Charles Kaufman

Integrable and Nonintegrable Classical Spin Clusters: Trajectories and Geometric Structure of Invariants

This study investigates the nonlinear dynamics of a pair of exchange-coupled spins with biaxial exchange and single-site anisotropy. It represents a Hamiltonian system with 2 degrees of freedom for which we have already established the (nontrivial) integrability criteria and constructed the integrals of the motion provided they exist. Here we present a comparative study of the phase-space trajectories for two specific models with the same symmetry properties, one of which (the XY model with exchange anisotropy) is integrable, and the other (the XY model with single-site anisotropy) nonintegrable. In the integrable model, the integrals of the motion (analytic invariants) can be reconstructed numerically by means of time averages of dynamical variables over all trajectories. In the nonintegrable model, such time averages over trajectories define nonanalytic invariants, where the nonanalyticities are associated with the presence of chaotic trajectories. A prominent feature in the nonintegrable model is the occurrence of very long time scales caused by the presence of low-flux cantori, which form “sticky” coats on the boundary between chaotic regions and regular islands or “leaky” walls between different chaotic regions. These cantori dominate the convergence properties of time averages and presumably determine the long-time asymptotic properties of dynamic correlation functions. Finally, we present a special class of integrable systems containing arbitrarily many spins coupled by general biaxial exchange anisotropy.

Integrable and Nonintegrable Classical Spin Clusters: Integrability Criteria and Analytic Structure of Invariants

The nonlinear dynamics is investigated for a system ofN classical spins. This represents a Hamiltonian system withN degrees of freedom. According to the Liouville theorem, the complete integrability of such a system requires the existence ofN independent integrals of the motion which are mutually in involution. As a basis for the investigation of regular and chaotic spin motions, we have examined in detail the problem of integrability of a two-spin system. It represents the simplest autonomous spin system for which the integrability problem is nontrivial. We have shown that a pair of spins coupled by an anisotropic exchange interaction represents a completely integrable system for any values of the coupling constants. The second integral of the motion (in addition to the Hamiltonian), which ensures the complete integrability, turns out to be quadratic in the spin variables. If, in addition to the exchange anisotropy also singlesite anisotropy terms are included in the two-spin Hamiltonian, a second integral of the motion quadratic in the spin variables exists and thus guarantees integrability, only if the model constants satisfy a certain condition. Our numerical calculations strongly suggest that the violation of this condition implies not only the nonexistence of a quadratic integral, but the nonexistence of a second independent integral of motion in general. Finally, as an example of a completely integrableN-spin system we present the Kittel-Shore model of uniformly interacting spins, for which we have constructed theN independent integrals in involution as well as the action-angle variables explicitly.

Hamiltonian chaos

Cartesian coordinates, generalized coordinates, canonical coordinates, and, if you can solve the problem, action-angle coordinates. That is not a sentence, but it is classical mechanics in a nutshell. You did mechanics in Cartesian coordinates in introductory physics, probably learned generalized coordinates in your junior year, went on to graduate school to hear about canonical coordinates, and were shown how to solve a Hamiltonian problem by finding the action-angle coordinates. Perhaps you saw the action-angle coordinates exhibited for the harmonic oscillator, and were left with the impression that you (or somebody) could find them for any problem. Well, you now do not have to feel badly if you cannot find them. They probably do not exist! Laplace said, standing on Newton’s shoulders, “Tell me the force and where we are, and I will predict the future!” That claim translates into an important theorem about differential equations—the uniqueness of solutions for given initial conditions. It turned out to be an elusive claim, but it was not until more than 150 years after Laplace that this elusiveness was fully appreciated. In fact, we are still in the process of learning to concede that the proven existence of a solution does not guarantee that we can actually determine that solution. In other words, deterministic time evolution does not guarantee predictability. Deterministic unpredictability or deterministic randomness is the essence of chaos. Mechanical systems whose equations of motion show symptoms of this disease are termed nonintegrable. Nonintegrability is not the result of insufficient brainpower or inadequate computational power. It is an intrinsic property of most nonlinear differential equations with three or more variables. In principle, Newton’s laws can predict the indefinite future of a mechanical system. But the distant future of a nonintegrable system must be “discovered” by numerical integration, one time step after another. The further into the future that prediction is to be made, or the more precise it is to be, the more precise must be our knowledge of the initial conditions, and the more precise must be the numerical integration procedure. For chaotic systems the necessary precision (for example, the number of digits to be retained) increases exponentially with time. The practical limit that this precision imposes on predictions made by real, necessarily finite, computation was emphasized in a recent article in this column. In the Hamiltonian formulation of classical dynamics, a system is described by a pair of first-order ordinary differential equations for each degree of freedom i. The dynamical variables are a canonical coordinate qi and its conjugate

Chaos in spin clusters: Quantum invariants and level statistics

The energy‐level sequence, whose spacings distribution is the most frequently invoked indicator of quantum chaos, can be derived (for systems with two degrees of freedom) from a two‐dimensional representation of quantum invariants by projection. In this representation, such properties of level sequences as effective randomness in integrable models and level repulsion in nonintegrable models can be more directly interpreted in terms of physical properties. In integrable models, anharmonicities convert quasiperiodic level sequences into effectively random sequences.

Acoustic fluctuations due to the temperature fine structure of the ocean

The effect of oceanic temperature fine structure on sound transmission is investigated. The model used assumes that the layered fine structure is advected horizontally and vertically by the internal waves. Taylor’s frozen turbulence hypothesis is then used to determine the space–time variations in sound speed. We compare our results to those of Ewart’s Cobb Seamount experiment [J. Acoust. Soc. Am. 60, 46 (1976)]. The agreement between the calculated spectrum of the log‐intensity fluctuations and the experiment is excellent except at low frequencies (ω≲0.3 cph) and extends, at high frequencies, even beyond the internal wave frequency range. Previous calculations based on the internal wave model of turbulence have consistently underestimated these fluctuations in this frequency range. The agreement between the observed and the previously calculated phase fluctuations is not affected; that is, the fine structure adds little to the phase fluctuations.

Classical spin clusters: Integrability and dynamical properties

A pair of exchange‐coupled classical spins with biaxial exchange and single‐site anisotropy represents a Hamiltonian system with two degrees of freedom for which the integrability question is nontrivial. We have found that such a system is completely integrable if the model parameters satisfy a certain condition. For the integrable cases, the second integral of the motion (in addition to the Hamiltonian), which guarantees integrability, is determined explicitly. It can be reconstructed numerically by means of time averages of dynamical variables over all trajectories. In the nonintegrable cases, the existence of the time averages is still guaranteed, but they no longer define an analytic invariant, and their determination is subject to long‐time anomalies. Our numerical calculation of time averages for two lines of initial conditions reveals a number of interesting features of such nonanalytic invariants.

Ultracold Neutron Depolarization in Magnetic Bottles

We analyze the depolarization of ultracold neutrons confined in a magnetic field configuration similar to those used in existing or proposed magneto-gravitational storage experiments aiming at a precise measurement of the neutron lifetime. We use an extension of the semi-classical Majorana approach as well as an approximate quantum mechanical analysis, both pioneered by Walstrom et al. [Nucl. Instr. Meth. Phys. Res. A 599, 82 (2009)]. In contrast with this previous work we do not restrict the analysis to purely vertical modes of neutron motion. The lateral motion is shown to cause the predominant depolarization loss in a magnetic storage trap. The system studied also allowed us to estimate the depolarization loss suffered by ultracold neutrons totally reflected on a non-magnetic mirror immersed in a magnetic field. This problem is of preeminent importance in polarized neutron decay studies such as the measurement of the asymmetry parameter A using ultracold neutrons, and it may limit the efficiency of ultracold neutron polarizers based on passage through a high magnetic field.

Hamiltonian Chaos IV

Does quantum chaos exist? If it exists, what is it? If it does not, what was it supposed to have been? Why should we associate chaos with a “third revolution in physics” if it fails to be expressible in terms of quantum mechanics, our most fundamental theory of physical reality? Is there something wrong with quantum mechanics? Or is chaos merely a mathematical construct relevant only for models in classical mechanics? Is the human mind doomed to interpret and understand quantum mechanics in classical terms? The list of unanswered questions in quantum chaos research itself shows symptoms of the phenomenon it attempts to grasp. How do we recognize classical Hamiltonian chaos? We have dealt with this question in our two previous columns. In Part I we discussed the implications of integrability and non-integrability for the phase-space trajectories of classical Hamiltonian systems with two degrees of freedom. We described the method of Poincare surfaces of section as a convenient and striking discriminant between the two possibilities. In Part II we searched for an alternate mode of representation for classical Hamiltonian systems, a mode that is an equally powerful indicator of chaos, but one that can be more directly translated into quantum mechanics than the Poincare map. We proposed a representation based on the construction of classical invariants via time averages of dynamical variables, a representation that can be employed under very general circumstances including integrable and nonintegrable models. In this column we shall construct quantum invariants, demonstrate the impact of non-integrability on these quantities, and discuss their properties in relation to their classical counterparts. In a future column we intend to introduce quantum Poincare surfaces of section for stationary states and bring full circle our survey of quantum and classical Hamiltonian chaos. Consider a quantized integrable model. Its eigenstates are labeled by quantum numbers. There are as many quantum numbers as there are degrees of freedom. Each quantum number represents a quantized action. In the still evolving language of quantum chaos, quantum numbers are said to have been “lost” in the transition from a quantum system whose classical counterpart is integrable to one whose counterpart is chaotic. The broken, or lost, tori of classical chaos become lost quantum numbers of quantum chaos. As we shall see, the loss of quantum numbers is vividly mirrored in the behavior of the quantum invariants. We begin by describing the construction of the quantum invariants for a

Chaos in spin clusters: Correlation functions and spectral properties

We investigate dynamic correlation functions for a pair of exchange‐coupled classical spins with biaxial exchange and/or single‐site anisotropy. This represents a Hamiltonian system with two degrees of freedom for which we have previously established the integrability criteria. We discuss the impact of (non‐)integrability on the autocorrelation functions and their spectral properties. We point out the role of long‐time anomalies caused by low‐flux cantori, which dominate the convergence properties of time averages and determine the long‐time asymptotic behavior of autocorrelation functions in nonintegrable cases.

Spin flip loss in magnetic confinement of ultracold neutrons for neutron lifetime experiments

We analyze the spin flip loss for ultracold neutrons in magnetic bottles of the type used in experiments aiming at a precise measurement of the neutron lifetime, extending the one-dimensional field model used previously by Steyerl