Walter F. Wreszinski

Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators

A Lyapunov function for the phase-locked state of the Kuramoto model of non-linearly coupled oscillators is presented. It is also valid for finite-range interactions and allows the introduction of thermodynamic formalism such as ground states and universality classes. For the Kuramoto model, a minimum of the Lyapunov function corresponds to a ground state of a system with frustration: the interaction between the oscillators,XY spins, is ferromagnetic, whereas the random frequencies induce random fields which try to break the ferromagnetic order, i.e., global phase locking. The ensuing arguments imply asymptotic stability of the phase-locked state (up to degeneracy) and hold for any probability distribution of the frequencies. Special attention is given to discrete distribution functions. We argue that in this case a perfect locking on each of the sublattices which correspond to the frequencies results, but that a partial locking of some but not all sublattices is not to be expected. The order parameter of the phase-locked state is shown to have a strictly positive lower bound (r ⩾ 1/2), so that a continuous transition to a nonlocked state with vanishing order parameter is to be excluded.

Stability theory for solitary-wave solutions of scalar field equations

We prove stability and instability theorems for solitary-wave solutions of classical scalar field equations.

The HeisenbergXXZ Hamiltonian with Dzyaloshinsky-Moriya interactions

The quantum Heisenberg chain with Dzyaloshinsky-Moriya interactions is solved by relating it to theXXZ Hamiltonian with a certain type of boundary conditions. Several properties of the ground state are derived which agree with the intuition derived from related soluble classical models. Implications to the model of known results from the theory of conformal invariance, as well as generalizations to higher spin, are briefly discussed.

On the mean-field Ising model in a random external field

We use a method developed by van Hemmen to obtain the free energy of the mean-field Ising model in a random external magnetic field. Some results of previous mean-field calculations are confirmed and generalized. The tricritical point in the global phase diagram is discussed in detail. We also consider different probability distributions of the random fields and provide some proofs regarding the conditions for the existence of a tricritical point.

Energy gap, clustering, and the Goldstone theorem in statistical mechanics

We prove a Goldstone-type theorem for a wide class of lattice and continuum quantum systems, both for the ground state and at nonzero temperature. For the ground state (T=0) spontaneous breakdown of a continuous symmetry implies no energy gap. For nonzero temperature, spontaneous symmetry breakdown implies slow clustering (noL1 clustering). The methods apply also to nonzero-temperature classical systems.

Models of Two-Level Atoms in Quasiperiodic External Fields

We prove that the spectrum of the generalized quasienergy operator of a plane-polarized two-level atom in a strong external quasiperiodic electromagnetic field with nonzero constant Fourier component is pure point, under Diophantine conditions on the frequency ratio, and excluding a small subset of resonant values. The widespread belief that there may be only pure point spectrum in such models is briefly discussed in Section 2 and the circularly polarized case—a well-known soluble model—is revisited from the point of view of the quasienergy operator..

Upper quantum Lyapunov exponent and parametric oscillators

We introduce a definition of upper Lyapunov exponent for quantum systems in the Heisenberg representation, and apply it to parametric quantum oscillators. We provide a simple proof that the upper quantum Lyapunov exponent ranges from zero to a positive value, as the parameters range from the classical system’s region of stability to the instability region. It is also proved that in the instability region the parametric quantum oscillator satisfies the discrete quantum Anosov relations defined by Emch, Narnhofer, Sewell, and Thirring.

Level statistics transitions in the spin-boson model

Abstract We present the nearest neighbour distribution (NND) and Δ 3 statistics for the spin-boson model. The results show a standard Poisson-Wigner (GOE) transition in the NND for large spin, but the most interesting feature is the existence of a transition between the two extreme types of fluctuation spectra: picket-fence or harmonic-oscillator type for S small to Poisson for S large.

Aspects of two-level systems under external time-dependent fields

The dynamics of two-level systems in time-dependent backgrounds is under consideration. We present some new exact solutions in special backgrounds decaying in time. On the other hand, following ideas of Feynman, Vernon and Hellwarth, we discuss in detail the possibility to reduce the quantum dynamics to a classical Hamiltonian system. This, in particular, opens the possibility to directly apply powerful methods of classical mechanics (e.g. KAM methods) to study the quantum system. Following such an approach, we draw conclusions of relevance for “quantum chaos” when the external background is periodic or quasi-periodic in time.

Spin waves in quantum ferromagnets

Low-temperature properties of Heisenberg quantum ferromagnets (“spin waves”) are derived within aconfiguration space formalism. Most of the work is done without explicitly assuming translational invariance. We provide a general criterion, classical domination, to decide about the nature and uniqueness of ground states for a large class of quantum ferromagnets. We also analyze and clarify the Dyson formalism and indicate why an energy gap between the physical ground state and the improper (unphysical) states does not exist. This is of particular relevance to the kinematical interaction. Using reflection positivity we provide upper and lower bounds to the contribution of the dynamical interaction to the free energy. In a certain approximation, these bounds imply that the dynamical interaction may be dropped if the inverse temperatureβ and the spin quantum numberS are large enough.