Devil's Staircase Research Articles

A Generalized Hysteresis Model of Antiferroelectric Liquid Crystals

In this paper, we shall put forward a generalized irreversible rotation model which takes account of the long-range couplings between the adjacent dipoles in each layer to explain the experimentally observable hysteresis curves in ferro-, antiferro- and ferrielectric phases with the corresponding wave numbers of the ferroelectric orderings along the layer normal. It is found that the long-range interaction between the dipoles critically affects on the dynamic response under an alternating electric field. In the present model, it may be possible to explain the experimentally found subphases of the antiferroelectric materials in a unified framework. That is, the dynamic response of the possible phases between the ferroelectric (SmC*) phase (the upper temperature) and the antiferroelectric (SmCA*) phase (the lower temperature) will be substantially explained by a unified irreversible model under an alternating electric field. In addition, the hysteresis curve with the devil's staircase, as seen in the field-induced phase transitions, is found to be critically affected by the coupling strength between the spontaneous polarizations in each layer. We shall also show that the present model may be considered as a candidate to explain the thresholdless, hysteresis-free, the V-shaped switching as has been experimentally found.

Critical fields and devil's staircase in superconducting ladders

We have used an intuitively simple method to determine the interaction between vortices and ground state for both a ladder array of Josephson junctions and a ladder of thin superconducting wires. The results are easily extended to cover the more complicated case of a Kagomé ladder.

V-shaped switching due to frustoelectricity in antiferroelectric liquid crystals

We have reinforced the assertion that the devil's staircase and the thresholdless, hysteresis free, V-shaped switching are two manifestations of the frustration between ferro-and antif-erro-electricity, emphasizing that the third nearest neighbor interaction parameter may become negative, i.e. J 3 < 0. Three important consequences are: (1) The subphase with q=1/6 becomes to exist stably even in the ground state. (2) A number of subphases emerge and may apparently look like a single subphase. (3) It is not easy to distinguish SC* from frustrated subphases with small q numbers. We have summarized compounds and mixtures for the Y-shaped switching that are described in Japanese Patent Gazettes open to public recently. The materials show an antiferroelectric, probably q=1/6, subphase or an apparently single subphase in a wide temperature range; some of them are characterized with novel chiral parts.

Unusual magnetic and transport properties of layered compound Gd 0.925La 0.075Mn 2Ge 2

In the layered compound Gd 0.925La 0.075Mn 2Ge 2, a reentrant ferrimagnetism having a para-( T C)-ferri-( T t1)-antiferro-( T t2)-ferri transition with decreasing temperature was observed. Between T t1 and T t2, a field-induced matamagnetism from collinear antiferromagnetism to ferrimagnetism appears, accompanied with a multi-step transition, the so-called “devil's stair-case”. In addition, a giant magnetoresistance effect (Δ ρ/ ρ∼15%) was also observed at the metamagnetic transition. These unusual features will be characteristic in layered antiferromagnetism.

Devil's staircase-type faceting of a cubic lyotropic liquid crystal

The faceting of monocrystals of the lyotropic cubic liquid crystals in equilibrium with a humid atmosphere is observed. Experiments reveal the presence of more than 60 different types of facets on the surface of a spherical crystal of radius R = 1 mm. The devil's staircase type of faceting has been predicted theoretically when the interaction between steps on the crystal surface is repulsive.

Origin of degeneracies and self-similar structure in the spectrum of quantum dots

We show that the appearance of ‘accidental’ degeneracies (degeneracies which persist beyond the harmonic potential) in the single-particle spectrum of quantum dots in a magnetic field can be understood by analyzing the evolution of the structure of the classical orbits of the system as a function of magnetic field. We demonstrate that these degeneracies are associated with closed orbits, which occur at particular magic ratios of the cyclotron energy to the confinement energy in the harmonic case. The spectrum of degenerate energies forms an interesting self-similar structure with an associated complete ‘devil's staircase’. Since the degeneracies are not lifted by deviations from parabolicity of the radial confinement potential of the dot, the devil's staircase structure that we predict should be detectable experimentally using available techniques.

QUASIPERIODICITY AND CHAOS IN THE DC–DC BUCK–BOOST CONVERTER

This paper is concerned with the study of nonlinear phenomena in a closed loop voltage-controlled DC–DC Buck–Boost converter when suitable parameters are varied. The dynamics is analyzed using both the continuous-time model and the numerically computed stroboscopic map. The analysis of the one-dimensional bifurcation diagram shows that Neimarck–Sacker bifurcation occurs at certain values of the parameters. Phase-locking periodic windows, the period-adding sequence, and transition from quasiperiodicity to period-doubling via torus breakdown are also obtained. The two-dimensional bifurcation diagram is carefully computed. This shows that phase-locking orbits produce so-called Arnold tongues in the parameter space. It is shown that the winding number plotted as a function of the bifurcation parameter is a devil's staircase. As typically occurs in general circle maps, the fine structures of the Arnold tongues and the devil's staircase show self-similarity.

Periodic Ground State Configurations in a One-Dimensional Hubbard Model of Statistical Mechanics

This paper considers an averaging procedure for the description of a particles arrangement in a Hubbard model with antiferromagnetic interactions. The arrangements are described by the devil's staircase. Completeness of the staircase is proved.

STABLE OSCILLATIONS AND DEVIL'S STAIRCASE IN THE VAN DER POL OSCILLATOR

A forced van der Pol relaxation oscillator is studied experimentally in the regime of stable oscillations. The variable parameter is chosen to be the driving frequency. For a range of parameter values, we show that the rotation number varies continuously from 0 to 1. This work provides experimental evidence that period-adding bifurcations to chaos previously reported by Kennedy and Chua are intimately connected to the existence of a regime of stable oscillations where the rotation number shows a Devil's-staircase structure.

Ground states of lattice gases with "almost" convex repulsive interactions.

To the best of our knowledge there is only one example of a lattice system with long-range two-body interactions whose ground states have been determined exactly: the one-dimensional lattice gas with purely repulsive and strictly convex interactions. Its ground-state particle configurations do not depend on any other details of the interactions and are known as the generalized Wigner lattices or the most homogeneous particle configurations. The question of the stability of this beautiful and universal result against certain perturbations of the repulsive and convex interactions is interesting in itself. Additional motivations for studying such perturbations come from surface physics (adsorption on crystal surfaces) and theories of correlated fermion systems (recent results on ground-state particle configurations of the one-dimensional spinless Falicov–Kimball model). As a first step, we studied a one-dimensional lattice gas whose two-body interactions are repulsive and strictly convex only from distance 2 on, while its value at distance 1 can be positive or negative, but close to zero. We showed that such a modification makes the ground-state particle configurations sensitive to the tail of the interactions; if the sum of the strengths of the interactions from the distance 3 on is small with respect to the strength of the interaction at distance 2, then particles form two-particle lattice-connected aggregates that are distributed in the most homogeneous way. Consequently, despite breaking of the convexity property, the ground state exhibits the feature known as the complete devil's staircase.

Magic Mountain and Devil's Staircase swapping problems

Two open questions concerning additivity problems in theoretical chemistry have been posed in this article. Each question involves swapping of the square root function (in the formula for the zero-point energy of a linear oscillator) with a highly irregular function. Global contextualization of molecular problems and function swapping are indispensable strategies in tackling additivity problems in theoretical chemistry.

Clustering and Devil's Staircase in a One-Dimensional Adsorbate System

Atoms adsorbed heteroepitaxially on a substrate interact with a short-range chemical cohesion - J and a long-range elastic repulsion u r -3 . In one dimension, adsorbed atoms can form clusters and arrange in periodic structures, whose periodicity shows a devil's staircase. On increasing J / u , the range of the chemical potential Δ H where periodic structures are achieved decreases, and vanishes at an upsilon point. For J larger than the upsilon point value, the phase transition turns out to be of a first-order.

Exact Renormalization Analysis of a Self-Similar Phase Diagram with Upsilon Points for an Ising Model

The ground-state phase diagram is analyzed for a one-dimensional Ising model with long range interaction in the presence of uniform and staggered magnetic fields; the interaction is antiferromagnetic and its strength decays exponentially with distance. The exact renormalization formulas for the interactions between interfaces are derived, which is used to calculate the phase diagram hierarchically. The phase diagram is self-similar, and it contains upsilon points and devil's staircases; this agrees with the result of Aubry, Axel and Vallet [J. Phys. C 18 (1985) 753] for an equivalent model obtained by a different method. A simple “phase rule” presenting what kind of phases appear and how they are located in the phase diagram is derived. The application of this rule to the phase diagram of betaine calcium chloride dihydrate is discussed.

Anisotropy and metamagnetism in the RNi 2Ge 2 (R=Y, La–Nd, Sm–Lu) series

Single crystals of a complete series of the RNi 2Ge 2 (R=Y, La–Nd, Sm–Lu) ternary intermetallics were grown and characterized by measurements of the anisotropic magnetization and in-plane resistivity in zero and applied magnetic field. The majority of moment-bearing members of the series order antiferromagnetically and have a series of transitions in zero applied magnetic field and there is a rough de Gennes scaling of Θ D and T N for the heavy rare-earth members of the series. Signatures of the metamagnetic transitions in the magnetically ordered state and magnetic transitions in zero field are observed for a number of the members of the series. TbNi 2Ge 2 was studied in more details. Angular-dependent magnetization measurements support the hypothesis that the spins in the ordered states are confined to the c-axis. Complex H–T (H‖c) phase diagram for this material was derived from the magnetization measurements at different temperatures ( M( H)) and applied fields ( M( T)). It is suggested that TbNi 2Ge 2 is a promising candidate for the `devil's staircase’ material. In addition, YbNi 2Ge 2 is suggested to have strong 4f-conduction electron hybridization and is characterized as a 4f-hole analog of the previously studied CeNi 2Ge 2 moderate heavy fermion compound.

Frustration between Ferroelectricity and Antiferroelectricity in Extremely Soft Chiral Smectic-C Like Phases of Liquid Crystals

In a group of compounds that exhibit antiferroelectric and/or ferrielectric liquid crystal phases, the bent molecular shape has been considered to cause unique interactions among smectic layers. The interactions are responsible for the successive phase transitions as described by the devil's staircase and the V-shaped switching, which are related with the interlayer ordering and the diminished tilting correlation, respectively. These two are apparently contradictory each other, but result from the same cause, i.e. frustration between ferroelectricity and antiferroelectricity in the Sm-C* like phase. The ANNNI model Yamashita and Miyazima introduced has been shown to describe the frustration appropriately provided that we undertake some refinements: the Ising spin corresponding to the tilting sense of the local in-plane director, the X-Y character of the spin, the tilt angle decrease toward Sm-A, and most importantly, the reassignment of the pseudo-spin to the biasing direction of the molecular rotation about their long axis.

Range of chaotic state in the Josephson junction driven by a pulse train signal

The parameter dependence of the onset of chaos which appears as the devil's staircase on the current-voltage curve in the Josephson junction driven by a pulse train signal is studied based on an analog circuit simulation. In this study, hysteresis parameter /spl beta/ and reduced frequency /spl Omega/ are regarded as significant. It is suggested that from the viewpoint of these parameter dependencies there is no remarkable difference in the range of chaotic states in the junction if compared to a monochromatic-signal-driven junction, meaning that operation above the junction plasma frequency is desirable to avoid chaos. The rf-amplitude dependence of the onset of chaos is also roughly surveyed.

Phase Boundary and Soliton Excitation in the XY Model with Competing Interactions in Twofold Field

The ground state of the XY model with competing interaction, which is reduced to the axial next nearest neighbour Ising model at a limit of infinite value of strength of twofold field, is studied from the viewpoint of solitons. Excitation energies of compressed and stretched solitons at phases with periods 6, 5 and 7 are obtained by numerical analysis of periodic states, from which phase boundaries are determined. Each of these phases is shown to have a finite stable range, which indicates that the ground state of the model has a type of devil's staircase structure.

FRACTAL GEOMETRY: A TOOL FOR FUZZY REASONING

A new procedure for construction of a fuzzy number is given, based on the covering of points by consequently contracting intervals. The approach leads to the membership function in the form of Devil's staircase, a borderline fractal finding many applications in the modern physics. The alternative definition of subsethood based on the self-similarity of the sets is also given.

Phase Locking, Devil's Staircases, Farey Trees, and Arnold Tongues in Driven Vortex Lattices with Periodic Pinning

Using numerical simulations, we observe phase locking, Arnold tongues, and Devil's staircases for vortex lattices driven at varying angles with respect to an underlying superconducting periodic pinning array. This rich structure should be observalble in transport measurments. The transverse $V(I)$ curves have a Devil's staircase structure, with plateaus occurring near the driving angles along symmetry directions of the pinning array. Each of the plateaus corresponds to a different dyanmical phase with a distinctive vortex structure and flow pattern.

A novel property caused by frustration between ferroelectricity and antiferroelectricity and its application to liquid crystal displays-frustoelectricity and V-shaped switching

We have studied the frustration between ferro- and antiferro-electricity in chiral smectic C like liquid crystalline phases, which is not only fundamentally interesting but also very attractive from an application point of view. It causes temperature induced successive phase transitions as characterized by a devil's staircase and the thresholdless, hysteresis-free, V-shaped switching induced by an applied electric field. The devil's staircase indicates some type of interlayer ordering, while the V-shaped switching suggests considerably diminished tilting correlation. These two are apparently contradictory to each other, but result from the same cause, i.e. the frustration. We have first summarized experimental facts regarding subphases and successive phase transitions observed in many compounds and mixtures, which we believe are related to one another and result from the frustration. We have introduced several different theoretical explanations for these observed facts, and shown that only the axial next nearest neighbor Ising (ANNNI) model can explain almost all of the facts, provided that it is unified with the XY model appropriately. The unified model can make a comprehensive explanation in the most natural way based on the most probable molecular interactions. We have then emphasised that there are several modes regarding the V-shaped switching, because the system becomes so soft with respect to the tilting direction and sense that any additional external or internal force modifies the in-plane local director alignments. For the practically usable ones, we have emphasised the need for some type of randomization in the molecular alignment at the tip of the V and/or the switching process. In particular, the two dimensional (ideally, cylindrically symmetric) azimuthal angle distribution of local in-plane directors around the smectic layer normal is most attractive. Such a randomized state at the tip of the V is thermodynamically unique under a given condition imposed by interfaces. It stays stable even when the smectic layer structure, such as a chevron, changes with temperature. Finally, we have summarized the so-far reported compounds and mixtures for the V-shaped switching and introduced some prototypes of LCDs using them.