When electrons are subject to a potential with two incommensurate periods, translational invariance is lost, and no periodic band structure is expected. However, model calculations based on nearly free one-dimensional electrons and experimental results from high-resolution photoemission spectroscopy on a quasi-one-dimensional material do show dispersing band states with signatures of both periodicities. Apparent band structures are generated by the nonuniform distribution of electronic spectral weight over the complex eigenvalue spectrum.

https://science.sciencemag.org/content/sci/290/5491/501.full.pdf

Electronic structure of solids with competing periodic potentials.

We develop a mathematical framework allowing to study anomalous transport in homogeneous solids. The main tools characterizing the anomalous transport properties are spectral and diffusion exponents associated to the covariant Hamiltonians describing these media. The diffusion exponents characterize the spectral measures entering in Kubo's formula for the conductivity and hence lead to anomalies in Drude's formula. We give several formulas allowing to calculate these exponents and treat, as an example, Wegner's n-orbital model as well as the Anderson model in coherent potential approximation.

Anomalous Transport: A Mathematical Framework

We present exact solutions for some eigenstates of hopping models on one- and two-dimensional quasiperiodic tilings and show that they are ``critical'' states, by explicitly computing their multifractal spectra. These eigenstates are shown to be generically present in 1D quasiperiodic chains, of which the Fibonacci chain is a special case. We then describe properties of the ground states for a class of tight-binding Hamiltonians on the 2D Penrose and Ammann-Beenker tilings. Exact and numerical solutions are seen to be in good agreement.

Critical eigenstates and their properties in one- and two-dimensional quasicrystals

We present a theoretical framework for understanding the wave functions and spectrum of an extensively studied paradigm for quasiperiodic systems, namely the Fibonacci chain. Our analytical results, which are obtained in the limit of strong modulation of the hopping amplitudes, are in good agreement with published numerical data. In the perturbative limit, we show a symmetry of wave functions under permutation of site and energy indices. We compute the wave-function renormalization factors and from them deduce analytical expressions for the fractal exponents corresponding to individual wave functions, as well as their global averages. The multifractality of wave functions is seen to appear at next-to-leading order in $\ensuremath{\rho}$. Exponents for the local spectral density are given, in extremely good accord with numerical calculations. Interestingly, our analytical results for exponents are observed to describe the system rather well even for values of $\ensuremath{\rho}$ well outside the domain of applicability of perturbation theory.

Fractal dimensions of wave functions and local spectral measures on the Fibonacci chain

The distinctive electronic properties of quasicrystals stem from their long range structural order, with invariance under rotations and under discrete scale change, but without translational invariance. This review introduces the multiple manifestations of multifractality in the Fibonacci chain, a 1D paradigm for quasicrystals. Quasiperiodic systems have been recently of interest furthermore for their topological properties, described in this review for the simplest 1D case. Some important generalizations of the basic tight-binding models, and closely related models for other quasiperiodic systems are also discussed.

The Fibonacci quasicrystal: Case study of hidden dimensions and multifractality

We solve the approximate renormalisation group found by Qiu Niu and Franco Nori(Phys. Rev. Lett. 57 2057(1986)) for a tight-binding hamiltonian on the Fibonacci chain. This enables us to characterize analytically as completely as possible the spectral properties of this model, such as the spectral measure, the bandwidth distribution, the Lebesgue measure exponent, the Hausdorff dimension, the multifractal scaling, the gaps distribution as well as the long time return probability. Our results, qualitatively and quantitatively complete and unify previous works on similar models.

Mourad Benakli

Papers

Analytical results for scaling properties of the spectrum of the Fibonacci chain.