Universalities in one-electron properties of limit quasiperiodic lattices

Abstract

We investigate one-electron properties of one-dimensional self-similar structures called limit quasiperiodic lattices. The trace map of such a lattice is nonconservative in contrast to the quasi-periodic case, and we can determine the structure of its attractor. It allows us to obtain the three new features of the present system: 1) The multi-fractal characters of the energy spectra are universal.2) The supports of the f (α)-spectra extend over the whole unit interval, [0, 1]. 3) There exist marginal critical states.

Limit-quasiperiodic Ammann bars and two-dimensional limit-quasiperiodic structures

“…Let L be one of them. Then, it is associated with an infinite sequence produced by a substitution rule (SR) on a pair of symbols S and L, which are identified with a shorter spacing and a longer one, respectively [11,12]. The SR is specified by a pair of words, e.g., (SLS, LSSL), where the two words are assumed palindromic so that L has the inversion symmetry.…”

“…The SR is specified by a pair of words, e.g., (SLS, LSSL), where the two words are assumed palindromic so that L has the inversion symmetry. The relevant substitution matrix to the SR is Frobenian, and L becomes a 1D LQPL iff its Frobenian eigenvalue, , is a Pisot number [11,12]. Moreover, |L|/|S| ¼ with (1, ) being the left Frobenian eigenvector.…”

“…Moreover, |L|/|S| ¼ with (1, ) being the left Frobenian eigenvector. Conversely, if a Pisot number in a quadratic field is given, there exist a finite number of the relevant Frobenian matrices, each of which yields a finite number of palindromic SRs [12]. We shall consider here the case of Pisot number 2 þ ffiffi ffi for them.…”

Spectral Classification of One‐Dimensional Binary Aperiodic Crystals: An Algebraic Approach

“…In turn, pure point lattice Fourier measures can be further split into four separate groups, namely, the so-called periodic ( 0 ), quasiperiodic ( I ), limit-quasiperiodic ( II ), and limit-periodic ( III ) pure point classes, respectively, by attending to …ner details in their related di¤raction patterns. [20,[22][23][24][25] The di¤raction spectra of 0 and I classes representatives both consist in Bragg peaks supported by a …nite Fourier module whose rank either equals the physical space dimension ( 0 class) or is larger than it ( I class). In the limit-quasiperiodic and limit-periodic classes II and III one also …nds a di¤raction spectrum consisting of a dense distribution of Bragg peaks.…”

Superquasicrystals: self-similar-ordered structures with non-crystallographic point symmetries

“…Discovering and/or synthesizing SQCs is a big challenge in material science, and its achievement will be a triumph of the generalized crystallography. Moreover, there exists strong evidence that the nature of the electronic states in an SQC is markedly different from that in a QC [17].…”