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Recent Progress in Many-Body Theories

Quantum Chaos and Fundamental Symmetry Violations in Nuclei and Other Complex Systems

Oleg Vorov

Published Jan 1, 1995• 0 citations
Publisher
DOI
10.1007/978-1-4615-1937-9_29

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Quantum mechanical systems with some degree of complexity due to multiple scattering behave as if their Hamiltonians were random matrices. Such behavior, while originally surmised for the interacting many-body system of highly excited atomic nuclei, was later discovered in a variety of situations including single-particle systems with disorder or chaos. A fascinating theme in this context is the emergence of universal laws for the fluctuations of energy spectra and transport observables. After an introduction to the basic phenomenology, the talk highlights the role of symmetries for universality, in particular the correspondence between symmetry classes and symmetric spaces that led to a classification scheme dubbed the 'Tenfold Way'. Perhaps surprisingly, the same scheme has turned out to organize also the world of topological insulators.

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A Projection-Neutral Theory of Quantum Chaos

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This paper presents a fundamental reinterpretation of quantum chaos, arguing that the connection between quantum complexity and operational failure—such as decoherence, information loss, or leakage from protected quantum states—has been poorly understood. The author proposes that quantum chaos should be viewed through the Trojan framework of thin and thick chaos, where projection onto operational variables plays the central role. The core insight is that a quantum system may display all the traditional signatures of chaos—Wigner-Dyson spectral statistics, rapid operator growth, entanglement spreading—yet remain operationally harmless. The key distinction is whether the projected quantum state, rather than the full quantum state, reaches a failure region. This is analogous to classical systems where local instability may remain confined (thin chaos) or may connect to failure (thick chaos). The paper establishes a Quantum Projection-Containment Theorem, which provides a rigorous sufficient condition for avoiding failure: as long as the projected quantum displacement remains smaller than the distance to the failure set, the system remains quantum-thin regardless of its spectral complexity. The Quantum Thinness Ratio is introduced as a quantitative diagnostic, comparing actual projected motion with the available safety margin. Several quantum protection mechanisms are identified that keep chaos thin: spectral gaps that suppress transitions from protected subspaces, symmetries and selection rules that create invariant sectors, localization that prevents delocalization failure, and quantum error-correcting codes that protect logical information even when physical degrees of freedom are chaotic. Quantum thickening occurs when these protections fail—through gap collapse, symmetry breaking, delocalization, or code failure—allowing quantum chaos to become operational failure. The paper provides a practical protocol for diagnosing quantum thinness and emphasizes that quantum chaos is not automatically decoherence. The difference between harmless quantum complexity and operational failure is reachability: chaos becomes thick only when information spreading or leakage reaches protected variables. This framework unifies classical and quantum understandings of stability while respecting the fundamental differences between trajectories and quantum states.

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Graduate texts in physics•Jan 1, 2022•0 citations

Complexity of quantum states in the two-dimensional pairing model

J. R. Armstrong, Sven Åberg, S. M. Reimann +1 more

It is known that many-fermion systems, such as complex atoms and nuclei, reveal (at some level of excitation energy) local signatures of quantum chaos similar to the predictions of random matrix theory. Here, we study the gradual development of such signatures in a model system of up to 16 fermions interacting through short-range pairing-type forces in a two-dimensional harmonic trap. We proceed from the simplest characteristics of the level spacing distribution to the complexity of eigenstates, strength, and correlation functions. For increasing pairing strength, at first, chaotic signatures gradually appear. However, when the pairing force dominates the Hamiltonian, we see a regression towards regularity. We introduce a "phase correlator" that allows us to distinguish the complexity of a quantum state that originates from its collective nature, from the complexity originating from quantum chaos.

Physical Review E•Dec 5, 2012•13 citations

Aspects of Quantum Chaos

Sandro Wimberger

Graduate texts in physics•Jan 1, 2014•0 citations
Quantum mechanical systems with some degree of complexity due to multiple scattering behave as if their Hamiltonians were random matrices. Such behavior, while originally surmised for the interacting many-body system of highly excited atomic nuclei, was later discovered in a variety of situations including single-particle systems with disorder or chaos. A fascinating theme in this context is the emergence of universal laws for the fluctuations of energy spectra and transport observables. After an introduction to the basic phenomenology, the talk highlights the role of symmetries for universality, in particular the correspondence between symmetry classes and symmetric spaces that led to a classification scheme dubbed the 'Tenfold Way'. Perhaps surprisingly, the same scheme has turned out to organize also the world of topological insulators.
This paper presents a fundamental reinterpretation of quantum chaos, arguing that the connection between quantum complexity and operational failure—such as decoherence, information loss, or leakage from protected quantum states—has been poorly understood. The author proposes that quantum chaos should be viewed through the Trojan framework of thin and thick chaos, where projection onto operational variables plays the central role. The core insight is that a quantum system may display all the traditional signatures of chaos—Wigner-Dyson spectral statistics, rapid operator growth, entanglement spreading—yet remain operationally harmless. The key distinction is whether the projected quantum state, rather than the full quantum state, reaches a failure region. This is analogous to classical systems where local instability may remain confined (thin chaos) or may connect to failure (thick chaos). The paper establishes a Quantum Projection-Containment Theorem, which provides a rigorous sufficient condition for avoiding failure: as long as the projected quantum displacement remains smaller than the distance to the failure set, the system remains quantum-thin regardless of its spectral complexity. The Quantum Thinness Ratio is introduced as a quantitative diagnostic, comparing actual projected motion with the available safety margin. Several quantum protection mechanisms are identified that keep chaos thin: spectral gaps that suppress transitions from protected subspaces, symmetries and selection rules that create invariant sectors, localization that prevents delocalization failure, and quantum error-correcting codes that protect logical information even when physical degrees of freedom are chaotic. Quantum thickening occurs when these protections fail—through gap collapse, symmetry breaking, delocalization, or code failure—allowing quantum chaos to become operational failure. The paper provides a practical protocol for diagnosing quantum thinness and emphasizes that quantum chaos is not automatically decoherence. The difference between harmless quantum complexity and operational failure is reachability: chaos becomes thick only when information spreading or leakage reaches protected variables. This framework unifies classical and quantum understandings of stability while respecting the fundamental differences between trajectories and quantum states.
It is known that many-fermion systems, such as complex atoms and nuclei, reveal (at some level of excitation energy) local signatures of quantum chaos similar to the predictions of random matrix theory. Here, we study the gradual development of such signatures in a model system of up to 16 fermions interacting through short-range pairing-type forces in a two-dimensional harmonic trap. We proceed from the simplest characteristics of the level spacing distribution to the complexity of eigenstates, strength, and correlation functions. For increasing pairing strength, at first, chaotic signatures gradually appear. However, when the pairing force dominates the Hamiltonian, we see a regression towards regularity. We introduce a "phase correlator" that allows us to distinguish the complexity of a quantum state that originates from its collective nature, from the complexity originating from quantum chaos.