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Aperiodic structures

The gap between perfect periodic crystals and disordered media is covered by a large class of systems with geometry belonging to the general framework of aperiodic order. The theoretical study of such systems, i.e. systems with order but without periodicity, has been increasingly developed, involving semiconductor heterostructures, photonic multilayers, acoustic waveguides or even DNA macromolecules and extends from physical properties, such as transmission spectra to more mathematical concepts as tiling theory and crystallography. The increased experimental and theoretical interest render the study of systems with aperiodic order a dynamic and promising research field, especially concerning the design of devices with new functionalities and increased control on their transport properties.

Local Symmetries and Symmetry Breaking

Recently we developed a theory of local symmetries, capable of describing aperiodic wave mechanical systems with symmetries valid in spatially restricted domains. We have found a systematic way to treat the breaking of discrete symmetries via the identification of nonlocal invariant currents and discovered a generalization of the well-known Bloch and parity theorems. Based on this theory, we have developed a construction principle for wave mechanical devices with prescribed perfect transmission properties. The latter allows for the design of specialized filters with possible technological applications. Additionally, we were able to extend the phase diagram from the PT-symmetric phase to the corresponding broken phase. This research direction reveals novel properties of quantum mechanical, acoustic and photonic structures such as the recently suggested classification of their perfectly transmitting resonances. The theory was recently experimentally verified employing acoustic waveguide setups, showing the existence of the spatially invariant currents in the presence of losses. Our results revealed the fundamental role of symmetries in restricted spatial domains and clearly indicate that completely locally symmetric devices constitute a promising class of setups, regarding the manipulation of wave propagation.