Incommensurate Case Research Articles

Energy relaxation in disordered charge and spin density waves

We investigate collective effects in the strong pinning model of disordered charge and spin density waves (CDWs and SDWs) in connection with heat relaxation experiments. We discuss the classical and quantum limits that contribute to two distinct contribution to the specific heat (a $C_v \sim T^{-2}$ contribution and a $C_v \sim T^{\alpha}$ contribution respectively), with two different types of disorder (strong pinning versus substitutional impurities). From the calculation of the two level system energy splitting distribution in the classical limit we find no slow relaxation in the commensurate case and a broad spectrum of relaxation times in the incommensurate case. In the commensurate case quantum effects restore a non vanishing energy relaxation, and generate stronger disorder effects in incommensurate systems. For substitutional disorder we obtain Friedel oscillations of bound states close to the Fermi energy. With negligible interchain couplings this explains the power-law specific heat $C_v \sim T^{\alpha}$ observed in experiments on CDWs and SDWs combined to the power-law susceptibility $\chi(T)\sim T^{-1+\alpha}$ observed in the CDW o-TaS$_3$.

Magnetism in the dilute Kondo lattice model

The one-dimensional dilute Kondo lattice model is investigated by means of bosonization for different dilution patterns of the array of impurity spins. The physical picture is very different if a commensurate or incommensurate doping of the impurity spins is considered. For the commensurate case, the obtained phase diagram is vertified using a non-Abelian density-matrix renormalization-group algorithm. The paramagnetic phase widens at the expense of the ferromagnetic phase as the f spins are diluted. For the incommensurate case, short-range antiferromagnetic correlations are found to dominate at low doping, which distinguishes the dilute Kondo lattice model from the standard Kondo lattice model.

Deconfinement in a 2D optical lattice of coupled 1D boson systems.

We show that a two-dimensional (2D) array of 1D interacting boson tubes has a deconfinement transition between a 1D Mott insulator and a 3D superfluid for commensurate fillings and a dimensional crossover for the incommensurate case. We determine the phase diagram and excitations of this system and discuss the consequences for Bose condensates loaded in 2D optical lattices.

Commensurate and incommensurate vortex lattice melting in periodic pinning arrays

We examine the melting of commensurate and incommensurate vortex lattices interacting with square pinning arrays through the use of numerical simulations. For weak pinning strength in the commensurate case we observe an order-order transition from a commensurate square vortex lattice to a triangular floating solid phase as a function of temperature. This floating solid phase melts into a liquid at still higher temperature. For strong pinning there is only a single transition from the square pinned lattice to the liquid state. For strong pinning in the incommensurate case, we observe a multi-stage melting in which the interstitial vortices become mobile first, followed by the melting of the entire lattice, consistent with recent imaging experiments. The initial motion of vortices in the incommensurate phase occurs by an exchange process of interstitial vortices with vortices located at the pinning sites. We have also examined the vortex melting behavior for higher matching fields and find that a coexistence of a commensurate pinned vortex lattice with an interstitial vortex liquid occurs while at higher temperatures the entire vortex lattice melts. For triangular arrays at incommensurate fields higher than the first matching field we observe that the initial vortex motion can occur through a novel correlated ring excitation where a number of vortices can rotate around a pinned vortex. We also discuss the relevance of our results to recent experiments of colloidal particles interacting with periodic trap arrays.

Upper critical field for anisotropic superconductivity: A tight-binding approach

We study the problem of the upper critical field ($H_{c2}$) for tight-binding electrons in a two-dimensional lattice. The external magnetic field is introduced into the model Hamiltonian both via the Peierls substitution and the Zeeman term. Carrying out calculations for finite systems we analyze the influence of the external field in the commensurable and incommensurable case on an equal footing. The upper critical field has been discussed for intrasite as well as anisotropic intersite pairing that, in the absence of magnetic field, has a $d_{x^2-y^2}$ symmetry. A comparison of $H_{c2}$ determined for different symmetries shows that the on-site pairing is more affected by the external field i.e., the critical temperature for the on-site pairing decreases with the increase of the magnetic field faster than in the anisotropic case. Moreover, we have shown that the tight-binding form of the Bloch energy can lead to the upward curvature of $H_{c2}$, provided that the Fermi level is close enough to the van Hove singularity.

Numerical Study of Dynamical Frictional Phenomena Based on the Two-Dimensional Frenkel-Kontorova Model with Impurities

We study numerically dynamical frictional phenomena using the Frenkel-Kontorova model with impurities. Even in the presence of randomness sliding states have a quasi long range order, which changes with the sliding velocity. Such an order affects the velocity dependence of the kinetic frictional force and produces peculiar transverse pinning effects. We discuss them in the relation with the moving Bragg glass theory developed for vortex lattices in type II superconductors. Frictional phenomena have common importance in various physical systems such as vortex lattices in type-II superconductors, charge and spin density waves, interfacial friction of solids and so on. 1),2) We investigate here dynamical frictional phenomena in two-dimensional elastic lattice systems based on the Frenkel-Kontorova (FK) model, 3) and focus especially on the kinetic friction and the transverse pinning effect. The FK model employed here consist of a discrete square lattice with harmonic interatomic force and a substrate lattice with impurities.The periodic potential due to the regular part of the substrate lattice has a two-dimensional sinusoidal form, of which principal axes coincide with those of the lattice on it.Impurities are distributed randomly in the substrate and the potential of each impurity has a Gaussian form.The amplitude of the periodic (impurity) potential is denoted by K(W ).We study overdamped dynamics of the FK model driven by an external force along one of the principal axes.Incommensurate systems are investigated here. The periodic potential pins the lattice even in an incommensurate case when the amplitude K is greater than a critical value Kc ≈ 0.23.The number of sites of a lattice is set to be N =8 9 2 (or 144 2 ) and the impurity density is fixed at about 0.21. A periodic boundary condition is imposed. The kinetic frictional force Fkin is plotted against the sliding velocity v in Fig.1. In the low velocity regime, v � 10, the kinetic frictional force is almost constant and not affected by the periodic potential.In the high velocity regime, however, it decreases with increasing velocity and depends on the periodic potential.The effectiveness of the periodic potential depends on the sliding velocity.This behavior will relate to the change of the order of the sliding lattice as mentioned later. We investigate next the response transverse to the sliding direction.According to the theory of the moving Bragg glass (MBG) developed by Giamarchi and Le Doussal, 4) a quasi long range order of the lattice exists in the transverse direction.

Resonance and non-recurrent rotationally ordered configuration in Frenkel-Kontorova models

We demonstrate, with a class of exactly solvable Frenkel-Kontorova models, the emergence of the non-recurrent rotationally ordered configurations whenever certain resonance conditions are satisfied among the openings in the hull function in both the commensurate and the incommensurate cases. If one insists on depicting them with hull functions, suitably defined extended numbers must be employed. The defect-mediated transition in the incommensurate case is also discussed.

Geometrical resonance analysis of chaos suppression in the bichromatically driven van der Pol oscillator

Elimination of chaotic behavior in the harmonically driven van der Pol oscillator by means of a comparatively weak additional forcing was studied through geometrical resonance analysis. We considered commensurate and incommensurate cases together with the effect of the phase difference between the forcings. The analysis provided parameter-space regions for regularization that were corroborated by numerical experiments, including instances with clearly large chaos-inducing forcing. A reinterpretation of a classical result, due to Cartwright and Littlewood [J. London Math. Soc. 20, 180 (1945)], was also derived by means of geometrical resonance analysis.

On the Genericity of Stabilizability for Time-Delay Systems

Conditions for the stabilizability of time-delay systems with incommensurable point delays by dynamic state feedback are known in the literature. In this paper it is shown that these conditions are satisfied generically. Although an algebraic approach is used to describe the class of all time-delay systems with point delays, the concept of genericity is formulated in a topological framework. In the metric space consisting of all parametrizations of time-delay systems, the subset of all stabilizable systems is an open and dense subset. The proof is given for the commensurable delay case first. It is shown that the incommensurable delay case is not significantly more difficult and that the same arguments prove also that systems with incommensurable time-delays are generically stabilizable.

Ground state of the Frenkel-Kontorova model with a transverse degree of freedom.

We study the ground state of a generalized Frenkel-Kontorova model with a transverse degree of freedom. The model describes a lattice of atoms with a fixed concentration, interacting by long-range repulsive forces, which is submitted to a two-dimensional substrate potential periodic (sinusoidal) in one direction and symmetric (parabolic) or asymmetric (Toda-like) in the transverse direction. When the magnitude of the interatomic repulsion increases, the ground state of the model undergoes a series of bifurcations. In particular, the first bifurcation leads to a zigzag ground state and results in drastic change of system properties, including a cusp in the average elastic constant. For incommensurate cases, the bifurcation can interplay with the Aubry transition from a pinned to a sliding state. A reentrant pinned state has, for instance, been found. The nature (continuous or discontinuous) of the next bifurcations depends on the symmetry of the substrate potential in the transverse direction. Finally, we discuss briefly the applicability of the model to describe conductivity of superionic conductors, surface diffusion, and crystal growth.

Influence of frustration on the persistent current of an interacting one-dimensional ring.

The persistent current for an arbitrarily filled strongly interacting one-dimensional ring is studied using variational and numerical exact diagonalization methods. For an ordered ring a difference between the commensurate and the incommensurate cases is observed. While for the commensurate case the current in the limit of strong interactions always tends to zero, in the incommensurate case there are situations (termed the frustrated cases) in which the current is finite. For a disordered ring the current always tends to zero in the limit of strong interactions. The implications on persistent current experiments are discussed

QUANTUM TUNNELING EFFECTS IN THE SSH MODEL OF ELECTRON-PHONON INTERACTIONS

We examine the Su-Schrieffer-Heeger (SSH) model of electron-phonon (e-p) interactions as an example of a theory in which commensurate and incommensurate bond-order wave (BOW) ground states are generated (in the quasi-one-dimensional (quasi 1-D) case) by an intersite e-p coupling. We focus primarily on the two-fold commensurate situation, in view of its applicability to the important quasi 1-D conducting polymer trans-polyacetylene (trans — (CH) x). Using a coherent state formalism to model the quantum fluctuation corrections to the adiabatic ground state, we study the quantum tunneling of the SSH “solitons” (viewed as discommensurations created in the two-fold commensurate ground state) and show that, as a result of this tunneling, the energy of the discommensurations can become negative for e-p couplings below a certain critical coupling. We argue that this negative defect energy is an indication of the instability of the underlying adiabatic commensurate BOW state. The critical e-p coupling for this transition depends on the phonon frequency and, for general commensurability, on the commensurability ratio. Thus, in real trans — (CH) x chains where the e-p coupling and the bare phonon frequency are fixed, the transition we describe can occur as a function of doping (which can change the effective commensurability) and may explain the BOW-to-quasimetal transition in doped (CH) x. For quasi-1-D systems, we also estimate the critical e-p couplings for the three-fold commensurate and incommensurate cases. We conclude by mentioning possible extensions of our approach and results to the new family of “fullerenes”, C n(n≥60).

Commensurability effects on the spectra of integrable systems

The eigenvalue spectra of two simple integrable systems-a three-dimensional anisotropic harmonic oscillator (AHO) and a particle in a cuboid-are studied as the aspect ratios are changed from rational to irrational values. For rational aspect ratios (commensurable case) there are large number-theoretic degeneracies which grow with energy. These degeneracies disappear when the aspect ratios are irrational (incommensurable case). However, for a given set of rational aspect ratios, commensurable behavior ensues only at large energies. The number of distinct eigenvalues per unit energy interval, which is indicative of the degree of degeneracies, is seen to approach a constant asymptotically for rational aspect ratios. The asymptotic constant is determined for all sets of aspect ratios for the AHO and is estimated semi-analytically for several sets of aspect ratios for the cuboid. The crossover from incommensurable to commensurable behaviour is studied by following a sequence of rational aspect ratios which approaches an irrational limit. A scaling form the crossover is suggested and explored numerically. A clear indication of scaling is seen for the AHO, while the evidence is suggestive but not definitive for the cuboid. The distribution of energy level spacings in the commensurable regime is determined for the AHO and a sequence of cuboids.

Explanation of the phase diagram of beta -di

The organic superconductor \ensuremath{\beta}-di[bis(ethylenedithio)tetrathiafulvalene]triiodide, \ensuremath{\beta}-(BEDT-TTF${)}_{2}$${\mathrm{I}}_{3}$, shows an unusual phase diagram in temperature (T) and pressure (P) with large metastability domains for the stabilization of an incommensurate modulation. This behavior is related to the T-P cycling around a tricritical point. The superconducting properties are drastically affected by the nature of the structural ground state. In this paper, we present a simple microscopic mean-field model that satisfactorily explains the experimental phase diagram of \ensuremath{\beta}-(BEDT-TTF${)}_{2}$${\mathrm{I}}_{3}$. This model takes explicitly into account the two nonsymmetrical configurations of one of the two ethylene groups of the BEDT-TTF molecule, under the form of pseudospins linearly coupled to the vibrational degrees of freedom involved in the structural modulation. The calculations are performed as a function of the reduced interaction v=V/${k}_{B}$${T}_{c}$, where V is half the energy difference between the two ethylene group configurations and ${k}_{B}$${T}_{c}$ is the coupling energy between the vibrational degrees of freedom. The antiferrodistortive (commensurate) case is first treated; then the incommensurate case is solved. The phase diagram obtained as a function of v exhibits a tricritical point and large metastability domains. The experimental phase diagram is well accounted for by assuming a linear temperature and pressure dependence of V in the vicinity of the tricritical point. The origin of the incommensurate modulation of \ensuremath{\beta}-(BEDT-TTF${)}_{2}$${\mathrm{I}}_{3}$ is briefly discussed.

Coulomb correlations in one-dimensional conductors with incommensurate band fillings and the semiconductor-metal transition in polyacetylene

The one-dimensional Hubbard and Hubbard-Peierls models are studied for various regimes of band filling. A variational ansatz for the ground state is worked out for small values (on the scale of the bandwidth) of the on-site Coulomb interaction. It is shown that this term enhances the lattice dimerization in the half-filled band case, but suppresses the lattice distortion in the incommensurate case. The semiconductor-metal transition in doped polyacetylene is tentatively attributed to this effect. Parameter values extracted from properties of the undoped material yield a critical dopant concentration of about 3%.

Leed theory for incommensurate overlayers: Application to graphite on Pt(111)

A dynamical LEED analysis is applied to study the structure of incommensurate overlayers. The previously described BSN method is extended to the incommensurate case, providing an efficient theoretical scheme for calculating the required LEED beam intensities. This method is used to investigate the structure of an incommensurate graphite overlayer on the Pt(111) surface by analyzing the specular diffraction beam intensities at various angles. The measured LEED intensities are well represented by a surface model consisting of a graphite layer 3.70 ± 0.05 Å above the Pt(111) surface, supported by at least a partial layer of “intercalated” carbidic carbon atoms chemisorbed in three-fold hollow sites 1.25 ± 0.10 Å above the Pt(111) surface. A “data subdivision method” is applied in the R-factor analysis to distinguish between the different R-factor minima.

LEED theory for incommensurate overlayers: Application to graphite on Pt(111)

Abstract A dynamical LEED analysis is applied to study the structure of incommensurate overlayers. The previously described BSN method is extended to the incommensurate case, providing an efficient theoretical scheme for calculating the required LEED beam intensities. This method is used to investigate the structure of an incommensurate graphite overlayer on the Pt(111) surface by analyzing the specular diffraction beam intensities at various angles. The measured LEED intensities are well represented by a surface model consisting of a graphite layer 3.70 ± 0.05 A above the Pt(111) surface, supported by at least a partial layer of “intercalated” carbidic carbon atoms chemisorbed in three-fold hollow sites 1.25 ± 0.10 A above the Pt(111) surface. A “data subdivision method” is applied in the R -factor analysis to distinguish between the different R -factor minima.

Incommensurability in crystals

Abstract The aim of this review is to present aperiodic crystals from a unifying point of view, showing why it is justified to call them crystals, despite the lack of a three-dimensional lattice periodicity, and to discuss in what sense they differ from periodic crystals in structure, symmetry and other physical properties. The extension of the concept of crystal has been based, during the last two decades, on investigation of incommensurate crystal phases. Among these, the most important ones are the modulated crystals. Their crystallographic nature, already apparent in the diffraction pattern, could be made explicit on the symmetry level by embedding in a higher-dimensional Euclidean space. The recent discovery of quasicrystal phases (representing a fairly different class of aperiodic crystals than the modulated ones) can also be approached in a similar way. Furthermore, it now appears that another class of incommensurate crystals, the so-called composite structures, represents a kind of intermediate case between the other two. In § 1 basic concepts are presented, together with a number of compounds given as illustration for typical incommensurate crystal phases. In § 2 we deal with the general formalism allowing a crystallographic symmetry characterization. It is intended as a first approach to crystal-structure determination, which justifies the emphasis on the diffraction pattern and on the modulated-crystal case. The appropriate generalization to the quasicrystal case is considered in § 3. The crystallographic nature of the incommensurate phases is apparent in their growth forms. It has been known for centuries that crystal morphology is essentially based on lattice periodicity. It is therefore fascinating to discover how nature solves the problem in the incommensurate crystal case; we discuss this in § 4. The origin of incommensurability is the subject of § 5 (on a phenomenological level) and of § 6 (on a microscopic level). In § 7 the basic concepts needed in crystal-structure determination are discussed in more detail than was appropriate in § 2. In § 8 we discuss those physical properties which are more closely related to incommensurability, fitting nevertheless into the general frame of crystal physics. In § 9 we deal with defects, again in particular with the additional ones due to the more complex structure of aperiodic crystals, making clear at the same time why defects play an even more important role than in periodic crystals.

Magnetic excitations in spin-density-wave systems.

We propose a versatile phenomenological model in which there is no magnetic anisotropy and yet which is suitable for a description of linearly polarized incommensurate magnetic structure (e.g., as in chromium) as well as ferromagnets, antiferromagnets, helical structures, and what we call even-commensurate and odd-commensurate (a sort of weak ferromagnet) structures. The frequencies and eigenvectors of the long-wavelength spin-wave excitations for all of these structures are obtained and compared; of particular interest is the linearly polarized incommensurate case for which well-defined long-wavelength spin-wave excitations having a linear dispersion relation are shown to exist.

Dislocations in layered systems

The properties of defects in two-dimensional systems are of importance to modern technology but are also of intrinsic physical interest. A case in point is an inert gas monolayer physisorbed onto a clean graphite surface. The gas-graphite interaction potential can stabilize a two-dimensional condensate, usually but not always incommensurate with the substrate; dislocations in such monolayers are believed to be important in phase transitions, and in particular the melting transition. Atomistic computer modelling methods have been used to study dislocations in the XeC and KrC systems, for both commensurate and incommensurate cases. Defects in both structures are found to have much in common with those in the f.c.c. bulk lattice, such as the observed stacking faults, the mode of dislocation dissociation, the dipole properties and the existence of stacking fault polygons. It is shown that, in spite of the similarities between the binding forces in the two systems, the thermodynamics of dislocation-domain wall interactions predict a very different behaviour, which can be linked to the state of expansion or compression of the gas monolayer and to the mechanics of the melting process.