Eigenvalue spectrum, density of states, and eigenfunctions in a two-dimensional quasicrystal

Real Space Theory for Electron and Phonon Transport in Aperiodic Lattices via Renormalization

Structural defects are inherent in solids at a finite temperature, because they diminish free energies by growing entropy. The arrangement of these defects may display long-range orders, as occurring in quasicrystals, whose hidden structural symmetry could greatly modify the transport of excitations. Moreover, the presence of such defects breaks the translational symmetry and collapses the reciprocal lattice, which has been a standard technique in solid-state physics. An alternative to address such a structural disorder is the real space theory. Nonetheless, solving 1023 coupled Schrödinger equations requires unavailable yottabytes (YB) of memory just for recording the atomic positions. In contrast, the real-space renormalization method (RSRM) uses an iterative procedure with a small number of effective sites in each step, and exponentially lessens the degrees of freedom, but keeps their participation in the final results. In this article, we review aperiodic atomic arrangements with hierarchical symmetry investigated by means of RSRM, as well as their consequences in measurable physical properties, such as electrical and thermal conductivities.

Acoustic directional source based on Penrose quasi-crystal arrangements of metallic rods embedded in a fluid

The directional emission properties of a circular device formed of a small number of metallic rods placed at the nodes of Penrose's tiling are investigated. At the center of the device a non-directional primary source emits monochromatic acoustic waves. In a wide frequency range, the beams come out of the device along preferential directions related to the symmetry of order 5 of the tiling and the non-periodicity in the radial direction. There are some frequencies for which the waves are completely trapped by the device. The shape, the distribution and the intensity of the emitted beams outside are closely dependent on the frequency. A comparison between numerical results and experiments is presented.

Simultaneous localization of photons and phonons within the transparency bands of LiNbO3 phoxonic quasicrystals

We report the properties of dual phononic-photonic band gaps and localized modes of eightfold lithium niobate (LiNbO₃) phoxonic quasicrystals (PhXQCs). Complete and large phoxonic band gaps are easily achieved despite the low refractive index of LiNbO₃ substrate. Point defect intentionally introduced can form localized modes within both forbidden and transparency bands over a wide range of geometric parameters. Further analysis indicates that the localized modes within transparency bands originate from the intrinsic high-order rotational symmetry of quasiperiodic structures, which resemble whispering gallery modes. LiNbO₃ PhXQCs provide a good candidate to enhance phononic-photonic interaction and show considerable advantage over the periodic counterparts.

Quasicrystalline and rational approximant wave patterns in hydrodynamic and quantum nested wells

The eigenfunctions of nested wells with an incommensurate boundary geometry, in both the hydrodynamic shallow water regime and quantum cases, are systematically and exhaustively studied in this Letter. The boundary arrangement of the nested wells consists of polygonal ones, square or hexagonal, with a concentric immersed, similar but rotated, well or plateau. A rich taxonomy of wave patterns, such as quasicrystalline states, their crystalline rational approximants, and some other exotic but well known tilings, is found in these mimicked experiments. To the best of our knowledge, these hydrodynamic rational approximants are presented here for the first time in a hydrodynamic-quantum framework. The corresponding statistical nature of the energy level spacing distribution reflects this taxonomy by changing the spectral types.

Quasiperiodic Bloch-like states in a surface-wave experiment

Bloch-like surface waves associated with a quasiperiodic structure are observed in a classic wave propagation experiment which consists of pulse propagation with a shallow fluid covering a quasiperiodically drilled bottom. We show that a transversal pulse propagates as a plane wave with quasiperiodic modulation, displaying the characteristic undulatory propagation in this quasiperiodic system and reinforcing the idea that analogous concepts to Bloch functions can be applied to quasicrystals under certain circumstances.

Scaling concepts in cellular and subcellular dynamics

After developing a map for certain states typical of hemopoietic stem cells (SCs), we identify a number of parameters (e.g. fractal dimensions, oscillatory behavior in a multistable landscape, etc.) that scale down to subcellular structures such as interphase genome, chromatids, chromatin entanglements and DNA segmental motions. A curious aspect is the continuous reappearance of recursive processes even at very small biologic scales. These iterations are relevant not only for the normal behavior of (hemopoietic) cells but also for amplifying hidden genomic singularities above some critical threshold. When that happens, there are sudden quali-quantitative and often clonal changes in the cell behavior. As illustrated by specific leukemic cases, many paradoxes of malignant growth seem best explained by the peculiar sensitivity to initial condition of (hemopoietic) cells. Interpreted as chaotic oscillators, these cells display a spectrum of disorders that, in spite of appearing as random, are in fact triggered by amplification of subtle preconditions with statistico-deterministic outcomes, that are predictable within certain time limits.