🔬 condensed matter

Chiral quantum chaos around exponentially many zero modes in the quantum breakdown model

This paper classifies the symmetry and quantum chaos of a zero-dimensional quantum breakdown model, revealing a $\mathbb{Z}_4$ periodicity, the realization of all five chiral symmetry classes with exponentially many protected zero modes, and distinct spectral signatures of chaos.

Original authors: Kohei Kawabata, Kinya Guan, Hosho Katsura

Published 2026-10-01

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CC BY 4.0

Original authors: Kohei Kawabata, Kinya Guan, Hosho Katsura

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 microscopic world of quantum physics, particles do not behave like the solid objects we see around us. Instead, they exist in a haze of possibilities, described by mathematical maps called energy levels. When a system of these particles is simple and predictable, its energy levels follow a quiet, orderly pattern, much like the distinct notes of a well-tuned instrument. But when the particles interact in a chaotic, random way, their energy levels begin to repel one another, refusing to sit too close together. This phenomenon, known as quantum chaos, is the rule for most complex materials, from the electrons in a metal to the stars in a galaxy. Physicists have long used a set of rules, based on how the system responds to time and symmetry, to predict exactly how these energy levels should behave. These rules act like a periodic table for chaos, sorting different types of random interactions into distinct families.

Recently, scientists have been exploring a specific type of chaotic system called the quantum breakdown model. This model was originally inspired by the way electricity suddenly surges through an insulator, a process where a single electron can trigger a cascade of others. In the version studied by researchers Kohei Kawabata, Kinya Guan, and Hosho Katsura, the system is stripped down to its bare essentials: a collection of particles that can randomly convert into one another without any spatial arrangement or movement through space. The researchers set out to understand how the fundamental symmetries of this system organize its chaotic behavior. They were particularly interested in a special kind of symmetry called chiral symmetry, which forces the energy levels to mirror each other around zero energy, and in the mysterious "zero modes," which are states where the system has no energy at all.

The team discovered that this simple model of interacting particles follows a repeating pattern of four, much like the days of the week. Depending on the number of particles in the system, the model falls into one of four different symmetry categories. What makes this finding remarkable is that the same basic rules of interaction can produce all five known types of quantum chaos that possess this special mirror-like symmetry. In most other famous models used to study chaos, the system is limited to just one or two of these types. Here, by simply changing the number of particles, the researchers could switch the entire system from one type of chaotic behavior to another, creating a complete laboratory for studying these different families of chaos.

Perhaps the most striking discovery was the behavior of the zero-energy states. In many quantum systems, having a state with exactly zero energy is rare or accidental. In this breakdown model, however, the researchers found that the system is forced to have a vast number of these zero-energy states. As the number of particles increases, the number of these zero-energy states grows exponentially, becoming incredibly large. Yet, despite their huge number, they still represent only a tiny fraction of all the possible states the system can be in. It is as if a massive crowd of people were standing perfectly still in the center of a stadium, while the rest of the stadium was filled with people running in a chaotic, random pattern. The stillness of the crowd does not stop the chaos around them; instead, it creates a distinct, empty zone that separates the quiet center from the noisy edges.

The researchers found that this massive collection of zero-energy states acts as a powerful organizer for the rest of the system. The presence of so many zero states pushes the nearby chaotic energy levels away, creating a wide, clear gap between the quiet center and the active chaos. The size of this gap depends on the specific symmetry type of the system. In some cases, the gap is so large that the chaotic levels are pushed far away, while in others, they sit much closer to the center. By measuring the distribution of these energy levels, the team confirmed that the chaotic parts of the system still follow the universal rules of randomness, even while being shaped by this massive central core. They showed that the system behaves exactly as predicted by the theory of random matrices, but with a unique twist: the rules change depending on how many zero states are present.

This work provides a clear map of how symmetry and chaos interact in a system that breaks the usual rules of particle conservation. It demonstrates that a system can host an enormous, exponentially growing number of perfectly still states without losing its chaotic nature. Instead, the chaos adapts, reorganizing itself around these still points in a way that is dictated by the fundamental symmetries of the universe. The findings suggest that the interplay between order and chaos is far more flexible than previously thought, with the ability to support vast numbers of zero-energy states while maintaining the complex, random fluctuations that define quantum chaos. This new understanding could help physicists better understand how complex materials behave and might offer new insights into the deep connections between quantum mechanics and the fundamental structure of space and time.

Technical Summary: Chiral Quantum Chaos around Exponentially Many Zero Modes in the Quantum Breakdown Model

Problem and Motivation
The paper addresses the classification of symmetry and quantum chaos in the quantum breakdown model, a system of randomly interacting fermions motivated by dielectric breakdown physics. While the Sachdev-Ye-Kitaev (SYK) model has established a prototypical link between symmetry and quantum chaos, the symmetry-resolved spectral statistics of the quantum breakdown model remain largely unexplored. Specifically, it is unclear how the breakdown interactions—which change particle number by two while preserving fermion parity—organize universal spectral statistics, particularly in the presence of chiral symmetry and many-body zero modes. The authors aim to establish a systematic classification of internal symmetries and characterize the resulting quantum chaotic correlations in both the spectral bulk and near the spectral origin.

Methodology
The authors investigate a zero-dimensional, all-to-all interacting version of the quantum breakdown model with $N$ complex fermionic modes. The Hamiltonian consists of random quartic interactions that convert one fermion into three (and vice versa), breaking particle-number conservation but preserving fermion parity.

Symmetry Analysis: The authors analytically derive the antiunitary (time-reversal and particle-hole) and unitary (chiral) symmetries of the Hamiltonian. They determine how these symmetries depend on the number of fermions $N$ and the fermion parity sector ( $(-1)^F = \pm 1$ ).
Zero Mode Counting: They analytically derive lower bounds for the number of exact many-body zero modes by calculating the chiral index (the dimensional imbalance between chiral subspaces) and accounting for rank deficiencies caused by isolated states (the vacuum and fully occupied states).
Numerical Simulation: Using exact diagonalization, the authors compute the energy spectra for $N$ $N$ up to 16 across multiple disorder realizations. They analyze:
- Bulk Statistics: Level-spacing ratios ( $r_n$ ) to identify Wigner-Dyson universality classes.
- Hard-Edge Statistics: The distribution of the smallest positive eigenenergy ( $E_{\min}$ ) to probe chiral symmetry effects.
- Spectral Scaling: The scaling of the spectral gap ( $\Delta$ ), mean level spacing ( $\delta$ ), and spectral width ( $W$ ) relative to the chiral index.

Key Contributions and Results

$Z_4$ Symmetry Classification:
The study establishes that the quantum breakdown model exhibits a fourfold periodicity with respect to $N$ (mod 4), similar to the complex SYK model. However, a unique feature is the emergence of chiral symmetry in the many-body Hamiltonian. By varying $N$ and fermion parity, the model realizes all five Altland-Zirnbauer symmetry classes possessing chiral symmetry: AIII, BDI, CI, CII, and DIII.
- For $N \equiv 0, 2 \pmod 4$ , the model exhibits time-reversal or particle-hole symmetries ( $P^2 = \pm 1$ ), leading to classes BDI/CI or DIII/CII depending on the fermion parity.
- For $N \equiv 1, 3 \pmod 4$ , only chiral symmetry remains, corresponding to class AIII.
Exponentially Many Zero Modes:
The authors demonstrate that the chiral index $\nu$ grows exponentially with $N$ (scaling as $2^{N/2}$ ). This enforces an exponentially large number of exact many-body zero modes in every disorder realization. While the absolute number of zero modes grows exponentially, they occupy an exponentially vanishing fraction of the total Hilbert space ( $O(2^{-N/2})$ ). The count is further refined by the presence of isolated states (vacuum and fully occupied), which introduce additional rank deficiencies, leading to an eightfold periodicity in the precise zero-mode count.
Bulk Spectral Statistics:
Numerical results confirm that the spectral bulk (away from zero energy) follows universal random-matrix statistics determined by the Dyson index $\beta$ .
- Classes BDI and CI exhibit $\beta=1$ (orthogonal).
- Class AIII exhibits $\beta=2$ (unitary).
- Classes DIII and CII exhibit $\beta=4$ (symplectic).
  Crucially, for even $N$ , the two fermion-parity subspaces can belong to different symmetry classes (e.g., BDI vs. CI) yet share identical bulk level-spacing distributions, indicating that bulk statistics alone cannot distinguish all chiral classes.
Hard-Edge Spectral Statistics:
The distribution of the smallest positive eigenenergy ( $E_{\min}$ ) resolves the distinctions missed by bulk statistics. The results show that the hard-edge statistics depend on both the Dyson index $\beta$ and the effective chiral index $\nu$ .
- For large $\nu$ , the distribution of $E_{\min}$ becomes sharply concentrated around the mean, consistent with chiral Gaussian ensembles (chGUE, chGOE, chGSE).
- Classes with the same bulk $\beta$ (e.g., BDI vs. CI) exhibit distinct hard-edge distributions and relative fluctuations, confirming that chiral symmetry and the zero-mode structure dictate the spectral behavior near the origin.
Spectral Gap Scaling:
The authors establish a hierarchy of energy scales. In subspaces with a large chiral index, a spectral gap $\Delta$ separates the zero modes from the nonzero spectrum.
- The gap scales linearly with the chiral index relative to the local mean level spacing: $\Delta \approx \nu \delta / \pi$ .
- Relative to the full spectral width $W$ , the gap scales as $\Delta/W \approx \nu / 4D$ , which decreases exponentially with $N$ because the Hilbert space dimension $D$ grows as $2^N$ while $\nu$ grows as $2^{N/2}$ .
- The density of states near the gap collapses onto a universal square-root profile consistent with chiral random matrix theory, despite the microscopic fluctuations retaining class dependence.

Significance
The paper claims to identify a distinct phenomenon where an exponentially large exact degeneracy of zero modes coexists with quantum chaotic spectral correlations in the bulk. The work demonstrates that chiral symmetry can support this subextensive zero-mode sector while reorganizing the surrounding chaotic spectrum on an energy scale much larger than the local level spacing. By providing a systematic symmetry classification across all five chiral classes and elucidating the "hard-edge" spectral statistics, the study connects exact zero-mode counting to the chaotic many-body energy spectrum. This offers a new platform for investigating symmetry-enriched quantum chaos, distinct from previous studies on quantum many-body scars or supersymmetric SYK models, and highlights the role of chiral random matrix theory in systems with macroscopic zero-mode sectors.

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