Incomplete Devil's Staircase Research Articles

Incomplete Devil’s Staircase in the Magnetization Curve ofSrCu2(BO3)2

We report on NMR and torque measurements on the frustrated quasi-two-dimensional spin-dimer system SrCu(2)(BO(3))(2) in magnetic fields up to 34 T that reveal a sequence of magnetization plateaus at 1/8, 2/15, 1/6, and 1/4 of the saturation and two incommensurate phases below and above the 1/6 plateau. The magnetic structures determined by NMR involve a stripe order of triplets in all plateaus, suggesting that the incommensurate phases originate from proliferation of domain walls. We propose that the magnetization process of SrCu(2)(BO(3))(2) is best described as an incomplete devil's staircase.

Devil’s staircases, quantum dimer models, and stripe formation in strong coupling models of quantum frustration

We construct a two-dimensional microscopic model of interacting quantum dimers that displays an infinite number of periodic striped phases in its $T=0$ phase diagram. The phases form an incomplete devil's staircase and the period becomes arbitrarily large as the staircase is traversed. The Hamiltonian has purely short-range interactions, does not break any symmetries of the underlying square lattice, and is generic in that it does not involve the fine-tuning of a large number of parameters. Our model, a quantum mechanical analog of the Pokrovsky-Talapov model of fluctuating domain walls in two-dimensional classical statistical mechanics, provides a mechanism by which striped phases with large periods compared to the lattice spacing can, in principle, form in frustrated quantum magnetic systems with only short-ranged interactions and no explicitly broken symmetries.

Critical ultrasonic behavior near phase transitions in BCCD crystals

Abstract The ultrasonic velocity and attenuation of longitudinal waves have been measured along the a, b and c directions as a function of frequency (10 – 70 MHz) and temperature (110 – 300 K) in single - crystal betaine calcium chloride dihydrate. The ultrasonic anomalies were observed at incomplete devil's staircase transition temperatures. The largest anomaly appeared at the normal - incommensurate phase transition at Ti ≈ 164 K Landau theory has been used to explain the experimental data below Ti. From attenuation data, the relaxation time of the amplitudon in the incommensurate phase has been estimated to be τ = (1.3 · 10−13) f 1, where t = (Ti-T)/Ti . In addition, the critical ultrasonic velocity behavior in the hydrodynamic region (ωτ≪ 1) was analyzed in terms of a scaling model. The critical - exponent value was found to be μ ≈ 0.3.

High-Resolution X-Ray Diffraction Study of Commensurate-Incommensurate Phase Transition in [N(CH3)4]2MnCl4under Pressure

Using synchrotron radiation, we have performed a high-resolution X-ray diffraction study of a phase transition between the incommensurate phase and the commensurate phase characterized by the wave vector q =3 c * /7 in [N(CH 3 ) 4 ] 2 MnCl 4 under pressure. The pressure dependence of the wave vector q exhibits an “incomplete devil's staircase.” The high-resolution capability of the synchrotron radiation reveals that the incommensurate satellite reflection shows an asymmetric and broad line shape along the modulation direction, while the q =3 c * /7 commensurate superlattice reflection is symmetric and its width is equal to the instrumental resolution. Such a profile of the incommensurate satellite reflection is found to depend on the history of the crystal.

The incomplete devil's staircase of betaine calcium chloride dihydrate: acoustic investigation at high pressures

The influence of the hydrostatic pressure on the temperature dependences of the ultrasonic velocities and attenuation coefficients is studied in the vicinity of the phase transition of BCCD crystals. The results obtained are considered within the framework of single-and two-soft-mode models. It is shown that acoustic anomalies are well explained only by the single-soft-mode model, when only one soft mode Pk, which transforms according to the two-dimensional irreducible representation A3 of the Pnma space group, is taken into account.

Quasicrystallinity in twist-grain-boundary phases.

We present a simplified polymer model for twist-boundary phases in liquid crystals. The thermodynamic phases of this model are characterized by an angle 2\ensuremath{\pi}\ensuremath{\alpha}\ifmmode \tilde{}\else \~{}\fi{}. There are incommensurate phases with \ensuremath{\alpha}\ifmmode \tilde{}\else \~{}\fi{} an irrational number and commensurate phases with \ensuremath{\alpha}\ifmmode \tilde{}\else \~{}\fi{}=P/Q, with P and Q relatively prime integers. The latter have quasicrystalline symmetry for Q=5 or Q>6. Equilibrium phases with all values of \ensuremath{\alpha}\ifmmode \tilde{}\else \~{}\fi{} can be produced by varying a control parameter \ensuremath{\alpha}. The curve \ensuremath{\alpha}\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\alpha}) is an incomplete devil's staircase with finite locking intervals about every rational \ensuremath{\alpha}\ifmmode \tilde{}\else \~{}\fi{}.

Structural phase transitions in betaine calcium chloride dihydrate

Abstract Models describing the incomplete devil's staircase in betaine calcium chloride dihydrate are briefly discussed. Experimental facts in favour of one or two condensed soft modes are pointed out. Phasons and amplitudons in a structure modulated by two incommensurate waves with the same wavevector are analyzed. Dielectric relaxation in the incommensurate phase between two lock-in improper ferroelectric phases is discussed.

Dielectric and thermal behavior of betainecalciumchloride dihydrate in an electric bias field

Abstract A study of the dielectric dispersion in the incommensurate phases reveals a simple relaxational behavior with characteristic frequencies in the microwave region. Electric bias fields shift the transition temperatures and also induce further commensurate ferroelectric phases. Most phase transitions of the incomplete devil's staircase of this substance show up in an investigation of the thermal expansion measured along the three orthorhombic axes.

On the incomplete devil's staircase of betaine calcium chloride dihydrate

Careful investigations of the temperature dependence of dielectric and thermal properties of betaine calcium chloride dihydrate establish the existence of numerous commensurate and incommensurate phases in addition to known ones. The wavevector of the modulation follows an incomplete devil's staircase, the discrete values of which are attributed to different phases by symmetry considerations.

Application of the method of effective potentials to a model for twinning in elastic materials.

We extend the method of effective potentials to systems with next-nearest-neighbor interactions, and apply it to a one-dimensional discrete model for twins in elastic materials. The energy of the system is given by $H=\mathbf{\ensuremath{\Sigma}}n\ensuremath{\alpha}{({u}_{n+1}\ensuremath{-}{u}_{n})}^{4}\ensuremath{-}\ensuremath{\beta}{({u}_{n+1}\ensuremath{-}{u}_{n})}^{2}+(\frac{\ensuremath{\gamma}}{2}){({u}_{n+1}\ensuremath{-}{2u}_{n}+{u}_{n\ensuremath{-}1})}^{2}\ensuremath{-}cos{u}_{n}$, where the first two terms model the elastic strain-dependent energy which we take to be in the form of a double well in the strains and the third term gives its dependence on the discretized strain gradients. The periodic potential in the last term is introduced to allow for additional interactions with a background such as a parent phase, grain boundaries, or another array of twins. We obtain the phase diagram and show that it consists of various modulated commensurate as well as incommensurate ground-state configurations. We find continuous phonon-driven transitions between the homogeneous and any modulated phase, an incomplete devil's staircase in a narrow region close to the homogeneous phase and first-order soliton-driven transitions between commensurate phases. The first-order transition lines end at triple points where three commensurate phases coexist. In contrast to other nonconvex models we do not find here any superdegenerate points. We give general arguments which exclude the existence of such points in the present model. Preliminary results obtained by driving the system are discussed. These consist of various metastable configurations exhibiting strong hysteretic variation with the driving force.

Devil's staircase in double helices self-organization

The structures of the ground state of a discrete model of two interacting helical chains are investigated by the discrete mapping method. When non-interacting chains have different spatial periods, the ground state structure of the model depends on the value of the interaction constant γ. We show the existence of a critical value γ c at which the chains form a common helix. Regular soliton structure is realized for γ⪡ γ c as a ground state. For γ→ γ c the system ground state passes through a sequence of transitions of incomplete devil's staircase type. Metastable chaotic structures are shown to exist.

Anderson localization for one-dimensional difference Schr�dinger operator with quasiperiodic potential

The Schrodinger difference operator considered here has the form $$(H_\varepsilon (\alpha )\psi )(n) = - (\psi (n + 1) + \psi (n - 1)) + V(n\omega + \alpha )\psi (n)$$ whereV is aC2-periodic Morse function taking each value at not more than two points. It is shown that for sufficiently smallɛ the operatorHɛ(α) has for a.e.α a pure point spectrum. The corresponding eigenfunctions decay exponentially outside a finite set. The integrated density of states is an incomplete devil's staircase with infinitely many flat pieces.

Chiral Potts model on a Cayley tree with complete and incomplete devil's staircase

The three-state chiral Potts model on a Cayley tree is analysed in the limit of infinitely large coordination number. The fractal dimensionalities of the wavenumber against chiral field curves are computed. It is shown that they change from a complete to an incomplete devil's staircase as the temperature is raised. Commensurate phase boundaries are determined analytically for high and low temperatures, and the discommensurations are shown to result from tangent bifurcations.

The Incommensurate Phase of Betaine Calciumchloride-Dihydrate

Betaine Calciumchloride-Dihydrate (BCCD), (CH3)3NCH2COO·CaCl2·H2O, is the first member of the (TGS, TSCC)-family which shows incommensurate behaviour. The phase transitions were investigated by conventional X-rays Double axis diffractometer “AMADEUS”, Saarbrücken) and by synchrotron radiation (two-axis-diffractometer Hasylab/DESY, Hamburg). Above 164 K there is a high temperature phase and below 43 K a ferroelectric phase In between, the incommensurate phase can be described as an “incomplete devil's staircase”. The temperature dependence of the satellite intensity can be fitted by a critical exponent 2β=0.85. The half width of the satellite reflections exhibits anomalous behaviour at the transition temperatures. The modulation vector is influenced in a very special manner by an external electric field.

Modulated phase of an ising system with competing interactions on a Cayley tree

A system with competing nearest-neighbour and next-nearest-neighbour interactions is considered on a Cayley tree. The phase diagram contains a modulated phase, as found for similar models on periodic lattices, but the multicritical Lifshitz point is at zero temperature. The variation of the wavevector with temperature in the modulated phase is studied in detail, it shows narrow commensurate steps between incommensurate regions (“incomplete devil's staircase”). The behaviour of the coherence length near the different transitions is also analyzed.