Integrable and nonintegrable classical spin clusters

Magyari, E.; Thomas, H.; Weber, R.; Kaufman, Charles; Müller, Gerhard

doi:10.1007/bf01303725

Cited by 41 publications

(32 citation statements)

References 14 publications

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“…, is equal to l and the other one is zero, in accordance with our analysis of the classical motion [17,18]. In the Ising case (J = 0), the situation is even simpler; here the canonical variables (II.7) are already action-angle variables.…”

Section: Quantum Numbers M1 M2supporting

confidence: 61%

“…Integrability of that system requires the existence of a 2 nd integral of the motion, I(S 1 , S 2 ), in addition to the energy function H(S 1 , S 2 ). In a previous study [17], we showed that with no single-site anisotropy, A x = A y = A z , the 2-spin system (II.2) is always integrable and the 2 nd analytic invariant has the form…”

Section: The 2-spin Clustermentioning

confidence: 99%

“…The present paper is, in fact, the fourth part of our study of the dynamics of integrable and nonintegrable spin clusters. In the first three parts, the focus was, respectively, on the following properties of classical 2-spin clusters: integrability criteria and analytic structure of invariants [17]; geometric structure of analytic and nonanalytic invariants [18]; time-dependent correlation functions and spectral properties [19,20].…”

Section: Introductionmentioning

confidence: 99%

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Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

Srivastava

Müller

1990

Z. Physik B - Condensed Matter

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We present a direct link between manifestations of classical Hamiltonian chaos and quantum nonintegrability effects as they occur in quantum invariants. In integrable classical Hamiltonian systems, analytic invariants (integrals of the motion) can be constructed numerically by means of time averages of dynamical variables over phase-space trajectories, whereas in near-integrable models such time averages yield nonanalytic invariants with qualitatively different properties. Translated into quantum mechanics, the invariants obtained from time averages of dynamical variables in energy eigenstates provide a topographical map of the plane of quantized actions (quantum numbers) with properties which again depend sensitively on whether or not the classical integrability condition is satisfied. The most conspicuous indicator of quantum chaos is the disappearance of quantum numbers, a phenomenon directly related to the breakdown of invariant tori in the classical phase flow. All results are for a system consisting of two exchange-coupled spins with biaxial exchange and single-site anisotropy, a system with a nontrivial integrability condition.

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Section: Quantum Numbers M1 M2supporting

confidence: 61%

Section: The 2-spin Clustermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

Srivastava

Müller

1990

Z. Physik B - Condensed Matter

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“…[14] The impact of nonintegrability on both the quantum invariant-web and the classical invariant-surface is conspicuously displayed in Fig. 3.…”

Section: Nonintegrable Two-spin Systemmentioning

confidence: 99%

“…[14] The value of the constant S is irrelevant for the issues discussed here. Hence it will be suppressed in any list of independent variables.…”

Section: Integrable and Nonintegrable Classical Spin Systemsmentioning

confidence: 99%

Quantum integrability and action operators in spin dynamics

Weigert¹,

Müller²

1995

Chaos, Solitons & Fractals

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A new formulation of the quantum integrability condition for spin systems is proposed. It eliminates the ambiguities inherent in formulations derived from a direct transcription of the classical integrability criterion. In the new formulation, quantum integrability of an N -spin system depends on the existence of a unitary transformation which expresses the Hamiltonian as a function of N action operators. All operators are understood to be algebraic expressions of the spin-components with no restriction to any finite-dimensional matrix representation. The consequences of quantum (non-)integrability on the structure of quantum invariants are discussed in comparison with the consequences of classical (non-)integrability on the corresponding classical invariants. Our results indicate that quantum integrability is universal for systems with N = 1 and contingent for systems with N ≥ 2.

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Bounded Quantum Systems

Reichl

2004

The Transition to Chaos

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Integrable and nonintegrable classical spin clusters

Cited by 41 publications

References 14 publications

Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

Quantum integrability and action operators in spin dynamics

Bounded Quantum Systems

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