Incommensurate Potential Research Articles

Energy spectrum, transmittance, and localization in an incommensurate nonanalytic potential.

The energy spectrum, localization properties of eigenstates, and transmittance are calculated for a one-dimensional incommensurate chain with a potential which is the absolute value of the cosine potential used in the Aubry model. This nonanalytical potential, which has cusps, has previously been proposed by Bardeen as a model for the effective pinning potential for the motion of charge-density waves. Results show that, contrary to what happens in the Aubry model, a sharp mobility edge exists for intermediate values of the modulation and separates extended low-energy states from high-energy localized states. The first three terms in the Fourier expansion of the studied potential correspond very closely to the Soukoulis-Economou potential. Compared to that later model, the effect of the nonanalyticity in the Bardeen potential, represented by the infinite rest of the above-mentioned Fourier expansion, is found to be small.

Discontinuous behavior of the localization properties in a finite-length incommensurate potential with weak applied electric fields.

Model calculations are performed for a tight-binding model with an incommensurate potential of finite length, in a weak electric field scrE, with site energies given by ${E}_{n}$=W cos(2n)-eascrEn (n=1,2,...,N). It is found that there is a discontinuous delocalizing behavior in that localization properties and the position of the localized wave function have sudden discontinuities as a function of the electric field, where the calculations are performed for values of W>${W}_{C}$, where ${W}_{C}$ is the value of W above which the states are localized in the absence of an external electric field. As W decreases, the discontinuities become more frequent, resulting in a ``noisy'' behavior near threshold.

Level statistics in a quasiperiodic system

The authors examine the level statistics of Harper's equation, a simple model of a one-dimensional quasiperiodic system, through the distribution of the spacings between adjacent eigenvalues, each normalised by the average local density of states. They present exact results for the distribution for both small and large amplitude alpha of the incommensurate potential, showing it has a simple form in each of these limits. They show numerically that these distributions are preserved as alpha approaches its critical value 2 from each side, and also that at the critical point the distribution of the normalised spacings has a distinct but simple form.

Electrons on one-dimensional quasi-periodic lattices

We study electronic states on one-dimensional quasi-periodic lattices numerically by using the tight binding Fibonacci model generated by the Fibonacci substitution rule. It is found that the electronic states are critical, that is, intermediate between the localized and extended states. This critical state is characterized by the branching rule of the energy spectra and the inverse power law behavior of the level spacings. The Harper model with an incommensurate cosine potential, and quasi-periodic lattice models generated by other substitution rules are examined. Phonon problem in the quasi-periodic lattice is also discussed.

Quantum energy spectra and one-dimensional quasiperiodic systems.

Statistical properties of energy spectra in one-dimensional quasiperiodic systems are studied numerically. We find three distinctive level distributions: the Poisson, inverse-power-law (IPL), and cosine-band-like behaviors in the Harper model with an incommensurate potential. These depend on whether the electronic state is localized, critical, or extended, respectively. Energy spectra of electrons on the quasiperiodic Fibonacci lattice are also characterized by the IPL irrespective of the strength of the modulation, indicating that the state is always critical.

Localization by electric fields in one-dimensional tight-binding systems

We show that upon including an electric field within the class of one-dimensional single-orbital, nearest-neighbor, tight-binding models for a general nonperiodic potential, all eigenstates are localized. Irrespective of the details of the potential, the energy eigenstates show factorial localization and the eigenvalue spectrum is discrete, characterizable as a Stark ladder with nonuniform spacing. As an example, all eigenstates of the Aubry model become localized by the electric field, whatever the strength epsilon-c/sub 0/ of the incommensurate potential, whereas in the field-free case this occurs only if epsilon-c/sub 0/ exceeds the Aubry critical value. We also present detailed results for the Koster-Slater single-impurity model in an electric field. We indicate briefly some necessary conditions for observing field-enhanced localization.

Quantized currents and topological invariants for electrons in incommensurate potentials.

It is proved that charge transport induced by adiabatic variations of an incommensurate potential acting on noninteracting electrons is quantized. The corresponding integers are topological invariants (Chern numbers of various kinds) which label the gaps. Similar results are obtained for electrons in a periodic potential, under the influence of an irrational magnetic field.

Electronic Phase Transition Out of Chaos

We have used the real-space renormalization group method to investigate the Aubry model of incommensurate potential. The exact numerically solved density of states exhibits the self-similarity property and is nowhere dense. From the behaviour of single-particle Green's function, we have shown that the localized-to-extended transition is energy dependent. The Avron-Simon singularity is interpreted physically as the extremely large fluctuation of the local density of states from one site to the other.

Long-Range Resonance in Anderson Insulators: Finite-Frequency Conductivity of Random and Incommensurate Systems

The low-frequency conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ of an exactly solvable model of an incommensurate potential is calculated for several classes of commensuration parameter and compared with the random case. The conductivity is related to the degree to which the eigenstates of the model occur in resonating multiplets. It is found that $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{x}$, where $x=2$ for the random case, $x=\ensuremath{\infty}$ for generic commensuration, and $\ensuremath{-}1lxl\ensuremath{\infty}$ for special commensurations. There are fewer resonances for large $x$, with no resonance at $x=\ensuremath{\infty}$.

Solvable model of quantum motion in an incommensurate potential

We solve a Schr\odinger equation with a potential having two periods, whose ratio $\ensuremath{\alpha}$ is arbitrary. This is one of very few cases for which the solution can be fully discussed. If $\ensuremath{\alpha}$ is rational, i.e., commensurate, the eigenfunctions are Bloch states, and the energy levels fall into a spectrum of continuous bands. If $\ensuremath{\alpha}$ is a typical irrational the eigenfunctions are localized with exponentially decaying tails, and each has a distinct center, just as in a random system. The spectrum (called pure point) covers all energies, but only a finite number of energies belong to wave functions appreciable in a given region. A third rarely encountered, but currently interesting, type of spectrum, the singular continuous, occurs when $\ensuremath{\alpha}$ is a Liouville number, a special irrational number infinitely close to rational numbers. This case is also concretely illustrated and interpolates between the other two possibilities. The time evolution of wave packets is also discussed.

Localisation with phase correlations and the effect of periodic cycles

This paper examines anomalies that arise in the transport properties of a disordered solid. The anomalies are associated with the period-r>or=2 marginally stable cycles of a keyphase recurrence relation. Their presence is signalled by divergences in standard nondegenerate perturbation theory. A 'degenerate' perturbation theory is developed within which anomalies can be computed by expanding about the cycles of interest. The low-order anomalies at energies corresponding to r=2 and 3 are evaluated and a possible extension to a system with an incommensurate potential is discussed.

CALCULATION OF WAVE FUNCTION OF A SINGLE IMPURITY IN A ONE DIMENSIONAL SYSTEM WITH INCOMMENSURATE LATTICE POTENTIAL

We study the effect of a single impurity in a one dimensional system with incommensurate lattice potential. The energy level and the wavefunction of the impurity State is derived with the Green's function technique. The energy level is found to be very sensitive to the impurity position in the region where the incommensurate potential is varying rapidly. The wavefunction of the impurity state decays exponentially with small oscillation of the amplitude. We also consider the effect of the impurity on the states in the band. For a system that is symmetric about the impurity, we find a large reduction of the amplitude of the wavefunction with even parity in the vicinity. For a system that is asymmetric about the impurity, a step-wise increase in the amplitude occurs at the impurity.

Electronic states in a one-dimensional system with incommensurate lattice potentials

The authors present some new and interesting results on the localisation of electronic states in a one-dimensional lattice with an incommensurate lattice potential. The system studied is represented by a tight-binding Hamiltonian with diagonal elements given by V0 cos(qn). V0 is the modulation strength and q is the wavenumber. They introduce the concept of quasilocalisation to clarify inconsistencies in previous work on the existence of mobility edges. Their technique allows very precise calculation of the resolvent operator without the problem of numerical instability. Thus they are able to present accurate results for the density of states even when the widths of the energy bands are extremely small.

Localization in an Incommensurate Potential: An Exactly Solvable Model

The exact eigenstates of a one-dimensional tight-binding model with a periodic diagonal potential that can be commensurate or incommensurate with the lattice are found. If the period is incommensurate, all the eigenstates are localized. The localization length and the density of states are identical to those of a related disordered system. The case of a commensurate potential, for which all the states are extended, and the approach to the incommensurate case are also discussed. The solution is achieved by mapping the model into a time-dependent quantum problem in which the potential is periodic in time.