" the spectrum (normalizing the mean level spacing to unity), the Krylov kernel of generic random-matrix ensembles is well-approximated by the analytically tractable kernel of a uniform lattice. This hypothesis is tested by comparing the overlap matrix between the polynomial bases of unfolded spectra and the uniform lattice, showing that the transformation is approximately local.
Analytic Derivation: Combining the universal uniform-lattice kernel with local spectral statistics, the authors derive a master formula (Eq. 1.1/5.6) expressing the ensemble-averaged spread complexity in terms of the Fourier transforms of the $k$ -th nearest-neighbour spacing distributions, $\Phi_k(t)$ .

Key Contributions and Results

Master Formula for Spread Complexity: The paper derives a simple analytic expression for the unfolded spread complexity:
$\frac{\langle C_{\text{unfold}}(t) \rangle}{D} \simeq \frac{1}{2} \left( 1 - 2 \sum_{k=1}^{\infty} \frac{\Phi_k(t)}{4k^2 - 1} \right)$
This formula explicitly connects the time evolution of complexity to the spectral statistics of the system. The universal kernel coefficients $(4k^2-1)^{-1}$ determine the relative weight of different neighbour separations, while all ensemble dependence enters solely through the spacing distributions $\Phi_k(t)$ .
Origin of the Complexity Peak: The analysis identifies the nearest-neighbour spacing distribution ( $k=1$ ) as the primary driver of the finite-time peak.
- For Gaussian Unitary Ensemble (GUE) and Gaussian Orthogonal Ensemble (GOE), retaining only the $k=1$ term (nearest-neighbour) reproduces the peak height and position with remarkable accuracy. The stronger level repulsion in GUE leads to a deeper minimum in the Fourier transform of the spacing distribution, resulting in a higher complexity peak compared to GOE.
- The framework quantitatively explains the variation in peak height across symmetry classes as a direct consequence of their distinct nearest-neighbour spacing distributions.
Absence of Peak in Integrable Systems: For Poisson statistics (integrable limit), the authors perform an exact resummation of the series. They prove analytically that the complexity increases monotonically from zero to its late-time plateau without developing a finite-time peak. This contrasts sharply with the chaotic case and confirms that the peak is a signature of level repulsion and short-range spectral correlations.
Numerical Validation: The authors validate their hypotheses by comparing the exact spread complexity of unfolded GUE and GOE spectra against the predictions of the uniform-lattice kernel. The agreement is excellent across the growth, peak, and plateau regimes. Furthermore, the leading-order analytic approximations (using only $k=1$ ) accurately predict the peak positions and heights observed in numerical simulations.

Significance
The paper claims to provide a microscopic framework that directly links Krylov dynamics to local spectral statistics. Its primary significance lies in:

Identifying the Spectral Origin: It establishes that the characteristic peak in spread complexity is controlled by short-range level correlations, specifically the nearest-neighbour spacing distribution.
Unifying Chaos and Integrability: The framework successfully describes both chaotic random-matrix ensembles and the integrable Poisson limit within a single analytic structure, explaining the presence or absence of the peak based on spectral statistics.
Simplification via Universality: By proposing the approximate kernel universality hypothesis, the authors demonstrate that the complex, system-specific Krylov dynamics can be effectively captured by a universal kernel (that of a uniform lattice) combined with local spectral data. This reduces the problem of calculating spread complexity to the evaluation of Fourier transforms of spacing distributions.
Diagnostic Potential: The work suggests that the detailed pattern of diagonal dominance in the Krylov kernel and the resulting complexity profile can serve as a diagnostic tool to distinguish between chaotic and integrable dynamics, complementing existing probes based on level statistics.

The authors note that while the connection to bulk geometry in holography (e.g., wormhole growth) is an area of interest, their current work focuses on establishing the direct connection between Krylov dynamics and spectral statistics in the boundary theory. They suggest future work could explore the extension of these results to sparse models and the potential for analytic bounds on complexity growth.