On the role of symmetries in the theory of photonic crystals

Nittis, Giuseppe De; Lein, Max

doi:10.1016/j.aop.2014.07.032

Cited by 18 publications

(26 citation statements)

References 42 publications

(23 reference statements)

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“…In the coming years, we expect the discovery of new topological mirrors, phases and invariants that could be classified with respect to different symmetries [73][74][75][76][77]. The topological phases of interacting photons [78][79][80] could be explored by considering nonlinearity [81] and entanglement.…”

Section: Discussionmentioning

confidence: 99%

Topological photonics

2014

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The application of topology, the mathematics studying conserved properties through continuous deformations, is creating new opportunities within photonics, bringing with it theoretical discoveries and a wealth of potential applications. This field was inspired by the discovery of topological insulators, in which interfacial electrons transport without dissipation even in the presence of impurities. Similarly, the use of carefully-designed wave-vector space topologies allows the creation of interfaces that support new states of light with useful and interesting properties. In particular, it suggests the realization of unidirectional waveguides that allow light to flow around large imperfections without back-reflection. The present review explains the underlying principles and highlights how topological effects can be realized in photonic crystals, coupled resonators, metamaterials and quasicrystals.Frequency, wavevector, polarization and phase are degrees of freedom that are often used to describe a photonic system. In the last few years, topology -a property of a photonic material that characterizes the quantized global behavior of the wavefunctions on its entire dispersion band-has been emerging as another indispensable ingredient, opening a path forward to the discovery of fundamentally new states of light and possibly revolutionary applications. Possible practical applications of topological photonics include photonic circuitry less dependent on isolators and slow light insensitive to disorder.Topological ideas in photonics branch from exciting developments in solid-state materials, along with the discovery of new phases of matter called topological insulators [1, 2]. Topological insulators, being insulating in their bulk, conduct electricity on their surfaces without dissipation or backscattering, even in the presence of large impurities. The first example was the integer quantum Hall effect, discovered in 1980. In quantum Hall states, two-dimensional (2D) electrons in a uniform magnetic field form quantized cyclotron orbits of discrete eigenvalues called Landau levels. When the electron energy sits within the energy gap between the Landau levels, the measured edge conductance remains constant within the accuracy of about one part in a billion, regardless of sample details like size, composition and impurity levels. In 1988, Haldane proposed a theoretical model to achieve the same phenomenon but in a periodic system without Landau levels [3], the so-called quantum anomalous Hall effect.Posted on arXiv in 2005, Haldane and Raghu transcribed the key feature of this electronic model into photonics [4,5]. They theoretically proposed the photonic analogue of the quantum (anomalous) Hall effect in photonic crystals [6], the periodic variation of optical materials, molding photons the same way as solids modulating electrons. Three years later, the idea was confirmed by Wang et al., who provided realistic material designs [7] and experimental observations [8]. Those studies spurred numerous subsequent theoretical [9...

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Section: Discussionmentioning

confidence: 99%

Topological photonics

2014

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“…That is because the particle-hole-type "symmetry" represents complex conjugation of classical waves, and since classical waves are necessarily real-valued, complex conjugation enters as a constraint rather than a symmetry that can be selectively broken. We have emphasized this point in earlier works on the Schrödinger formalism of classical waves [14] and the topological classification of electromagnetic media [5].…”

Section: Introductionmentioning

confidence: 99%

Krein-Schrödinger formalism of bosonic Bogoliubov–de Gennes and certain classical systems and their topological classification

Lein

Sato

2019

Phys. Rev. B

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To understand recent works on classical and quantum spin equations and their topological classification, we develop a unified mathematical framework for bosonic BdG systems and associated classical wave equations; it applies not just to equations that describe quantized spin excitations in magnonic crystals but more broadly to other systems that are described by a BdG hamiltonian. Because here the generator of dynamics, the analog of the hamiltonian, is para-aka Krein-hermitian but not hermitian, the theory of Krein spaces plays a crucial role. For systems which are thermodynamically stable, the classical equations can be expressed as a "Schrödinger equation" with a hermitian "hamiltonian". We then proceed to apply the Cartan-Altland-Zirnbauer classification scheme: to properly understand what topological class these equations belong to, we need to conceptually distinguish between symmetries and constraints. Complex conjugation enters as a particle-hole constraint (as opposed to a symmetry), since classical waves are necessarily real-valued. Because of this distinction only commuting symmetries enter in the topological classification. Our arguments show that the equations for spin waves in magnonic crystals are a system of class A, the same topological class as quantum hamiltonians describing the Integer Quantum Hall Effect. Consequently, the magnonic edge modes first predicted by Shindou et al.[1] are indeed analogs of the Quantum Hall Effect, and their net number is topologically protected.

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“…In this subsection, we briefly review the Floquet-Bloch theory for the operator M W when W (x) is Λ-periodic, see for example [9,18,25]. The spectrum of M W can be obtained by solving the following L 2 k (Λ)-eigenvalue problem…”

Section: Floquet-bloch Theorymentioning

confidence: 99%

Linear and nonlinear electromagnetic waves in modulated honeycomb media

Zhu

2019

Stud Appl Math

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Wave dynamics in topological materials has been widely studied recently. A striking feature is the existence of robust and chiral wave propagations that have potential applications in many fields. A common way to realize such wave patterns is to utilize Dirac points which carry topological indices and is supported by the symmetries of the media. In this work, we investigate these phenomena in photonic media. Starting with Maxwell's equations with a honeycomb material weight as well as the nonlinear Kerr effect, we first prove the existence of Dirac points in the dispersion surfaces of transverse electric and magnetic Maxwell operators under very general assumptions of the material weight. Our assumptions on the material weight are almost the minimal requirements to ensure the existence of Dirac points in a general hexagonal photonic crystal. We then derive the associated wave packet dynamics in the scenario where the honeycomb structure is weakly modulated. It turns out the reduced envelope equation is generally a two-dimensional nonlinear Dirac equation with a spatially varying mass. By studying the reduced envelope equation with a domain-wall-like mass term, we realize the subtle wave motions which are chiral and immune to local defects. The underlying mechanism is the existence of topologically protected linear line modes, also referred to as edge states. However, we show that these robust linear modes do not survive with nonlinearity. We demonstrate the existence of nonlinear line modes, which can propagate in the nonlinear media based on high-accuracy numerical computations. Moreover, we also report a new type of nonlinear modes which are localized in both directions.

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On the role of symmetries in the theory of photonic crystals

Cited by 18 publications

References 42 publications

Topological photonics

Topological photonics

Krein-Schrödinger formalism of bosonic Bogoliubov–de Gennes and certain classical systems and their topological classification

Linear and nonlinear electromagnetic waves in modulated honeycomb media

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