Photonic topological insulators

Topological insulators are a new phase of matter, with the striking property that conduction of electrons occurs only on their surfaces. In two dimensions, electrons on the surface of a topological insulator are not scattered despite defects and disorder, providing robustness akin to that of superconductors. Topological insulators are predicted to have wide-ranging applications in fault-tolerant quantum computing and spintronics. Substantial effort has been directed towards realizing topological insulators for electromagnetic waves. One-dimensional systems with topological edge states have been demonstrated, but these states are zero-dimensional and therefore exhibit no transport properties. Topological protection of microwaves has been observed using a mechanism similar to the quantum Hall effect, by placing a gyromagnetic photonic crystal in an external magnetic field. But because magnetic effects are very weak at optical frequencies, realizing photonic topological insulators with scatter-free edge states requires a fundamentally different mechanism-one that is free of magnetic fields. A number of proposals for photonic topological transport have been put forward recently. One suggested temporal modulation of a photonic crystal, thus breaking time-reversal symmetry and inducing one-way edge states. This is in the spirit of the proposed Floquet topological insulators, in which temporal variations in solid-state systems induce topological edge states. Here we propose and experimentally demonstrate a photonic topological insulator free of external fields and with scatter-free edge transport-a photonic lattice exhibiting topologically protected transport of visible light on the lattice edges. Our system is composed of an array of evanescently coupled helical waveguides arranged in a graphene-like honeycomb lattice. Paraxial diffraction of light is described by a Schrödinger equation where the propagation coordinate (z) acts as 'time'. Thus the helicity of the waveguides breaks z-reversal symmetry as proposed for Floquet topological insulators. This structure results in one-way edge states that are topologically protected from scattering.

Topological insulators(TIs) constitute a novel state of quantum matter which possesses non-trivial topological properties. Although discovered only in the recent few years, TIs have attracted intensive interest among the community of condensed matter physics and material science. TIs are insulating in the bulk but have conductive gapless edge or surface states on the boundaries, which have their origin in the nontrivial bulk band topology that is induced by the strong spin-orbital interactions in the materials. Existing in all dimensions, TIs exhibit a variety of exotic physics such as quantum spin Hall effect, momentum-spin locked surface states, Dirac fermion transport, quantized anomalous Hall effect, Majorana fermions, etc. In this thesis,\n\nI study the transport properties of 2D and 3D TIs by numerical approaches. As an introduction, a brief review of TIs is given. A detailed description of the numerical methods is also presented. The results can be summarized in four aspects. First, disorder is found be able to induce a non-trivial TI from an originally trivial band insulator, where the conductance of a two terminal device drops to nearly zero and then rises to form an anomalous plateau as disorder strength is increased, and finally all the states become localized. The real space Chern number calculation as well as the effective medium theory suggests that disorder is fundamentally responsible for the emerging of the extended helical edge states in this system. We also present a levitation and pair annihilation picture of the extended states for this model. Second, by making the 2D TIs into singly connected quantum point contacts(QPCs), I show a coherent and fast Aharonov-Bohm oscillation of conductance caused by the quantum interference of the helical edge states. This oscillation not only happens against weak magnetic field but also against the gate voltage in the zero-field condition.\n\nThis results in a giant edge magnetoresistance of the device in weak magnetic fields. The amplitude of the magnetoresistance is controllable by adjusting either the\n\nQPCs' slit width or the interference loop size in the device. The oscillation is found robust against disorder. Third, by applying a uniform spin-splitting Zeeman field in the bulk of the 3D TI whose surface states can be viewed as massless Dirac fermions,\n\nI find chiral edge states on the gapped surfaces of the 3D TI, which can be considered as interface states between domains of massive and massless Dirac fermions.\n\nEffectively these states are result of splitting of a perfect interface conducting channel. This picture is confirmed by the Landauer-B?ttiker calculations in four-terminal Hall bars. Finally, I propose the concept of topological semi-metals. By calculating the local density of states on the surfaces, I demonstrate that surface states and the gapless\n\nDirac cone already exist in the system although the bulk is not gapped. We show how the uni-axial strain induces an insulating band gap and turn the semi-metal into true TI. We predict existence of quantum spin Hall effect in the thin films made of these materials, which can be significantly enhanced by disorders.

Sonic and photonic topological insulators that host topological edge states offer promising potentials for the resilient control of acoustic and electromagnetic waves, respectively. Despite the great progress on sonic or photonic topological insulators, the research of their integration, i.e., the phoxonic topological insulator, is less explored. In this work, we propose a phoxonic topological insulator that hosts acoustic and dual-polarization photonic helical edge states simultaneously. In specific, we first design a glide-symmetric phoxonic crystal with concurrent sonic and dual-polarization photonic bandgaps via the topology optimization method. Then by choosing two different unit cells from the optimized phoxonic crystal and assembling them to create a domain-wall interface, a phoxonic topological insulator that supports two pairs of gapless helical edge states within both the sonic and photonic bandgaps is constructed. Pseudospin-locked unidirectional transmissions and robust manipulations of helical edge states are demonstrated for acoustic and dual-polarization electromagnetic waves simultaneously in the proposed phoxonic topological insulator. The designed phoxonic topological insulator opens new avenues for developing topological photoacoustic devices, enabling the reliable management of both acoustic and electromagnetic waves, as well as the investigation of their interplay. Published by the American Physical Society 2024

Topological insulators are a new kind of quantum state of the matters. Similar to the traditional insulators, topological insulators also possess the bulk gap and the insulating bulk states. The difference is that topological insulators have the conducting edge states on their surface or boundary at the same time. These conducting edge states are immune to the local disorders and perturbations since they are protected by the nonlocal topological invariant. The different values of the topological invariant usually correspond to different topological phases. The topological protection of the topological invariant originates from the topological structure of the energy bands in momentum space, which leads that the topological insulator has many potential applications in quantum information processing and quantum computing. The robust quantum state transfer can be realized with a high fidelity via the edge channel of the topological insulator, in which the process of state transfer is robust to the local disorders and perturbations since the edge channel is protected by the energy gap. Also, the topological quantum computing can be implemented by dint of the non-Abelian anyons and Majorana zero modes. These potential applications make the topological insulator have many significant research values in quantum information processing and quantum computing. However, the experimental realization of the topological insulators based on the traditional electronic system has many challenges both in experimental techniques and experimental detections. From the perspectives of energy spectrum and Bloch theorem, electron, photon, and phonon present certain similarity. So, in the field of bosons with integral spin, the quantized topological states of light, sound, mechanical motion should also be implemented similarly, which provides opportunity to conveniently realize the topological insulators in experiment based on different bosonic optical systems. With the fast-developing fields of micro-nano manufacturing and materials processing technology in recent years, all kinds of quantum optical platforms become reliable candidates for the simulations and mappings of the topological insulators, which are the so called photonic topological insulators. Based on the bosonic statistical properties in these platforms, the relevant topological issues can be easily realized and even detected directly. For example, the circuit-QED lattice, consisting of a series of superconducting resonators and qubits, is becoming a more and more appealing and reliable candidate for the mappings and the simulations of all kinds of topological insulators due to its advantages of tunability and scalability. Many topological issues based on the one dimensional circuit-QED system have been investigated, such as the mapping of the high-dimensional topological Chern insulator, the topological edge state and the topological invariant, the detection of topological edge states, etc. Another promoting quantum optical platform is the optomechanical array, which is composed by single optomechanical system. The optomechanical array contains the photons and the phonons at the same time, which has many potential advantages in the mapping and the detection of all kinds of topological issues. However, the topological phenomenon based on the optomechanical array is still rarely investigated. Thus, we summarize the development in the field of the topological insulators and review the latest research progress of photonic topological insulators in circuit-QED lattice and optomechanical array.

A topological insulator is a material with an insulating interior but time-reversal symmetry-protected conducting edge states. Since its prediction and discovery almost a decade ago, such a symmetry-protected topological phase has been explored beyond electronic systems in the realm of photonics. Electrons are spin-1/2 particles, whereas photons are spin-1 particles. The distinct spin difference between these two kinds of particles means that their corresponding symmetry is fundamentally different. It is well understood that an electronic topological insulator is protected by the electron's spin-1/2 (fermionic) time-reversal symmetry [Formula: see text] However, the same protection does not exist under normal circumstances for a photonic topological insulator, due to photon's spin-1 (bosonic) time-reversal symmetry [Formula: see text] In this work, we report a design of photonic topological insulator using the Tellegen magnetoelectric coupling as the photonic pseudospin orbit interaction for left and right circularly polarized helical spin states. The Tellegen magnetoelectric coupling breaks bosonic time-reversal symmetry but instead gives rise to a conserved artificial fermionic-like-pseudo time-reversal symmetry, Tp ([Formula: see text]), due to the electromagnetic duality. Surprisingly, we find that, in this system, the helical edge states are, in fact, protected by this fermionic-like pseudo time-reversal symmetry Tp rather than by the bosonic time-reversal symmetry Tb This remarkable finding is expected to pave a new path to understanding the symmetry protection mechanism for topological phases of other fundamental particles and to searching for novel implementations for topological insulators.

Topological insulators are a new phase of matter, with the striking property that conduction of electrons occurs only on the surface. In two dimensions, surface electrons in topological insulators do not scatter despite defects and disorder, providing robustness akin to superconductors. Topological insulators are predicted to have wideranging applications in fault-tolerant quantum computing and spintronics. Recently, large theoretical efforts were directed towards achieving topological insulation for electromagnetic waves. One-dimensional systems with topological edge states have been demonstrated, but these states are zero-dimensional, and therefore exhibit no transport properties. Topological protection of microwaves has been observed using a mechanism similar to the quantum Hall effect, by placing a gyromagnetic photonic crystal in an external magnetic field. However, since magnetic effects are very weak at optical frequencies, realizing photonic topological insulators with scatterfree edge states requires a fundamentally different mechanism - one that is free of magnetic fields. Recently, a number of proposals for photonic topological transport have been put forward. Specifically, one suggested temporally modulating a photonic crystal, thus breaking time-reversal symmetry and inducing one-way edge states. This is in the spirit of the proposed Floquet topological insulators, where temporal variations in solidstate systems induce topological edge states. Here, we propose and experimentally demonstrate the first external field-free photonic topological insulator with scatter-free edge transport: a photonic lattice exhibiting topologically protected transport of visible light on the lattice edges. Our system is composed of an array of evanescently coupled helical waveguides arranged in a graphene-like honeycomb lattice. Paraxial diffraction of light is described by a Schrödinger equation where the propagation coordinate acts as 'time'. Thus the waveguides' helicity breaks zreversal symmetry in the sense akin to Floquet Topological Insulators. This structure results in scatter-free, oneway edge states that are topologically protected from scattering.

Spin–orbit coupling, a fundamental mechanism underlying topological insulators, has been introduced to construct the latter’s photonic analogs, or photonic topological insulators (PTIs). However, the intrinsic lack of electronic spin in photonic systems leads to various imperfections in emulating the behaviors of topological insulators. For example, in the recently demonstrated three-dimensional (3D) PTI, the topological surface states emerge, not on the surface of a single crystal as in a 3D topological insulator, but along an internal domain wall between two PTIs. Here, by fully abolishing spin–orbit coupling, we design and demonstrate a 3D PTI whose topological surface states are self-guided on its surface, without extra confinement by another PTI or any other cladding. The topological phase follows the original Fu’s model for the topological crystalline insulator without spin–orbit coupling. Unlike conventional linear Dirac cones, a unique quadratic dispersion of topological surface states is directly observed with microwave measurement. Our work opens routes to the topological manipulation of photons at the outer surface of photonic bandgap materials.

Photonic crystal topological insulators host protected states at their edges. In the band structure these edge states appear as continuous bands crossing the photonic band gap. They allow light to propagate unidirectionally and without scattering. In practice it is essential to make devices relying on these effects as miniature as possible. Here we study all-dielectric photonic topological insulator particles (finite crystals) which do not require a magnetic field. In such particles the edge states' frequencies are discrete. Nevertheless, the discrete states support pseudospin-dependent unidirectional propagation. They allow light to bend around sharp corners similarly to the continuous edge states and act as topologically protected whispering gallery modes which can store and filter light as well as manipulate its angular momentum. In addition, they explain multiple experimental observations of discrete transmission peaks in photonic topological insulators.

The photonic valley-Hall topological insulator can support valley-polarized edge states at non-trivial domain walls. Similar to dielectric photonic crystal topological insulator, the designer surface plasmon (DSP) periodic structure can also realize the valley-Hall topological phase transitions However, the construction and choice for the DSP unit cells of photonic topological insulators lack guidance with intuitive physical significance. The DSPs can be achieved by texturing metal layer into periodic structure, and the periodic cell can be analyzed by using the theory of characteristic modes. We believe that the unit cell structures in some mode bands can realize the photonic topological phase transitions (PTPT) by selecting special degeneracy characteristic modes. In this paper, the hexagonal conductor element is constructed by co-locating two equilateral triangle layers to form a DSP periodic unit structure. The cell structure dimensions of topological phase transitions are determined by characteristic mode analysis (CMA). Energy band analysis shows that the proposed DSP crystal can achieve the valley-Hall PTPT. The corresponding domain wall is constructed, and the robust valley-polarized chiral transmission characteristics are experimentally verified. The photonic PTPT waveguide based on periodic metal structure is easily compatible to traditional microwave circuits and to be manufactured, which has a wider application.

Two-dimensional topological photonic systems

Photonic topological crystalline insulators (PTCIs) with helical edge states provide an alternative way to achieve robust electromagnetic wave transport and processing. However, most existing PTCIs only involve a single topological bandgap, and generally support a pair of gapped helical edge states, restricting the scope of applications in various fields such as multiband waveguides, filters, and communication systems. Here, we design dual-band PTCIs, in which multiple helical edge modes appear within two distinct bulk gaps, for transverse electric (TE) and transverse magnetic (TM) modes, respectively, by introducing the topology optimization method into the photonic crystals with glide symmetry. For PTCIs with TE modes, the mismatched frequency ranges of edge modes hosted by two orthometric boundaries offer an opportunity to realize a photonic demultiplexer. For PTCIs with TM modes, we show the enhanced second harmonic (SH) generation through the coupling of multiband edge modes by matching the frequency ranges of edge modes within the first and second bandgaps to fundamental and SH waves, respectively. This work provides a new way for designing multiband PTCIs with helical edge states, having promising potentials in developing multiband topological photonic devices for both linear and nonlinear applications.

The preliminary study reported here investigates if unit-cell inclusion-symmetries may be broken in time-reversalinvariant topological insulator designs, while maintaining the desired global behaviour of pseudo-spin-dependent edge state based bi-directional, back-scattering robust, energy propagation. By allowing symmetries to be broken additional geometrical design freedom is attained, which may turn out to enable an improvement of various performance measures such as bandwidth and field confinement. The particular study considers a time-reversal-invariant acoustic topological insulator design, designed using a modified version of our recently proposed topology optimization based method for designing photonic and acoustic topological insulators.1 This method relies on a carefully constructed model system combined with the application of density based topology optimization to design two carefully interfaced crystal phases to maximize the flow of energy through the system. Through simple modifications of the method, we demonstrate that it is possible to design structures with different symmetry conditions from those that have previously been investigated using the method.

The photonic topological insulator has become an important research topic with a wide range of applications. Especially the higher-order topological insulator, which possesses gapped edge states and corner or hinge states in the gap, provides a new scheme for the control of light in a hierarchy of dimensions. In this paper, we propose a heterostructure composed of ordinary-topological-ordinary (OTO) photonic crystal slabs. Two coupled edge states (CESs) are generated due to the coupling between the topological edge states of the ordinary-topological interfaces, which opens up an effective way for high-capacity photonic transport. In addition, we obtain a new band gap between the CESs, and the two kinds of coupled corner states (CCSs) appear in the OTO bend structure. In addition, the topological corner state is also found, which arises from the filling anomaly of a lattice. Compared with the previous topological photonic crystal based on C-4 lattice, CESs, CCSs, and the topological corner state are all directly observed in experiment by using the near-field scanning technique, which makes the manipulation of the electromagnetic wave more flexible. We also verify that the three corner states are all robust to defects. Our work opens up a new way for guiding and trapping the light flow and provides a useful case for the coupling of topological photonic states.

We present Floquet fractal topological insulators: photonic topological insulators in a fractal-dimensional lattice consisting of helical waveguides. The helical modulation induces an artificial gauge field and leads to a trivial-to-topological phase transition. The quasi-energy spectrum shows the existence of topological edge states corresponding to real-space Chern number 1. We study the propagation of light along the outer edges of the fractal lattice and find that wavepackets move along the edges without penetrating into the bulk or backscattering even in the presence of disorder. In a similar vein, we find that the inner edges of the fractal lattice also exhibit robust transport when the fractal is of sufficiently high generation. Finally, we find topological edge states that span the circumference of a hybrid half-fractal, half-honeycomb lattice, passing from the edge of the honeycomb lattice to the edge of the fractal structure virtually without scattering, despite the transition from two dimensions to a fractal dimension. Our system offers a realizable experimental platform to study topological fractals and provides new directions for exploring topological physics.

Topological insulators are found in materials that have elements with strong spin orbit interaction. However, electron Coulomb repulsion also potentially generates the topological insulators as well as Chern insulators by the mechanism of spontaneous symmetry breaking, which is called topological Mott insulators. The quantum criticality of the transition to the topological Mott insulators from zero-gap semiconductors follows unconventional universality distinct from the Landau-Ginzburg-Wilson scenario. On the pyrochlore lattice, the interplay of the electron correlation and the spin orbit interaction provides us in a rich phase diagram not only with simple topological insulators but also with Weyl semimetal and topologically distinct antiferromagnetic phases. Magnetic domain wall of the all-in-all-out type antiferromagnetic order offers a promising candidate of magnetically controlled transport, because, even when the Weyl points disappears, the domain wall maintains robust gapless excitations with Fermi surfaces around it embedded in the bulk insulator and bears uniform magnetization simultaneously. The ingap state is protected by a mechanism similar to the solitons in polyacetylene. Puzzling experimental results of pyrochlore iridates are favorably compared with the prediction of the domain wall theory.