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

The quantum breakdown model, a description of randomly interacting fermions motivated by dielectric breakdown physics, exhibits rich internal symmetry and quantum chaos that are deeply connected to random matrix theory. This work establishes a systematic classification of symmetry and spectral statistics for this model, revealing how chiral symmetry organizes an exponentially large number of many-body zero modes and dictates the universal spectral correlations in both the bulk and near zero energy.

Symmetry Classification of the Model

The paper develops a systematic classification based on internal symmetries, finding that the same breakdown interactions realize all five Altland-Zirnbauer classes possessing chiral symmetry: AIII, BDI, CI, CII, and DIII. This classification is organized by the number of complex fermionic modes, denoted as N (mod 4), summarized in Table I. The interplay between fermion parity and antiunitary symmetries determines the specific class realized for a given N. For instance, for N ≡ 0 (mod 4), the antiunitary symmetry P is respected within each subspace of fermion parity (−1)F and satisfies P2 = +1, leading to classes BDI or CI depending on the chiral symmetry.

Chiral Symmetry and Zero Modes

A crucial feature identified is the emergence of chiral symmetry in the many-body Hamiltonian, which pairs nonzero energy E and −E. This structure underlies the emergence of many-body zero modes protected by this index. The paper shows that varying N and fermion parity reveals that the same breakdown interactions realize all the five Altland-Zirnbauer classes possessing chiral symmetry. Furthermore, the dimensional imbalance between the two chiral subspaces can grow exponentially with N, which enforces an exponentially large number of manybody eigenstates with exact zero energy in every disorder realization.

Spectral Statistics in the Bulk

The study investigates bulk spectral correlations using level-spacing ratios, defined as [46, 47] as:

r n:= min (En+1 − En, En − En−1) / max (En+1 − En, En − En−1). The classification in Section III determines the appropriate Dyson index β for the bulk statistics: BDI and CI correspond to β = 1, class AIII to β = 2, and classes CII and DIII to β = 4. The numerical analysis confirms that the bulk level-spacing ratios follow the three Wigner-Dyson distributions, demonstrating quantum chaotic correlations in the spectral bulk despite the presence of exact many-body zero modes.

Hard-Edge Statistics and Chiral Index Dependence

To resolve features beyond the bulk, hard-edge statistics are analyzed by studying the minimum positive eigenenergy, Emin. The probability density exhibits asymptotic behavior pmin (Emin) ∝ E α min (E → 0), where α = β(ν + 1) − 1. This exponent α characterizes the strength of level repulsion from the origin. The results show that the hard-edge random-matrix statistics remain applicable even when the exponent α becomes subextensive, as it is found to be O (2N/2) for large N. The analysis also shows that while bulk statistics probe time-reversal symmetry within each subspace, the hard-edge spectral statistics elucidate the symmetry- and index-dependent distributions across all the five classes.

Spectral Gap Scaling

The separation between the many-body zero modes and the nonzero spectrum is characterized by three scales: spectral gap ∆, mean level spacing δ, and spectral width W. The paper derives a random-matrix relation for these scales: ∆/W ≃ ν/4D (for ν ≪ D), where D = p + q is the Hilbert space dimension. This implies that a large chiral index ν ≫ 1 pushes the nonzero spectrum away from zero energy, on the scale of the local level spacing. The scaling behavior confirms an exponential separation of energy scales, with ∆/W decreasing exponentially with N.

Many-Body Zero Modes

The number of many-body zero modes is determined by two constraints: the dimensional imbalance between chiral subspaces and additional rank deficiency arising from isolated states. The paper analytically derives lower bounds on the number of zero modes summarized in Table IV, which are saturated for all cases with N ≤ 16. A unique feature is that the chiral index ν can grow exponentially with N, leading to an exponentially large number of zero modes protected by this index, although these modes occupy a subextensive scaling relative to the Hilbert-space dimension. This coexistence of exact degeneracy and random-matrix universality constitutes a distinctive feature identified in disordered interacting fermions.

How it works

1. The Hamiltonian is defined for N complex fermionic modes with random all-to-all interactions, where each interaction term involves four distinct fermionic modes, changing the total fermion number by two but preserving fermion parity: [H,(−1)F] = 0.

Here are specific improvements for AI systems derived from the scientific insights in this paper, categorized by capability:

) Improvements for AI Systems

The core value of this research lies in understanding how internal symmetries (like chiral symmetry) and their interplay with quantum chaos govern the spectral statistics of complex, interacting fermion systems. This knowledge can be directly applied to developing more robust and theoretically informed machine learning models.

Here are the specific improvements:

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2. Improved AI System Capability: Symmetry-Aware Quantum Simulation & Model Selection

3. The improved system can perform highly accurate simulations of complex quantum many-body systems (like those modeled by the Quantum Breakdown Model or SYK) by automatically selecting and classifying the appropriate random matrix ensembles (GUE, GOE, GSE) based on the system's underlying symmetries.

4. Specific Function: Given a Hamiltonian structure, the AI can analyze its constraints (fermion parity, time-reversal symmetry signs) to determine if it belongs to a specific Altland-Zirnbauer class (AIII, BDI, CI, CII, DIII). This allows the AI to select the correct spectral unfolding method and theoretical prediction library for spectral statistics.

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6. Improved AI System Capability: Zero-Mode Prediction & Topological Feature Detection

7. The system can predict the existence and scaling of many-body zero modes in disordered interacting fermion systems, moving beyond simple finite-size scaling arguments used in current methods (like those based on conventional random matrix theory).

8. Specific Function: The AI can calculate the exponential growth rate of the chiral index and determine if a specific Hamiltonian realization is likely to host an exponentially large number of exact zero modes, which is crucial for identifying topologically non-trivial phases or protected states in disordered systems.

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10. Improved AI System Capability: Spectral Feature Discrimination (Bulk vs. Hard-Edge)

11. The system can distinguish between bulk spectral correlations and hard-edge statistics (the distribution of the smallest positive eigenenergy, Emin), providing a more complete picture of quantum chaos in interacting systems.

12. Specific Function: When analyzing the spectrum, the AI can determine whether observed level repulsion or fluctuations are dictated by bulk symmetry (Dyson index β) or by chiral symmetry-dependent hard-edge statistics (exponent α). This enables the system to correctly diagnose subtle phase transitions or crossovers that bulk statistics alone might miss.

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14. Improved AI System Capability: Dynamic Energy Scale Characterization

15. The system can characterize the separation between characteristic energy scales—the spectral gap (∆), mean level spacing (δ), and spectral width (W)—as a function of system parameters like the chiral index ν and fermion parity.

16. Specific Function: The AI can predict how these scales scale with system size or parameter variations, allowing it to quantify the separation between zero modes and the surrounding spectrum. This is vital for understanding thermalization, scrambling dynamics, and response functions in quantum simulators.

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18. Improved AI System Capability: Universal Scaling Law Application

19. The system can apply universal scaling laws derived from chiral random matrix theory (e.g., Marchenko-Pastur law or the scaling of the density of states near a gap) to predict spectral features in novel, high-dimensional or interacting quantum models where exact diagonalization is infeasible.

20. Specific Function: Given a system's symmetry class and effective chiral index, the AI can estimate the resulting density of states profile near zero energy (e.g., the collapse onto a common square-root profile), providing rapid diagnostic information about the spectral structure without needing full diagonalization.

) Summary of Enhanced AI System Capabilities

The resulting AI system will be a sophisticated, symmetry-aware quantum dynamics simulator capable of:

1. Predicting the exact many-body zero mode count and its exponential scaling based on internal symmetries (Chiral Index).

2. Classifying complex Hamiltonians into the ten Altland-Zirnbauer symmetry classes in real-time.

3. Diagnosing whether observed spectral correlations are due to bulk time-reversal symmetry or chiral symmetry, using hard-edge statistics as a diagnostic tool.

4. Quantifying the energy separation between zero modes and the chaotic spectrum, providing insights into thermalization and scrambling rates in strongly correlated quantum systems.

The quantum breakdown model is a model of randomly interacting fermions, motivated by the physics of dielectric breakdown. Here, we establish classification of symmetry and quantum chaos in a zero-dimensional, all-to-all-interacting version of the quantum breakdown model. It exhibits the Z 4 periodicity with respect to the number of fermionic modes, reminiscent of the symmetry classification of the Sachdev-Ye-Kitaev model. A unique feature of the quantum breakdown model is that it realizes all the five classes with chiral symmetry and hosts an exponentially large number of many-body zero modes protected by the chiral index. We elucidate the separation and scaling of the spectral gap and demonstrate the symmetry-enriched hard-edge spectral statistics as signatures of quantum chaos in the chiral symmetry classes.