Quasiperiodic Chains Research Articles

Use of the trace map for evaluating localization properties

The use of Lyapunov exponents for evaluating localization lengths of wave functions in one-dimensional lattices is discussed. As a result, it is shown that it is more practical to calculate this length by using the scaling properties of the trace map of the transfer matrix. This leads to a relationship between localization and the fixed points of the map, which is considered as a dynamical system. The localization length is then defined by a Lyapunov exponent, used in the sense of chaos theory. All these results are discussed for periodic, disordered, and quasiperiodic chains. In particular, the Fibonacci quasiperiodic chain is studied in detail.

FROM MULTI-SITE TO ON-SITE TRANSFER MATRIX MODELS FOR SELF-SIMILAR CHAINS

We show how transfer matrix models on chains that are self-similar (renormalizable) with respect to a substitution rule can be transformed from multi-site models in which transfer matrices depend on the nature of a finite number of neighboring sites, to on-site models in which transfer matrices depend on the nature of one site only. We present sufficient conditions and show that these conditions are satisfied in the case of quasiperiodic chains of two symbols that are renormalizable with respect to an invertible substitution rule. We illustrate the application of our results to tight-binding Schrodinger equations modeling the electronic behavior of self-similar chains of atoms and to models describing the transmission of light through self-similarly stacked multilayers.

Two interacting electrons in a quasiperiodic chain

We study numerically the effect of on-site Hubbard interaction U between two electrons in the quasiperiodic Harper's equation. In the periodic chain limit by mapping the problem to that of one electron in two dimensions with a diagonal line of impurities of strength U we demonstrate a band of resonance two particle pairing states starting from E=U. In the ballistic (metallic) regime we show explicitly interaction-assisted extended pairing states and multifractal pairing states in the diffusive (critical) regime. We also obtain localized pairing states in the gaps and the created subband due to U, whose number increases when going to the localized regime, which are responsible for reducing the velocity and the diffusion coefficient in the qualitatively similar to the non-interacting case ballistic and diffusive dynamics. In the localized regime we find propagation enhancement for small U and stronger localization for larger U, as in disordered systems.

Universal spectral properties of spatially periodic quantum systems with chaotic classical dynamics

We consider a quasi one-dimensional chain of N chaotic scattering elements with periodic boundary conditions. The classical dynamics of this system is dominated by diffusion. The quantum theory, on the other hand, depends crucially on whether the chain is disordered or invariant under lattice translations. In the disordered case, the spectrum is dominated by Anderson localization whereas in the periodic case, the spectrum is arranged in bands. We investigate the special features in the spectral statistics for a periodic chain. For finite N, we define spectral form factors involving correlations both for identical and non-identical Bloch numbers. The short-time regime is treated within the semiclassical approximation, where the spectral form factor can be expressed in terms of a coarse-grained classical propagator which obeys a diffusion equation with periodic boundary conditions. In the long-time regime, the form factor decays algebraically towards an asymptotic constant. In the limit N → ∞, we derive a universal scaling function for the form factor. The theory is supported by numerical results for quasi one-dimensional periodic chains of coupled Sinai billiards.

Ordered structures in Fe3O4 kerosene-based ferrofluids

The patterns of kerosene-based ferrofluid films in applied parallel and perpendicular magnetic fields were studied. The time dependence of quasiperiodic chains in a parallel magnetic field, the perfect hexagonal crystal structure in a weaker perpendicular field, and the labyrinthine pattern in a stronger perpendicular field were observed. The Fe3O4 kerosene-based ferrofluids that we used in this study were synthesized by the coprecipitation method. As the parallel external magnetic field was applied to a ferrofluid film, the ordered quasiperiodic chains were obtained. On the other hand, the initial nonequilibrium disordered quantum columns were formed in an applied perpendicular magnetic field, and an equilibrium hexagonal structure with columns occupying its vortices was formed after two hours. The distance between these columns decreases as the strength of the applied magnetic field increases, and hence, the patterns change gradually from the hexagonal structure to a labyrinthine pattern with the strength of the perpendicular field above a critical value. Thus, a phase transition exists in the ferrofluid film system as the field strength increases.

Analytical results for multifractal properties of spectra of quasiperiodic Hamiltonians near the periodic chain

The multifractal properties of the electronic spectrum of a general quasiperiodic chain are studied in first order in the quasiperiodic potential strength. Analytical expressions for the generalized dimensions are found and are in good agreement with numerical simulations. These first order results do not depend on the irrational incommensurability.

Transmission fingerprints of quasi-periodic chains

We propose a method to characterize quasi-periodic chains, based on the transmission probabilities. We consider finite layers which are building up following the Fibonaci, Thue-Morse and Cantor sequences. The spectra show an undoubtedly fractal behavior, with a distinct appearence for each chain. Furthermore, we construct return maps which present the same characteristic attractor for all peaks of energies within the same kind of chain.

Phonon localization in one-dimensional quasiperiodic chains.

Quasiperiodic long-range order is intermediate between spatial periodicity and disorder, and the excitations in one-dimensional (1D) quasiperiodic systems are believed to be transitional between extended and localized. These ideas are tested with a numerical analysis of two incommensurate 1D elastic chains: Frenkel-Kontorova (FK) and Lennard-Jones (LJ). The ground-state configurations and the eigenfrequencies and eigenfunctions for harmonic excitations are determined. Aubry's transition by breaking the analyticity is observed in the ground state of each model, but the behavior of the excitations is qualitatively different. Phonon localization is observed for some modes in the LJ chain on both sides of the transition. The localization phenomenon apparently is decoupled from the distribution of eigenfrequencies since the spectrum changes from continuous to Cantor-set-like when the interaction parameters are varied to cross the analyticity-breaking transition. The eigenfunctions of the FK chain satisfy the "quasi-Bloch" theorem below the transition, but not above it, while only a subset of the eigenfunctions of the LJ chain satisfy the theorem.

Self-similarity and scaling of wave function for binary quasiperiodic chains associated with quadratic irrationals

By establishing the correspondence between the substitution rule (a→anb; b→a) and the transformation on the value of ω=tan φ byω→1/(n+ω) in the cut-and-project (CP) method, it is proved that the necessary and sufficient conditions for a binary quasiperiodic (QP) sequence made by the CP method to be self-similar is that ω is a quadratic irrational (QI) number. And, vice versa, the necessary condition for a binary self-similar sequence generated by the substitution rule to be obtainable by the CP method is that the corresponding substitution rule can be rewritten as a simple composition of transformations with the type (a→anb; b→a). To illustrate some physical properties of the self-similar QP chains associated with QI numbers, we analyze the scaling behaviour of the wave function atE=0 for an off-diagnonal tight-binding model. It is shown that the wave function atE=0 grows at most by a power-law for the QP lattices, characterized by a special class of QI numbers. For the QP chains associated with general QI numbers, with the same logic, the typical scaling behaviour of the wave function is conjectured to be the same.

Renormalization-group analysis of extended electronic states in one-dimensional quasiperiodic lattices.

We present a detailed analysis of the nature of electronic eigenfunctions in one-dimensional quasi-periodic chains based on a clustering idea recently introduced by us [Sil et al., Phys. Rev. {\bf B 48}, 4192 (1993) ], within the framework of the real-space renormalization group approach. It is shown that even in the absence of translational invariance, extended states arise in a class of such lattices if they possess a certain local correlation among the constituent atoms. We have applied these ideas to the quasi-periodic period-doubling chain, whose spectrum is found to exhibit a rich variety of behaviour, including a cross-over from critical to an extended regime, as a function of the hamiltonian parameters. Contrary to prevailing ideas, the period-doubling lattice is shown to support an infinity of extended states, even though the polynomial invariant associated with the trace map is non-vanishing. Results are presented for different parameter regimes, yielding both periodic as well as non-periodic eigenfunctions. We have also extended the present theory to a multi-band model arising from a quasi-periodically arranged array of $\delta$-function potentials on the atomic sites. Finally, we present a multifractal analysis of these wavefunctions following the method of Godreche and Luck [ C. Godreche and J. M. Luck, J. Phys. A :Math. Gen. {\bf 23},

A natural class of generalized Fibonacci chains

In this paper we propose a class of substitution rules that generate quasiperiodic chains sharing their typical properties with the quasiperiodic Fibonacci chain. For a subclass we explicitly construct the atomic surface. Moreover, scaling properties of the energy spectrum are discussed in relation to the dynamics of trace maps.

Fibonacci chain as a periodic chain with discommensurations

The quasiperiodic Fibbonacci chain can be obtained by finite displacements from a periodic chain if one introduces ordered defects that are similar to discommensurations in incommensurate crystal phases. It is shown that the chain with discommensurations can also be constructed by means of a substitution rule having the Pisot property. In superspace the defect structure is a superstructure for the two-dimensional array that corresponds to the Fibonacci chain.

The unusual electronic spectrum of an infinite quasiperiodic chain: extended signature of all eigenstates

We have shown in an analytical way supported by intuitive arguments that all the eigenstates of an infinite quasiperiodic chain can be of extended nature in an unusual way. The well known copper mean chain provides such an example. Earlier works, based on calculations with systems having finite sizes, showed the co-existence of localized and extended states for this particular system. Within the framework of the real space renormalization group scheme we transform a perfectly ordered chain and a copper mean chain into a period doubling sequence for which an area-preserving dynamical map is already in existence. The recursion relations for the Hamiltonian parameters of the effective period doubling sequences generated from an ordered chain and a copper mean chain are then compared to extract information regarding the true nature of the eigenstates of the latter.

Dynamical systems for quasiperiodic chains and new self-similar polynomials

Dynamical systems in SL(2, R) or SL(2, C) naturally appear in the transfer matrix method for quasiperiodic chains characterized by arbitrary irrational numbers. We show new subdynamical systems and invariants that are related to full diagonal and off-diagonal components of the transfer matrices; they are analogous to formulae of Chebyshev polynomials of the first and second kinds. Applying them to an electronic problem on the Fibonacci chain, we obtain sets of self-similar polynomials, quasiperiodic extension of the Chebyshev polynomials of the first and second kinds with self-similar properties. Two scaling factors of the self-similarities coincide with ones obtained by the perturbative decimation renormalization group method.

Renormalization group of random Fibonacci chains.

A renormalization method is proposed to analyze the electronic band structure of disordered Fibonacci chains. The perfect (quasiperiodic) chain is contemplated as a particular case. Both bond and on-site problems are considered. The method is computationally efficient and suitable to deal with large chains in real space. It can be useful for the study of electronic properties of random tilings.

Extended states in one-dimensional lattices: Application to the quasiperiodic copper-mean chain.

The question of the conditions under which one-dimensional systems support extended electronic eigenstates is addressed in a very general context. Using real-space renormalization-group arguements we discuss the precise criteria for determining the entire spectrum of extended eigenstates and the corresponding eigenfunctions in disordered as well as quasiperiodic systems. For purposes of illustration we calculate a few selected eigenvalues and the corresponding extended eigenfunctions for the quasiperiodic copper-mean chain. So far, for the infinite copper-mean chain, only a single energy has been numerically shown to support an extended eigenstate [J. Q. You, J. R. Yan, T. Xie, X. Zeng, and J. X. Zhong, J. Phys.: Condens. Matter 3, 7255 (1991)]: we show analytically that there is in fact an infinite number of extended eigenstates in this lattice which form fragmented minibands.

The nature of the atomic surfaces of quasiperiodic self-similar structures

Quasiperiodic self-similar chains generated by substitutions (i.e. deterministic concatenation rules) and their diffraction spectra are analysed in a systematic fashion, from the viewpoint of the superspace formalism. A substitution acting on n objects generates quasiperiodic chains if, and only if, the associated substitution matrix fulfils two arithmetic conditions (Pisot property and unit determinant). The structures thus obtained can be alternatively built as sections of periodic patterns in an n-dimensional superspace, which are regular repetitions of an atomic surface. The authors derive a general algorithm to construct this atomic surface. It is a compact set of the (n-1)-dimensional internal space, which is a unit cell for a lattice of translations. The atomic surface is nevertheless not necessarily connected, and its boundary is generically an anisotropic self-similar fractal. The dimension dB of this boundary is shown to govern the anomalously slow fall-off of the intensities of Bragg diffractions, and therefore to influence physical properties.

Selfsimilarity in a Class of Quadratic-Quasiperiodic Chains

We prove that quasiperiodic chains associated with a class of quadratic irrational numbers have an inflation symmetry and can be generated from a regular chain by a hyperinflation. We devise the explicit method to find the hyperinflation symmetry and discuss the properties of such a class of quasiperiodic sequences.

Electrons and structure of quasiperiodic one-dimensional chains

The electronic properties of a quasiperiodic one-dimensional chain with a Fibonacci-sequence of atomic sites are studied with a modified Su-Schrieffer-Heeger (SSH) model. The optimal geometrical structure is calculated automatically by means of the forces. In order to quantify the localization/delocalization of the electronic orbitals we define a degree of localization based on an autocorrelation function. Our main interest is the differences in the electronic properties between a neutral and a doped system. It is demonstrated that, if the system is rigid enough, the degree of the localization and the total density of the stats (DOS) are relatively insensitive to the site-relaxation caused by doping. Furthermore, the unoccupied states of a neutral system have the tendency to be more localized after doping. If the system is more flexible the self-similar structure of the DOS is disturbed and there is a larger difference in the studied properties.

Electronic stability of quasicrystals: application to flux phases

The energetic stability of disordered binary alloys in which electrons are interacting via a tight-binding Hamiltonian is considered. A phenomenological repulsive interatomic potential is also added to the Hamiltonian. For a weak potential, the gain in energy compared to the periodic linear chain is derived as a function of the electronic filling factor ν, and the disorder. For a given ν, the most stable structure is a quasiperiodic chain associated with ν and the atomic concentration. These results, which are exact for any concentration and filling factor, are illustrated on the quasiperiodic Fibonacci chain. As an application, we show that the preceding results favor the stability of uniform flux phases with a flux per plaquette Φ = ν.