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Physical Review Letters

Dynamical Symmetry Breaking and Quantum Nonintegrability

Wei-Min Zhang, Craig C. Martens, D.H. Feng, Jian‐Min Yuan

Published Nov 7, 1988• 41 citations
Publisher

Abstract

In this Letter, we describe the connection between the classical concepts of nonintegrability and chaos and the quantum concept of dynamical symmetry breaking. The existence of unbroken dynamical symmetry implies integrability of the mean-field motion in a quantum phase space, defined as a symplectic coherent-state parametrization of the coset space of the overall dynamical group. Broken dynamical symmetry leads to nonintegrability, and thus chaotic solutions to Hamilton's equations in the quantum phase space. We illustrate the general ideas with results obtained for a model of two coupled spins.

DOI
10.1103/physrevlett.61.2167

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In this Letter, we describe the connection between the classical concepts of nonintegrability and chaos and the quantum concept of dynamical symmetry breaking. The existence of unbroken dynamical symmetry implies integrability of the mean-field motion in a quantum phase space, defined as a symplectic coherent-state parametrization of the coset space of the overall dynamical group. Broken dynamical symmetry leads to nonintegrability, and thus chaotic solutions to Hamilton's equations in the quantum phase space. We illustrate the general ideas with results obtained for a model of two coupled spins.
In this paper we discuss the concepts of quantum integrability and nonintegrability. Based on the concept of a complete set of commuting observables and the Hilbert-space structure of a quantum system, the definitions are given for the quantum-dynamical degrees of freedom and quantum phase space from which the quantum integrability is defined. A criterion for quantum integrability then emerges; the system is integrable if it possesses dynamical symmetry. Breaking of dynamical symmetry is connected with the nonintegrability of systems and thus is the inherent mechanism of chaotic motion. A number of examples are discussed.
In this paper, the dynamical symmetry concept is used to investigate the quantum integrability of nuclear many-body systems. It is found that such systems are generally nonintegrable due to the complicated inherent interactions. In fact, in certain regions of the intrinsic parameters, the time-dependent mean-field solution can be chaotic. However, when subdynamical symmetries, e.g., certain excitation modes, dominate, the corresponding dynamics will be rendered regular, thus signaling that the nuclear many-body system is not fully chaotic. Chaotic behavior emerges in the transitional region between the excitation modes where subdynamical symmetries are broken and accompanied by structural phase transitions.
Based on the concepts of integrability and nonintegrability of a quantum system presented in a previous paper [Zhang, Feng, Yuan, and Wang, Phys. Rev. A 40, 438 (1989)], a realization of the dynamics in the quantum phase space is now presented. For a quantum system with dynamical group scrG and in one of its unitary irreducible-representation carrier spaces ${\mathit{gerh}}_{\mathrm{\ensuremath{\Lambda}}}$, the quantum phase space is a 2${\mathit{M}}_{\mathrm{\ensuremath{\Lambda}}}$-dimensional topological space, where ${\mathit{M}}_{\mathrm{\ensuremath{\Lambda}}}$ is the quantum-dynamical degrees of freedom. This quantum phase space is isomorphic to a coset space scrG/scrH via the unitary exponential mapping of the elementary excitation operator subspace of scrg (algebra of scrG), where scrH (\ensuremath{\subset}scrG) is the maximal stability subgroup of a fixed state in ${\mathit{gerh}}_{\mathrm{\ensuremath{\Lambda}}}$. The phase-space representation of the system is realized on scrG/scrH, and its classical analogy can be obtained naturally. It is also shown that there is consistency between quantum and classical integrability. Finally, a general algorithm for seeking the manifestation of ``quantum chaos'' via the classical analogy is provided. Illustrations of this formulation in several important quantum systems are presented.
The concept of integrability of a quantum system is developed and studied. By formulating the concepts of quantum degree of freedom and quantum phase space, a realization of the dynamics is achieved. For a quantum system with a dynamical group G in one of its unitary irreducible representative carrier spaces, the quantum phase space is a finite topological space. It is isomorphic to a coset space G/R by means of the unitary exponential mapping, where R is the maximal stability subgroup of a fixed state in the carrier space. This approach has the distinct advantage of exhibiting consistency between classical and quantum integrability. The formalism will be illustrated by studying several quantum systems in detail after this development.
It is known that the development and deep understanding of modern interaction theory and classical mechanics are made through geometry and symmetry. Yet, quantum mechanics which was regarded to be the microscopic theory of classical mechanics and achieved the crowning success in interpreting the entire microscopic world was developed purely from algebraic methods. In this thesis, we will study the geometry and dynamical symmetry in quantum systems, from which the question of integrability and chaos are explicitly addressed. First of all, the quantum dynamical degrees of freedom and quantum integrability are precisely defined and the inherent geometrical structure of quantum systems is explored from the fundamental structure of quantum theory. Such a geometrical structure can provide a framework to simultaneously build quantum and classical mechanics. The quantum-classical correspondence is then explicitly deduced. The dynamics of quantum system before it reaches the classical limit is formulated. Thus, the classical chaos is proven to be a special limiting phenomena of quantum systems and the dynamics before the system reaches its classical chaos is explored. The latter is the first step to seek the quantum manifestation of chaos. The relationship between integrability and dynamical symmetry are studied and some universal properties are discovered: a dynamical system (both quantum and classical) is integrable if it possesses a dynamical symmetry. Chaos will occur if the system undergoes a dynamical symmetry breaking and is accompanied by a structural phase transition. Thus, the concept of dynamical symmetry can be used to predict the general behaviors of a system. The theoretical underpinnings developed in this thesis are verified by many basic quantum mechanical examples. In the second part, we have applied the theory to construct the geometrical description for fermionic systems and then use this general formulation to study the geometrical properties, integrability and chaos in nuclear collective motions. The study of geometry for the fermion systems clearly show how the pure quantum properties (e.g. Pauli principle) are exhibited in geometry and how these geometrical manifestations can interpret experimental facts in nuclear collective motion (e.g. the satiation of nuclear deformation and narrow oblate window). The dynamical symmetry and dynamical symmetry breaking in nuclear collective motion are extensively studied.