Aperiodic Modulation Research Articles

Log-periodic corrections to scaling: exact results for aperiodic Ising quantum chains

Log-periodic amplitudes of the surface magnetization are calculated analytically for two Ising quantum chains with aperiodic modulations of the couplings. The oscillating behaviour is linked to the discrete scale invariance of the perturbations. For the Fredholm sequence, the aperiodic modulation is marginal and the amplitudes are obtained as functions of the deviation from the critical point. For the other sequence, the perturbation is relevant and the critical surface magnetization is studied.

Transitions to chaos in a forced jet: intermittency, tangent bifurcations and hysteresis

Experimental studies of a forced transitional jet at moderate Reynolds numbers reveal interesting transitions between low-dimensional states, namely tangent bifurcations, intermittency and hysteresis. The experiments were carried out in an axisymmetric air jet with a laminar top-hat exit profile in the low-noise ambient of a large anechoic chamber, using forcing amplitude and frequency as control parameters. Tangent bifurcations are seen to occur in two different transitions from periodicity to chaos: (i) from stable pairing to nearly periodic modulations of pairing and (ii) from stable double pairing to a quarter-harmonic chaotic attractor. In case (i), an empirically derived mapping closely reproduces the temporal dynamics of one chaotic attractor at a point just after the tangent bifurcation. In case (ii), the intermittency was characterized by estimating the scaling exponent of the p.d.f. of the periodic durations, which was found to be close to the characteristic value for type-II intermittency. Hysteresis is seen at higher Strouhal numbers in the transitions between aperiodic modulations and the periodic/chaotic double pairing states. Based on simultaneous flow visualization and velocimetry, the hysteresis appears to be associated with intermittent tilting of (otherwise) axisymmetric vortices. These transitions are explained in terms of feedback-driven dynamics from vortex roll-ups and pairings, which can be phase-locked (periodic) or unlocked (nearly quasi-periodic and chaotic). The observed transitions connect large regions of deterministic behaviour in the phase diagram, confirming the existence of a low-dimensional dynamical system in transitional jets-an open flow of technological relevance.Two-point coherence measurements indicate that spatial coupling (and, hence, coherent motion from pairing dynamics) extends for five to eight diameters from the exit, well beyond the locations of pairing and double pairing. This justifies the use of single-point measurements and confirms our hypothesis that the dynamics in this convectively unstable flow are primarily temporal rather than spatio-temporal.

Anisotropic scaling in layered aperiodic Ising systems

The influence of a layered aperiodic modulation of the couplings on the critical behaviour of the two-dimensional Ising model is studied in the case of marginal perturbations. The aperiodicity is found to induce anisotropic scaling. The anisotropy exponent z, given by the sum of the surface magnetization scaling dimensions, depends continuously on the modulation amplitude. Thus these systems are scale invariant but not conformally invariant at the critical point.

Surface magnetization and critical behavior of aperiodic Ising quantum chains.

We consider semi-infinite two-dimensional layered Ising models in the extreme anisotropic limit with an aperiodic modulation of the couplings. Using substitution rules to generate the aperiodic sequences, we derive functional equations for the surface magnetization. These equations are solved by iteration and the critical exponent ${\mathrm{\ensuremath{\beta}}}_{\mathit{s}}$ can be determined exactly. The method is applied to three specific aperiodic sequences, which represent different types of perturbation, according to a relevance-irrelevance criterion. On the Thue-Morse lattice, for which the modulation is an irrelevant perturbation, the surface magnetization vanishes with a square-root singularity, like in the homogeneous lattice. For the period-doubling sequence, the perturbation is marginal and ${\mathrm{\ensuremath{\beta}}}_{\mathit{s}}$ is a continuous function of the modulation amplitude. Finally, the Rudin-Shapiro sequence, which corresponds to the relevant case, displays an anomalous surface critical behavior which is analyzed via scaling considerations. Depending on the value of the modulation, the surface magnetization either vanishes with an essential singularity or remains finite at the bulk critical point, i.e., the surface phase transition is of first order.

Critical behavior of the aperiodic quantum Ising chain in a transverse magnetic field

We consider the quantum spin-1/2 Ising chain in a uniform transverse magnetic field, with an aperiodic sequence of ferromagnetic exchange couplings. This system is a limiting anisotropic case of the classical two-dimensional Ising model with an arbitrary layered modulation. Its formal solution via a Jordan-Wigner transformation enables us to obtain a detailed description of the influence of the aperiodic modulation on the singularity of the ground-state energy at the critical point. The key concept is that of thefluctuation of the sums of any number of consecutive couplings at the critical point. When the fluctuation isbounded, the model belongs to the “Onsager universality class” of the uniform chain. The amplitude of the logarithmic divergence in the specific heat is proportional to the velocity of the fermionic excitations, for which we give explicit expressions in most cases of interest, including the periodic and quasiperiodic cases, the Thue-Morse chain, and the random dimer model. When the couplings exhibit anunbounded fluctuation, the critical singularity is shown to be generically similar to that of the disordered chain: the ground-state energy has finite derivatives of all orders at the critical point, and an exponentially small singular part, for which we give a quantitative estimate. In themarginal case of a logarithmically divergent fluctuation, e.g., for the period-doubling sequence or the circle sequence, there is a negative specific heat exponentα, which varies continuously with the strength of the aperiodic modulation.

Long-range proximity effect in aluminum thin films with regions of nearly identical transition temperatures.

We have investigated the resistive transition of dirty aluminum thin films whose transition temperature is spatially modulated by \ensuremath{\sim}2--4 % along the current path. At current densities as high as 600 A/${\mathrm{cm}}^{2}$, the system exhibits a single homogeneous transition for modulation lengths up to 50 \ensuremath{\mu}m, an order of magnitude longer than predicted by current theories. In the limit that the modulation length is 50 \ensuremath{\mu}m, the ${\mathit{T}}_{\mathit{c}}$ depends sensitively and monotonically on the ratio of the lengths of the etched and the unetched regions (${\mathit{d}}_{1}$/${\mathit{d}}_{2}$), but only weakly on their absolute magnitudes. This behavior is reproduced for ${\mathit{T}}_{\mathit{c}}$ modulation parallel or perpendicular to the current flow. For samples with a controlled aperiodic (Fibonacci) modulation, the ${\mathit{T}}_{\mathit{c}}$ is compatible with that of periodically modulated films. The dependence of the ${\mathit{T}}_{\mathit{c}}$ on ${\mathit{d}}_{1}$/${\mathit{d}}_{2}$ can be fitted to a simple Ginzburg-Landau theory which assumes that the effective pair coherence length is long in comparison with the modulation period. Despite the good fit, data for the full range of experimentally accessible modulation length scales are not consistent with a more detailed microscopic theory. We believe this long-range proximity effect is not the consequence of pair tunneling, but is a manifestation of the long distance over which quasiparticle phase coherence is maintained in a two-dimensional metal film.

Quantized electrostatic field effects in the disordered two-dimensional systems of superconducting small particles

The conductance modulation of highly resistive granular superconductor thin films by applying of electrostatic field has been measured for the purpose of studying electrostatic field effects in the two-dimensional arrays of small superconductor tunnel junctions. We observed reproducible quasiperiodic and aperiodic conductance modulation. The influence of the fixed charge trapped in the insulator-grain interface is also investigated.

Periodicity and aperiodicity in solar magnetic activity

Solar activity varies irregularly with an 11-year period whereas the magnetic cycle has a period of 22 years. Similar cycles of activity are seen in other slowly rotating late-type stars. The only plausible theory for their origin ascribes them to a hydromagnetic dynamo operating at, or just below, the base of the convective zone. Linear (kinematic) dynamo models yield strictly periodic solutions with dynamo waves propagating towards or away from the equator. Nonlinear (magnetohydrodynamic) dynamo models allow transitions from periodic to quasi-periodic to chaotic behaviour, as well as loss of spatial symmetry followed by the development of complex spatial structure. Results from simple models can be compared with the observed sunspot record over the past 380 years and with proxy records extending over 9000 years, which show aperiodic modulation of the 11-year cycle.

The Curie temperature of multilayer films with aperiodic compositional modulation

The influence of random fluctuations in modulation amplitude, thickness of layers, and width of interdiffusion zones on the Curie temperature of compositionally modulated magnetic films is studied in the molecular field approximation. The Curie temperature has been found to be most sensitive to the fluctuations in the modulation amplitude, which give rise to some kind of localized magnetism in the film. >

Method of, and circuit arrangement for, function control of ultrasonic alarm installations

In an ultrasonic alarm installation ultrasonic waves are continuously emitted into a monitored region and a frequency shift due to a moving object, e.g. an intruder, by virtue of the Doppler effect is evaluated for giving an alarm signal. The function control in this installation occurs during a test phase by means of brief aperiodic modulation of the emitted ultrasonic waves. The resulting brief frequency broadening generates in the same evaluating circuit a signal if the installation is functioning correctly. The time duration of the aperiodic modulation is selected to be so short that no standing waves can form, so that the function control can also work faultlessly if the installation, because of vectorial addition of the received ultrasonic waves, accidentally is in an insensitive state. Thus even coverings of the ultrasonic transmitter during a sabotage attempt can be recognized and distinguished from an insensitive state.

Asynchronous whirl in a rotating cylinder partially filled with liquid

Experimental and analytical results are presented for the self-excited oscillations that occur in a partially filled centrifuge when centrifugal forces interact with shallow-water waves. Periodic and aperiodic modulations of the basic whirl phenomena are both observed and calculated. The surface waves are found to be hydraulic jumps, undular bores or solitary waves.