Electronic states of pseudospin-1 fermions in dice lattice ribbons

Boundary conditions for the two-dimensional fermions in ribbons of the hexagonal lattice are studied in the dice model whose energy spectrum in infinite system consists of three bands with one completely flat band of zero energy. Like in graphene the regular lattice terminations are of the armchair and zigzag types. However, there are four possible zigzag edge terminations in contrast to graphene where only one type of zigzag termination is possible. Determining the boundary conditions for these lattice terminations, the energy spectra of pseudospin-1 fermions in dice model ribbons with zigzag and armchair boundary conditions are found. It is shown that the energy levels for armchair ribbons display the same features as in graphene except the zero energy flat band inherent to the dice model. In addition, unlike graphene, there are no propagating edge states localized at zigzag boundary and there are specific zigzag terminations which give rise to bulk modes of a metallic type in dice model ribbons. We find that the existence of the flat zero-energy band in the dice model is very robust and is not affected by the zigzag and armchair boundaries.Comment: 16 pages, 7 figure

Algorithms for Bohemian Matrices

This thesis develops several algorithms for working with matrices whose entries are multivariate polynomials in a set of parameters. Such parametric linear systems often appear in biology and engineering applications where the parameters represent physical properties of the system. Some computations on parametric matrices, such as the rank and Jordan canonical form, are discontinuous in the parameter values. Understanding where these discontinuities occur provides a greater understanding of the underlying system. Algorithms for computing a complete case discussion of the rank, Zigzag form, and the Jordan canonical form of parametric matrices are presented. These algorithms use the theory of regular chains to provide a unified framework allowing for algebraic or semi-algebraic constraints on the parameters. Corresponding implementations for each algorithm in the Maple computer algebra system are provided. In some applications, all entries may be parameters whose values are limited to finite sets of integers. Such matrices appear in applications such as graph theory where matrix entries are limited to the sets {0, 1}, or {-1, 0, 1}. These types of parametric matrices can be explored using different techniques and exhibit many interesting properties. A family of Bohemian matrices is a set of low to moderate dimension matrices where the entries are independently sampled from a finite set of integers of bounded height. Properties of Bohemian matrices are studied including the distributions of their eigenvalues, symmetries, and integer sequences arising from properties of the families. These sequences provide connections to other areas of mathematics and have been archived in the Characteristic Polynomial Database. A study of two families of structured matrices: upper Hessenberg and upper Hessenberg Toeplitz, and properties of their characteristic polynomials are presented

Topologically protected elastic waves in phononic metamaterials

Topological states of quantum matter exhibit unique disorder-immune surface states protected by underlying nontrivial topological invariants of the bulk. Such immunity from backscattering makes topological surface or edge states ideal carriers for both classical and quantum information. So far, topological matters have been explored only in the realms of electronics and photonics, with limited range of bulk properties and largely immutable materials. These constraints thus impose severe performance trade-offs in experimentally realizable topologically ordered states. In contrast, phononic metamaterials not only provide access to a much wider range of material properties, but also allow temporal modulation in the non-adiabatic regime. Here, from the first-principles we demonstrate numerically the first phononic topological metamaterial in an elastic-wave analogue of the quantum spin Hall effect. A dual-scale phononic crystal slab is used to support two effective spins of phonon over a broad bandwidth, and strong spin-orbit coupling is realized by breaking spatial mirror symmetry. By preserving the spin polarization with an external load or spatial symmetry, phononic edge states are shown to be robust against scattering from discrete defects as well as disorders in the continuum. Our system opens up the possibility of realizing topological materials for phonons in both static and time-dependent regimes.Comment: 19 pages, 6 figure

Nonlinearity, topology and PT symmetry in Photonic Lattices

Since over a decade, there is an ever-growing interest of scientific community towards the hot topics of so-called parity-time (PT ) symmetry and topological phases of matter, historically originating from non-Hermitian extensions of Quantum Mechanics and phase transitions without symmetry breaking in Condensed Matter Physics, respectively. Recent technological advancements in Photonics allowed one to study and fruitfully develop some of the most peculiar aspects of PT symmetry and topological matter on both theoretical and experimental levels. PT -symmetric photonic structures, with a judicious tailoring of gain and loss bulk regions, became a new paradigm in controlling the ow of light in unconventional manner, thereby paving the way to novel applications in laser physics, synthetic optical materials, optical sensing and so on. Likewise, fundamental ideas of topology rapidly emerged in the field of Photonics and brought about new possibilities for harnessing light, such as robust backscattering-free transport and Thouless pumping, to name a few. Owing to universality of the topological principles, a wide range of experimental platforms became feasible, including waveguides, metamaterials, optical crystals, optomechanics, silicon-based photonics, cavities and circuit QED. Most of the aspects of PT symmetry and topology in Photonics are very well understood in the linear regime, where light particles, photons, do not interact with each other. In contrast, up to date, they remain hardly explored in nonlinear optical regimes, characterized by self-interaction and self-localization of light in nonlinear media. In that regard, the aim of this thesis is to extend those powerful ideas further on in the direction of nonlinear light, in order to eventually discover and experimentally observe novel phenomena and interplays between nonlinearity, PT symmetry and topology. For that, we study both experimentally and theoretically 1D and 2D Discrete Photonic Lattices with synthetic dimensions, mimicking the celebrated Discrete Quantum Walks and experimentally based on the extremely versatile and interferometrically robust technique, called time-multiplexing. The set-ups essentially consist of optical fiber loops, mutually coupled via passive or active in-fiber beam splitters. In particular, we discover and experimentally observe novel and fascinating aspects of non-Hermitian discrete solitons in PT -symmetric environments and topological chiral edge states under the action of optical Kerr nonlinearity.Bereits seit über einem Jahrzehnt besteht ein stetig wachsendes Interesse der Wissenschaftsgemeinde an den hochaktuellen Themen von sogenannter Parität-Zeit (PT ) Symmetrie und topologischen Phasen der Materie, die historisch aus nicht-hermitischen Erweiterungen der Quantenmechanik bzw. Phasenübergängen ohne Symmetrieeinbruch in der Physik der kondensierten Materie stammen. Jüngste technologische Fortschritte im Forschungsfeld der Photonik ermöglichten es, einige der besondersten Aspekte der PT Symmetrie und der topologischen Materie sowohl theoretisch als auch experimentell zu untersuchen und weiter zu entwickeln. PT -symmetrische, photonische Strukturen mit einer gezielten Anpassung von Verstärkungs- und Verlustbereichen wurden zu einem neuen Paradigma für die unkonventionelle Steuerung des Lichtusses, und ebneten damit den Weg für neuartige Anwendungen in der Laserphysik, der synthetischen optischen Materialien, Lichtsensoren und so weiter. Ebenso entstanden im Bereich der Photonik schnell grundlegende Topologie-Ideen, die neue Möglichkeiten der Lichtnutzung eröffneten, wie zum Beispiel einen robusten, rückstreuungsfreien Transport und ein Thouless Pumping, um nur Einige zu nennen. Aufgrund der Universalität der topologischen Prinzipien wurde eine Vielzahl von experimentellen optischen Plattformen realisierbar, darunter optischeWellenleiter, Metamaterialien, optische Kristalle, Optomechanik, Photonik auf Siliziumbasis, optische Resonatoren und Schaltkreis-QED. Die meisten Aspekte der PT Symmetrie und Topologie sind im linearen Bereich, in dem Lichtteilchen, Photonen, nicht miteinander interagieren, sehr gut erforscht. Im Gegensatz dazu sind sie in nichtlinearen optischen Systemen, die durch Selbstinteraktion und Selbstlokalisierung von Licht in nichtlinearen Medien gekennzeichnet sind, bislang kaum erforscht. In dieser Hinsicht ist das Ziel dieser Arbeit, diese einussreichen Ideen in Richtung nichtlineares Licht weiter auszudehnen, um schließlich neue Phänomene und Wechselwirkungen zwischen Nichtlinearität, PT Symmetrie und Topologie zu entdecken und experimentell zu beobachten. Dazu untersuchen wir sowohl experimentell als auch theoretisch diskrete, photonische 1D- und 2D-Gitter mit synthetischen Dimensionen, die die berühmten diskreten Quantenwanderungen imitieren und auf der extrem vielseitigen und interferometrisch robusten experimentellen Technik namens Zeitmultiplex basieren. Die experimentellen Aufbauten bestehen im Wesentlichen aus Lichtleiterschleifen, die über passive oder aktive Lichtleiterstrahlteiler miteinander gekoppelt sind. Unsere besonderen Aktivitäten betreffen die Entdeckung und experimentelle Beobachtung neuartiger und faszinierender Aspekte nicht-hermitischer diskreter Solitonen in PT -symmetrischen Umgebungen und topologischen chiralen Randzuständen unter der Einwirkung optischer Kerr-Nichtlinearität

Topological Photonics

Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.Comment: 87 pages, 30 figures, published versio

Nonlinear topological photonics

Rapidly growing demands for fast information processing have launched a race for creating compact and highly efficient optical devices that can reliably transmit signals without losses. Recently discovered topological phases of light provide a novel ground for photonic devices robust against scattering losses and disorder. Combining these topological photonic structures with nonlinear effects will unlock advanced functionalities such as nonreciprocity and active tunability. Here we introduce the emerging field of nonlinear topological photonics and highlight recent developments in bridging the physics of topological phases with nonlinear optics. This includes a design of novel photonic platforms which combine topological phases of light with appreciable nonlinear response, self-interaction effects leading to edge solitons in topological photonic lattices, nonlinear topological circuits, active photonic structures exhibiting lasing from topologically-protected modes, and harmonic generation from edge states in topological arrays and metasurfaces. We also chart future research directions discussing device applications such as mode stabilization in lasers, parametric amplifiers protected against feedback, and ultrafast optical switches employing topological waveguides.Comment: 21 pages, 12 figure