Electronic properties of one-dimensional quasiperiodic lattices: Green's function renormalization group approach

Using the inflation-deflation symmetry, we have developed a new real-space decimation approach to study the electronic properties of one-dimensional quasiperiodic lattices. The key result is the construction of a compact renormalization group that allows simple calculation of the average Green's function and the average density of states to any degree. The Fibonacci and the generalized Fibonacci lattices are used to demonstrate the method. Numerical results for the average density of states of these lattices show a good agreement with the results obtained by other methods. This confirms the validity and the efficiency of the approach.     

The properties of one-dimensional quasiperiodic lattice's phonon spectrum

A numerical method with renormalization group transformation is used to study the scaling properties of phonon spectrum and its relevant state of one-dimensional quasiperiodic lattice which is constructed by reduced map. We find that the phonon spectrum at finite gaps' edges in the binary chain Fibonacci model is a Cantor-like set spectrum. The spectrum is singularly continuous and the state is a critical state.     

Electronic properties of one-dimensional systems with long-range correlated binary potentials

We study numerically the electronic properties of one-dimensional systems with long-range correlated binary potentials.The potentials are mapped from binary sequences with a power-law power spectrum over the entire frequency range,which is characterized by correlation exponent β.We find the localization length ξ increases with β.At system sizes N →∞,there are no extended states.However,there exists a transition at a threshold β c.When β > β c,we obtain ξ > 0.On the other hand,at finite system sizes,ξ≥ N may happen at certain β,which makes the system "metallic",and the upper-bound system size N (β) is given.     

Electronic properties of one-dimensional quasilattices

We study the on-site model of a new class of one-dimensional qusiperiodic lattices, for which the substitution rules areBBA, andABAB. By means of the renormalization-group approach, a interesting multifractal specrral behavior has been found, which has been confirmed by numerical simulation. A Cantor-like energy spectra is obtained by using the Kohmoto-Kadanaff-Tang (KKT) renormalization-group method. Three kinds of wave-function behavior (extended, localized, and intemediate states) are definitely found.     

Statistical mechanics of a quasiperiodic one-dimensional magnetic chain

In this paper we study the thermodynamic properties of the Ising model on a linear chain in which the sites are generated according to the Fibonacci sequence. We calculate the partition function, the specific heat and the q-dependent magnetic susceptibility.     

Electronic states of the icosahedral quasiperiodic systems in magnetic fields

The electronic properties of the icosahedral quasiperiodic system in a tile-dependent or uniform magnetic field is studied by quasi-Bloch scheme and the general solutions are obtained for the electronic states. The behavior of the three-dimensional (3D) non-interacting electrons in an icosahedral quasiperiodic system may be treated as the projection of that of the non-interacting pseudo-electrons in 6D. In the presence of the tile-dependent magnetic field, the non-interacting electrons are quasi-Bloch electrons, while they are partial quasi-Bloch electrons when the magnetic field is uniform.     

First passage time for a class of one-dimensional stochastic systems

We consider the time evolution of a class of stochastic systems of finite size with polynomial nearest neighbor transition rates. We obtain analytical expressions for the first passage time (FPT) and its moments. We show that the mean FPT, averaged over a uniform initial distribution, shows a simple asymptotoc behavior with the system size and the parameters of the transition rates.     

Electronic spectra of one-dimensional nano-quasi-periodic systems under bias

We investigate the properties of the energy spectra of quantum one-dimensional nano-quasi-crystals in the presence of external electric fields. These systems are modelled by means of finite sequences, ordered according to a Fibonacci rule which constituted of two blocks A$A$ (constant potentials of different heights defined on finite intervals) and B$B$ (delta potentials of different intensities). We use the electric field ability of producing Stark ladders in periodic systems to obtain well separated energy levels and to study the evolution of these levels when disorder is introduced. We show that this effect also allows us to predict the approximate position of the levels in the disordered system, in spite of its chaotic appearance at first view. We show, against the usual belief, that the n$n$th Stark ladder in general is not formed exclusively from the levels of the n$n$th band. The disorder is introduced in two different ways: by changing the distribution of the blocks or by changing the values of the delta potential intensities. In both cases we start from electrified periodic structures which are gradually perturbed to obtain electrified quasi-periodic structures. We show that the use of Fibonacci sequences as a particular case is not crucial and one can use the electric field to analyze any other type of quasi-periodic systems.     

Statistical properties of chaos demonstrated in a class of one-dimensional maps

One-dimensional maps with complete grammar are investigated in both permanent and transient chaotic cases. The discussion focuses on statistical characteristics such as Lyapunov exponent, generalized entropies and dimensions, free energies, and their finite size corrections. Our approach is based on the eigenvalue problem of generalized Frobenius-Perron operators, which are treated numerically as well as by perturbative and other analytical methods. The examples include the universal chaos function relevant near the period doubling threshold. Special emphasis is put on the entropies and their decay rates because of their invariance under the most general class of coordinate changes. Phase-transition-like phenomena at the border state of chaos due to intermittency and super instability are presented.     

Localization properties of driven disordered one-dimensional systems

We generalize the definition of localization length to disordered systems driven by a time-periodic potential using a Floquet-Green function formalism. We study its dependence on the amplitude and frequency of the driving field in a one-dimensional tight-binding model with different amounts of disorder in the lattice. As compared to the autonomous system, the localization length for the driven system can increase or decrease depending on the frequency of the driving. We investigate the dependence of the localization length with the particle's energy and prove that it is always periodic. Its maximum is not necessarily at the band center as in the non-driven case. We study the adiabatic limit by introducing a phenomenological inelastic scattering rate which limits the delocalizing effect of low-frequency fields.     

Femtosecond laser writing of subwave one-dimensional quasiperiodic nanostructures on a titanium surface

One-dimensional quasiperiodic structures whose period is much smaller than the wavelength of exciting radiation have been obtained on a titanium surface under the multipulse action of linearly polarized femtosecond laser radiation with various surface energy densities. As the radiation energy density increases, the one-dimensional surface nanorelief oriented perpendicularly to the radiation polarization evolves from quasiperiodic ablation nanogrooves to regular lattices with subwave periods (100–400 nm). In contrast to the preceding works for various metals, the period of lattices for titanium decreases with increasing energy density. The formation of the indicated surface nanostructures is explained by the interference of the electric fields of incident laser radiation and a surface electromagnetic wave excited by this radiation, because the length of the surface electromagnetic wave for titanium with significant interband absorption decreases with an increase in the electron excitation of the material.     

Electronic properties of quasiperiodic Fibonacci chain including second-neighbor hopping in the tight-binding model

We present an exact real-space renormalization group (RSRG) scheme for the electronic Green's functions of one-dimensional tight-binding systems having both nearest-neighbor and next-nearest-neighbor hopping integrals, and determine the electronic density of states for the quasiperiodic Fibonacci chain. This RSRG method also gives the Lyapunov exponents for the eigenstates. The Lyapunov exponents and the analysis of the flow pattern of hopping integrals under renormalization provide information about the nature of the eigenstates. Next we develop a transfer matrix formalism for this generalized tight-binding system, which enables us to determine the wave function amplitudes. Interestingly, we observe that like the nearest-neighbor tight-binding Fibonacci chain, the present generalized tight-binding system also have critical eigenstates, Cantor-set energy spectrum and highly fragmented density of states. It indicates that these exotic physical properties are really the characteristics of the underlying quasiperiodic structure. Received 5 April 1999     

New class of exactly solvable one-dimensional disordered electron hopping systems

Two types of disordered chains are presented, which allow for the exact calculation of the (configurational averaged) density of states in terms of a continued fraction. The first model contains a certain type of site-diagonal disorder and is a generalization of Lloyd's model; it refers to a substitutional alloy.The second model contains site-off-diagonal (hopping) disorder and may refer to a generalized alloy—analog treatment of a Hubbard chain.