🌀 nonlinear sciences

Complex spacing ratio statistics in the partially open asymmetric quantum baker map

This paper investigates the complex eigenvalue statistics of the asymmetric quantum baker map with partial projective openings, revealing a smooth crossover from quasi-1D to 2D Ginibre-like regimes governed by the number of open channels and reflectivity, which aligns with partially truncated circular unitary ensemble predictions and suggests universal behavior in open quantum chaotic systems.

Original authors: Leonardo Ermann, Pablo Sesin, Alejandro M. F. Rivas, Pablo D. Bergamasco, Gabriel G. Carlo

Published 2026-07-10

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CC BY 4.0

Original authors: Leonardo Ermann, Pablo Sesin, Alejandro M. F. Rivas, Pablo D. Bergamasco, Gabriel G. Carlo

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 a chaotic dance floor where particles (eigenvalues) are trying to find their partners. In a perfectly closed room, they dance in a perfect circle. But what happens when you poke holes in the walls? Do the dancers suddenly scatter into a messy, two-dimensional crowd, or do they stay huddled in a line?

This paper explores that exact question using a mathematical model called the "asymmetric quantum baker map." Think of this map as a chaotic machine that stretches and folds a square of dough (representing the system's state) over and over again. The "asymmetric" part means the machine doesn't treat the left and right sides equally, which keeps the dance interesting and free of boring, repetitive patterns.

The Main Discovery: A Smooth Slide, Not a Sudden Jump

The researchers wanted to see how the dancers behave when the room is partially open. They introduced two "knobs" to control the chaos:

The Number of Holes ( $M$ ): How many channels are open for the particles to escape? They tested from just 1 hole up to 1,000 holes.
The Reflectivity ( $\rho$ ): How "sticky" are the walls? If $\rho = 0$ , the walls are perfectly open (particles vanish instantly). If $\rho = 1$ , the walls are closed (particles bounce back). If $\rho = 0.5$ , the walls are half-open, letting some escape but reflecting others back.

The team measured the "spacing ratio" between the dancers. This is a fancy way of asking: "How far is a dancer from their nearest neighbor compared to their next-nearest neighbor?" This ratio tells us if the dancers are lined up in a single file (1D) or scattered all over the floor (2D).

Here is the big finding: In the systems they studied, there is no sudden switch.

When the room is almost closed (few holes or high reflectivity), the dancers huddle near the edge of the circle, behaving like a one-dimensional line. As you open more holes or make the walls less sticky, the dancers don't suddenly jump into a 2D scatter. Instead, they smoothly glide from that tight line into a full, two-dimensional spread across the complex plane.

The paper found no evidence of an "abrupt transition" in the systems they simulated. You won't find a point where the system suddenly snaps from 1D to 2D. It's a continuous crossover, like a dimmer switch slowly brightening a room rather than a light switch flipping on. However, the authors note that whether this smooth behavior holds true for an infinitely large system ( $N \to \infty$ ) remains an interesting open question.

The Three Ways to Poke Holes

The researchers tested three different ways to make holes in the wall:

Localized: A single block of holes right next to each other.
Random: Holes scattered randomly.
Uniform: Holes spaced out evenly like fence posts.

They found that for a small number of holes (like $M=10$ ), the "Localized" method behaves differently than the others. It's harder for the dancers to escape because the holes are clumped together, creating a "traffic jam" effect. However, as the number of holes grows (around $M=100$ and beyond), all three methods start to look the same. They all converge to the same statistical pattern predicted by a random matrix model called the "Partially Truncated Circular Unitary Ensemble" (PTCUE).

What the Numbers Tell Us

The team ran simulations with a system size of $N = 5001$ . They looked at the "power-law exponent" ( $\beta$ ), a number that describes how the dancers repel each other.

In the 1D "huddled" regime (few holes or high reflectivity like $\rho = 0.9$ ), this exponent is low, near 0.
In the 2D "scattered" regime (many holes or low reflectivity like $\rho = 0$ ), the exponent rises to about 2.5.

They found that even with a single open channel ( $M=1$ ), if you make the walls very reflective ( $\rho = 0.9$ ), the system stays in that 1D huddle. But if you lower the reflectivity to $\rho = 0$ , the system starts to spread out. The reflectivity knob ( $\rho$ ) is a powerful tool: it can control the crossover even if you only have one hole open.

How Sure Are They?

These results come from simulations and numerical experiments, not a physical experiment in a lab. The authors ran their models 20 times for the baker map and 50 times for the random matrix benchmark to ensure the patterns weren't just flukes. They used a statistical tool called the "Bhattacharyya coefficient" to measure how similar their results were to the theoretical predictions.

They found that for large numbers of holes ( $M \gtrsim 100$ ), their simulated results matched the theoretical "PTCUE" prediction with a similarity score greater than 0.975. This suggests a very strong agreement, but it remains a simulation-based finding. The authors suggest that this smooth crossover is likely a universal behavior for many chaotic systems, but they note that whether this holds true for an infinitely large system ( $N \to \infty$ ) is still an open question.

The Takeaway

The paper suggests that the transition from a "one-dimensional" chaotic state to a "two-dimensional" one is a gentle, continuous process controlled by how open the system is and how much it reflects. It's not a sudden explosion of chaos, but a gradual unfolding. This insight helps physicists understand how energy leaks out of complex systems like optical cavities or microwave billiards, showing that even a tiny amount of reflectivity can keep a system "one-dimensional" until the opening gets large enough to let the chaos truly spread out.

Technical Summary: Complex Spacing Ratio Statistics in the Partially Open Asymmetric Quantum Baker Map

Problem Statement
The study addresses the statistical properties of resonance spectra in open quantum systems with chaotic classical dynamics. When such systems couple to the environment, their eigenvalues migrate into the complex plane, characterized by finite lifetimes. While the Grobe-Haake-Sommers (GHS) conjecture posits that fully chaotic open systems follow complex Ginibre random matrix statistics, the behavior of systems with partial leakage (finite reflectivity) remains less understood. Specifically, there is a need to characterize the crossover between quasi-one-dimensional (quasi-1D) spectral regimes, where eigenvalues cluster near the unit circle, and fully two-dimensional (2D) Ginibre-like regimes. Furthermore, the influence of specific opening geometries (localized vs. extended) on these statistics in the presence of partial reflectivity requires detailed investigation.

Methodology
The authors investigate the asymmetric quantum baker map, a system chosen for its fully chaotic classical limit, lack of reflection symmetry (avoiding spectral degeneracies), and exact quantization.

Model: The classical map is defined on a unit torus with a discontinuity at $q = 2/3$ . The quantum propagator is constructed using the Saraceno-Voros quantization with antiperiodic boundary conditions.
Partial Opening: Instead of standard projective openings (where probability is removed entirely), the authors implement partial projective openings. This is achieved by multiplying $M$ $M$ columns of the unitary propagator (corresponding to open channels in position space) by an amplitude reflectivity parameter $\rho \in [0, 1]$ $ρ \in [0, 1]$ .
- $\rho = 0$ : Fully open (standard projective leakage).
- $\rho = 1$ : Closed system.
- $0 < \rho < 1$ : Partially open.
Opening Geometries: Three distinct channel configurations are tested:
1. Localized: $M$ contiguous channels.
2. Random: $M$ channels chosen uniformly at random.
3. Uniform: $M$ equispaced channels.
Benchmark: The Partially Truncated Circular Unitary Ensemble (PTCUE) serves as the random matrix theory benchmark, constructed by applying the same partial projection to a CUE matrix.
Observable: The primary metric is the complex spacing ratio $z_i = (\lambda_{NN,i} - \lambda_i) / (\lambda_{NNN,i} - \lambda_i)$ , where $\lambda_{NN}$ and $\lambda_{NNN}$ are the nearest and next-nearest neighbors of an eigenvalue $\lambda_i$ in the complex plane. The study analyzes the joint distribution $P(|z|, \theta_z)$ of the modulus and phase of $z$ , avoiding the need for spectral unfolding.
Parameters: Simulations were performed with Hilbert space dimension $N=5001$ , varying the number of open channels $M$ and reflectivity $\rho$ . Eigenvalues with $|\lambda| < 0.1$ were filtered out to focus on long-lived resonances.

Key Results

Smooth Crossover: The system exhibits a smooth crossover from a quasi-1D spectral regime to a 2D Ginibre-like regime.
- Quasi-1D Regime (Small $M$ or Large $\rho$ ): Eigenvalues cluster near the unit circle. The phase distribution $P(\theta_z)$ is peaked at $\theta_z = 0, \pm\pi$ , and the power-law exponent $\beta$ in $P(|z|) \sim |z|^\beta$ is near zero (indicating weak level repulsion in the $z$ -variable).
- 2D Ginibre Regime (Large $M$ or Small $\rho$ ): Eigenvalues fill the complex disk. $P(\theta_z)$ becomes nearly uniform (with a dip at $\theta_z=0$ due to level repulsion), and $\beta$ converges to $\approx 2.5$ , consistent with the Ginibre Unitary Ensemble (GinUE).
Role of Reflectivity ( $\rho$ ): The parameter $\rho$ acts as a universal control knob for this crossover. High reflectivity ( $\rho \approx 0.9$ ) delays the transition to 2D statistics, requiring significantly larger $M$ to reach the Ginibre limit compared to the fully open case ( $\rho=0$ ). Notably, a system with $\rho=0.9$ and $M=100$ exhibits statistics as quasi-1D as a system with $\rho=0$ and $M=1$ .
Geometry Dependence:
- At large $M$ , all three opening geometries (localized, random, uniform) converge to the PTCUE statistics.
- Differences are most pronounced at small $M$ . The localized model shows the slowest convergence to the PTCUE limit, consistent with its lower classical escape rate and higher fractal repeller dimension compared to random or uniform openings.
No Abrupt Transition: No evidence of an abrupt phase transition is found for any combination of $M$ and $\rho$ . The transition is continuous.
Statistical Similarity: Using the Bhattacharyya coefficient, the authors quantify the similarity between the baker map models and the PTCUE. While deviations exist at small $M$ and high $\rho$ , the similarity $S$ exceeds 0.975 for $M \gtrsim 100$ in the fully open case.

Significance and Claims
The paper claims that the transition from quasi-1D to 2D spectral statistics in open chaotic systems is not a singularity dependent solely on the opening size, but rather a continuous crossover controlled jointly by the number of open channels ( $M$ ) and the amplitude reflectivity ( $\rho$ ).

Universal Behavior: The results suggest a universal behavior where the specific details of the opening geometry are washed out at large $M$ , converging to the PTCUE prediction.
Experimental Relevance: The findings are directly relevant to open quantum and wave-chaotic experiments (e.g., optical microcavities, microwave billiards) where boundaries possess finite reflectivity. The ability to tune $\rho$ offers a route to access different statistical regimes even with a fixed number of open channels.
Methodological Value: The study demonstrates the utility of the full 2D joint distribution $P(|z|, \theta_z)$ over marginal distributions alone, as it captures structural features (such as the evolution of the phase peak and forbidden regions) that are invisible in 1D projections.
Symmetry Breaking: By using the asymmetric baker map, the authors isolate the universal convergence to PTCUE statistics from the effects of dynamical symmetries (such as the AI $^\dagger$ class found in the leaky standard map), confirming that the 1D-to-2D crossover is a robust feature of chaotic resonance spectra.

The authors conclude that while the crossover is smooth in the finite- $N$ regime studied, the existence of a sharper transition in the $N \to \infty$ limit for fixed $M/N$ and $\rho$ remains an open question. They also propose that the observed crossover may be described by universal effective analytical expressions.

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