Nontrivial Edge Research Articles

Topologically protected quantum entanglement emitters

Entanglement and topology portray nature at the fundamental level but differently. Entangled states of particles are intrinsically sensitive to the environment, whereas the topological phases of matter are naturally robust against environmental perturbations. Harnessing topology to protect entanglement has great potential for reliable quantum applications. Generating topologically protected entanglement, however, remains a significant challenge, requiring the operation of complex quantum devices in extreme conditions. Here we report topologically protected entanglement emitters that emit a topological Einstein–Podolsky–Rosen state and a multiphoton entangled state from a monolithically integrated plug-and-play silicon photonic chip in ambient conditions. The device emulating a photonic anomalous Floquet insulator allows the generation of four-photon topological entangled states at non-trivial edge modes, verified by the observation of a reduced de Broglie wavelength. Remarkably, we show that the Einstein–Podolsky–Rosen entanglement can be topologically protected against artificial structure defects by comparing the state fidelities of 0.968 ± 0.004 and 0.951 ± 0.010 for perfect and defected emitters, respectively. Our topologically protected devices may find applications in quantum computation and in the study of quantum topological physics.

Topological One-Way Edge States in an Air-Hole Honeycomb Gyromagnetic Photonic Crystal

Topological one-way edge states have attracted increasing attention because of their intriguing fundamental physics and potential applications, particularly in the realm of photonics. In this paper, we present a theoretical and numerical demonstration of topological one-way edge states in an air-hole honeycomb gyromagnetic photonic crystal biased by an external magnetic field. Localized horizontally to the edge and confined in vertical direction by two parallel metallic plates, these unique states possess robust one-way propagation characteristics. They are strongly robust against various types of defects, imperfections and sharp corners on the path, and even can unidirectionally transport along the irregular edges of arbitrary geometries. We further utilize the one-way property of edge states to overcome entirely the issue of back-reflections and show the design of topological leaky wave antennas. Our results open a new door towards the observation of nontrivial edge states in air-hole topological photonic crystal systems, and offer useful prototype of robust topological photonic devices, such as geometry-independent topological energy flux loops and topological leaky wave antennas.

A piezoelectric quantum spin Hall insulator VCClBr monolayer with a pure out-of-plane piezoelectric response.

The combination of piezoelectricity with a nontrivial topological insulating phase in two-dimensional (2D) systems, namely piezoelectric quantum spin Hall insulators (PQSHI), is intriguing for exploring novel topological states toward the development of high-speed and dissipationless electronic devices. In this work, we predict a PQSHI Janus monolayer VCClBr constructed from VCCl2, which is dynamically, mechanically and thermally stable. In the absence of spin orbital coupling (SOC), VCClBr is a narrow gap semiconductor with a gap value of 57 meV, which is different from Dirac semimetal VCCl2. The gap of VCClBr is due to a built-in electric field caused by asymmetrical upper and lower atomic layers, which is further confirmed by the external-electric-field induced gap in VCCl2. When including SOC, the gap of VCClBr is increased to 76 meV, which is larger than the thermal energy of room temperature (25 meV). The VCClBr is a 2D topological insulator (TI), which is confirmed by Z2 topological invariant and nontrivial one-dimensional edge states. It is proved that the nontrivial topological properties of VCClBr are robust against strain (biaxial and uniaxial cases) and external electric fields. Due to broken horizontal mirror symmetry, only an out-of-plane piezoelectric response can be observed, when a biaxial or uniaxial in-plane strain is applied. The predicted piezoelectric strain coefficients d31 and d32 are -0.425 pm V-1 and -0.219 pm V-1, respectively, and they are higher than or compared with those of many 2D materials. Finally, Janus monolayer VCFBr and VCFCl (dynamically unstable) are also constructed, and they are still PQSHIs. Moreover, the d31 and d32 of VCFBr and VCFCl are higher than those of VCClBr, and the d31 (absolute value) of VCFBr is larger than one. According to out-of-plane piezoelectric coefficients of VCXY (X ≠ Y = F, Cl and Br), CrX1.5Y1.5 (X = F, Cl and Br; Y = I) and NiXY (X ≠ Y = Cl, Br and I), it is concluded that the size of the out-of-plane piezoelectric coefficient has a positive relation with the electronegativity difference of X and Y atoms. Our studies enrich the diversity of Janus 2D materials, and open a new avenue in the search for PQSHI with a large out-of-plane piezoelectric response, which provides a potential platform in nanoelectronics.

Four-band non-Abelian topological insulator and its experimental realization

Very recently, increasing attention has been focused on non-Abelian topological charges, e.g., the quaternion group Q8. Different from Abelian topological band insulators, these systems involve multiple entangled bulk bandgaps and support nontrivial edge states that manifest the non-Abelian topological features. Furthermore, a system with an even or odd number of bands will exhibit a significant difference in non-Abelian topological classification. To date, there has been scant research investigating even-band non-Abelian topological insulators. Here, we both theoretically explore and experimentally realize a four-band PT (inversion and time-reversal) symmetric system, where two new classes of topological charges as well as edge states are comprehensively studied. We illustrate their difference in the four-dimensional (4D) rotation sense on the stereographically projected Clifford tori. We show the evolution of the bulk topology by extending the 1D Hamiltonian onto a 2D plane and provide the accompanying edge state distributions following an analytical method. Our work presents an exhaustive study of four-band non-Abelian topological insulators and paves the way towards other even-band systems.

Quantum transport signatures of non-trivial topological edge states in a ring-shaped Su–Schrieffer–Heeger double-chain system

In the ring-shaped Su–Schrieffer–Heeger (SSH) double-chain, the quantum interference between the two different electron tunneling paths of the upper and lower chains has an important influence on the electron transport properties of non-trivial topological edge states. Here, we have studied the electron transport signatures of non-trivial topological edge states in a ring-shaped SSH double-chain system based on the wave-guide theory and transfer-matrix method. In the ring-shaped SSH double-chain with the upper chain being different from the lower one, it is demonstrated that the electron transmission probability displays the four and two resonance peaks associated with the non-trivial topological edge states in the weak and strong coupling regimes, respectively. Whereas in the case of the upper chain being the same as the lower one, the two transmission resonance peaks associated with the non-trivial topological edge states in the weak coupling regime are only found, and that in the strong coupling regime disappear that originated from the destructive interference between the two different electron tunneling paths of the upper and lower chains. Consequently, the variation of the number of transmission resonance peaks associated with the non-trivial topological edge states in the weak and strong coupling regimes suggests that an alternative scheme for detecting non-trivial topological edge states in the ring-shaped SSH double-chain system.

Biphoton topology in a quadratic nonlinear waveguide array under the Su-Schrieffer-Heeger model

We demonstrate the generation of a stable topologically nontrivial biphoton state with a high degree of quantum entanglement by the bosonic Bogoliubov Hamiltonian in a quadratic nonlinear waveguide array (QNWA) under the Su-Schrieffer-Heeger model. Analysis of its energy spectrum and eigenmode spectra verifies that this biphoton state is a topological nontrivial edge state characterized by the nonzero Berry phase at 1/4 and 3/4 fillings. The changes of Berry phase at 1/4 and 3/4 fillings indicate the topological quantum transition. The topological robustness of the biphoton edge state can be demonstrated by the spatial correlation under random disorders in propagation constant of the QNWA. It is also shown that the R\'enyi entanglement entropy of this topologically nontrivial biphoton state can approach high values. The robustness of this topologically nontrivial property suggests the promising application of this highly entangled biphoton state in quantum computing and information.

Engineering antiferromagnetic topological insulator by strain in two-dimensional rare-earth pnictide EuCd2Sb2

Antiferromagnetic topological insulator (AFM TI) provides an important platform to explore prominent physical phenomena and innovative design of topological spintronics devices, but very few high-quality candidate materials are known especially in two dimensions with intrinsic magnetism. Here, we propose an intrinsic two-dimensional (2D) AFM insulator and present a strain-engineered topological phase transition that realizes the 2D AFM TI phase in EuCd2Sb2 with in-plane magnetization. On the basis of first-principles calculations, the bandgaps of EuCd2Sb2 quintuple layers (QLs) are identified to be tunable, and a bandgap closing and reopening process is revealed with a small critical tensile strain of 2%. With opened bandgap, the topologically nontrivial characteristics of strained EuCd2Sb2 QLs are confirmed by the direct calculation of the spin Chern number CS, ℤ2 topological invariant, and the nontrivial topological edge states. Remarkably, while the previously proposed magnetic topological states may be heavily deformed by fragile magnetism, the obtained 2D AFM TI phase is highly robust against magnetic configurations, including ferromagnetic and AFM coupling with both in-plane and out-of-plane directions. Our results, thus, not only reveal the high possibility for engineering the 2D AFM TI state but also provide a very promising platform to uncover the complex interaction between magnetism and topology.

Time-reversal symmetric topological metal

Topological metals possess gapless band structures accompanied by nontrivial edge states. Topological metals are created from the topological insulators by adjusting the magnetic flux. Here we propose the time-reversal symmetry protected topological metal without employing the magnetic flux. Topological metallic phase presents although the Chern number vanishes. In the topologically nontrivial phase, three different cases in the metallic phase are distinguished from the band-touching and in-gap features of the topological edge states. These findings shed light on the time-reversal symmetric topological metals and greatly simplify the realization of topological metals.

Symmetry-protected dissipative preparation of matrix product states

We propose and analyze a method for efficient dissipative preparation of matrix product states that exploits their symmetry properties. Specifically, we construct an explicit protocol that makes use of driven-dissipative dynamics to prepare a many-body quantum state that features symmetry-protected topological order and nontrivial edge excitations. The preparation protocol is protected from errors that respect the symmetry, allowing for robust experimental implementation without fine-tuned control. Numerical simulations show that the preparation time scales polynomially in system size $n$. Furthermore, we demonstrate that this scaling can be improved to $O({log}^{2}n)$ by using parallel preparation of individual segments and fusing them via quantum feedback. A concrete scheme using excitation of trapped neutral atoms into the Rydberg state via electromagnetically induced transparency is proposed and generalizations to a broader class of matrix product states are discussed.

Control of plasmons in topological insulators via local perturbations

We use a fully quantum mechanical approach to demonstrate control of plasmonic excitations in prototype models of topological insulators by molecule-scale perturbations. Strongly localized surface plasmons are present in the host systems, arising from the topologically nontrivial single-particle edge states. A numerical evaluation of the random phase approximation equations for the perturbed systems reveals how the positions and the internal electronic structure of the added molecules affect the degeneracy of the locally confined collective excitations, i.e., shifting the plasmonic energies of the host system and changing their spatial charge density profile. In particular, we identify conditions under which significant charge transfer from the host system to the added molecules occurs. Furthermore, the induced field energy density in the perturbed topological systems due to external electric fields is determined.

Open spin chain realization of a topological defect in a one-dimensional Ising model: Boundary and bulk symmetry

We study the realizations of topological defects in a one-dimensional quantum Ising model with an open boundary condition at criticality. Applying the construction discussed in M. Hauru, G. Evenbly, W. W. Ho, D. Gaiotto, and G. Vidal, Phys. Rev. B 94, 115125 (2016), we prove that the Ising model on an open chain with multiple topological defects can be transformed to the same model with boundary magnetic fields and noninteracting boundary degrees of freedom. This results in the appearance of a linear combination of Cardy states, which can be interpreted as an edge state of the spin or fermion chain. We show that this edge state with the large boundary entropy can be protected under bulk perturbation, whereas it is fragile to a boundary perturbation. Our formulation suggests the existence of nontrivial edge physics under the existence of topological defects and opens many interesting questions for future analysis related to boundary and bulk physics.

Multihole models for deterministically placed acceptor arrays in silicon

We compute the electronic structure of acceptor clusters in silicon by using three methods to include electron correlations: the full configuration interaction, the Heitler-London approximation, and the unrestricted Hartree-Fock method. We show both the HL approach and the UHF method are good approximations to the ground state of the full CI calculation for a pair of acceptors and for finite linear chains. The total energies for finite linear chains show the formation of a 4-fold degenerate ground state when there is a weak bond at the end of the chain, which is shown to be a manifold of topological edge states. We identify a change in the angular momentum composition of the ground state at a critical pattern of bond lengths and show it is related to a crossing in the Fock matrix eigenvalues. We also test the symmetry of the UHF solution and compare it to the full CI; the symmetry is broken under almost all the arrangements by the formation of a magnetic state in UHF, and we find further broken symmetries for some particular arrangements related to crossings or potential crossings between the Fock-matrix eigenvalues. We also compute the charge distributions across the acceptors obtained from the eigenvectors of the Fock matrix: with weak bonds at the chain ends, two holes are localized at either end of the chain while the others have a nearly uniform distribution over the middle; this implies the existence of the non-trivial edge states. We apply the UHF method to treat an infinite linear chain with periodic boundary conditions. We find the band structures in the UHF approximation and compute the Zak phases for the occupied Fock-matrix eigenvalues; we find they do not correctly predict the topological edge states in this interacting system. We find direct study of the quantum numbers characterising the edge states provides a better insight into their topological nature.

Robust large-gap topological insulator phase in transition-metal chalcogenide ZrTe4Se

Based on density functional theory, we investigate the electronic properties of bulk and single-layer ZrTe4Se. The band structure of bulk ZrTe4Se can produce a semimetal-to-topological insulator (TI) phase transition under uniaxial strain. The maximum global band gap is 0.189 eV at the 7% tensile strain. Meanwhile, the Z 2 invariants (0; 110) demonstrate conclusively it is a weak topological insulator. The two Dirac cones for the (001) surface further confirm the nontrivial topological nature. The single-layer ZrTe4Se is a quantum spin Hall insulator with a band gap 86.4 meV and Z 2 = 1, the nontrivial metallic edge states further confirm the nontrivial topological nature. The maximum global band gap is 0.211 eV at 8% uniaxial tensile strain along the [100] direction. When the compressive strain is more than 1%, the band structure of single-layer ZrTe4Se produces a TI-to-semimetal transition. These theoretical analysis may provide a method for searching large band gap TIs and platform for topological nanoelectronic device applications.

Comparative study of Hermitian and non-Hermitian topological dielectric photonic crystals

The effects of gain and loss on the band structures of a bulk topological dielectric photonic crystal (PC) with ${C}_{6v}$ symmetry and the PC-air-PC interface are studied based on first-principle calculation. To illustrate the importance of parity-time (PT) symmetry, three systems are considered, namely the PT-symmetric, PT-asymmetric, and lossy systems. We find that the system with gain and loss distributed in a PT symmetric manner exhibits a phase transition from a PT exact phase to a PT broken phase as the strength of the gain and loss increases, while for the PT-asymmetric and lossy systems, no such phase transition occurs. Furthermore, based on the Wilson loop calculation, the topology of the PT-symmetric system in the PT exact phase is demonstrated to keep unchanged as the Hermitian system. At last, different kinds of edge states in Hermitian systems under the influences of gain and loss are studied and we find that while the eigenfrequencies of nontrivial edge states become complex conjugate pairs, they keep real for the trivial defect states.

Zak phases of chiral photonic crystals designed via transformation optics

We study a class of inhomogeneous chiral photonic crystals with topologically nontrivial bands and edge modes using the approach of non-coordinate transformation optics. By transforming the chiral photonic crystals into achiral binary photonic crystals, the closed-form solutions to the photonic bands of the chiral photonic crystals are obtained. We show that the topological edge modes of these chiral photonic crystals can be accurately predicted by the Zak phases developed from the topological band theory for achiral binary photonic crystals.

Epitaxial Growth of Quasi-One-Dimensional Bismuth-Halide Chains with Atomically Sharp Topological Non-Trivial Edge States.

Quantum spin Hall insulators (QSHIs) have one-dimensional (1D) spin-momentum locked topological edge states (ES) inside the bulk band gap, which can serve as dissipationless channels for the practical applications in low consumption electronics and high performance spintronics. However, obtaining the clean and atomically sharp ES which serves as ideal 1D spin-polarized nondissipative conducting channels is demanding and still a challenge. Here, we report the formation of the quasi-1D Bi4I4 nanoribbons on the surface of Bi(111) with the support of the graphene-terminated 6H-SiC(0001) and the direct observation of the topological ES at the step edges by the scanning tunneling microscopy (STM) and spectroscopic-imaging results. The ES reside surround the edge of Bi4I4 nanoribbons and exhibits noteworthy robustness against nontime reversal symmetry (non-TRS) perturbations. The theoretical simulations verify the topological nontriviality of 1D ES, which is retained after considering the presence of the underlying Bi(111). Our study supports the existence of topological ES in Bi4I4 nanoribbons, benefiting to engineer the topological features by using the 1D nanoribbons as building blocks.

Distribution of the time of the maximum for stationary processes

We consider a one-dimensional stationary stochastic process of duration T. We study the probability density function (PDF) of the time at which reaches its global maximum. By using a path integral method, we compute for a number of equilibrium and nonequilibrium stationary processes, including the Ornstein-Uhlenbeck process, Brownian motion with stochastic resetting and a single confined run-and-tumble particle. For a large class of equilibrium stationary processes that correspond to diffusion in a confining potential, we show that the scaled distribution , for large T, has a universal form (independent of the details of the potential). This universal distribution is uniform in the “bulk”, i.e., for and has a nontrivial edge scaling behavior for (and when ), that we compute exactly. Moreover, we show that for any equilibrium process the PDF is symmetric around , i.e., . This symmetry provides a simple method to decide whether a given stationary time series is at equilibrium or not.

Tendency of Gap Opening in Semimetal 1T′‐MoTe2 with Proximity to a 3D Topological Insulator

AbstractMonolayer (ML) 1T′‐MoTe2 has attracted intensive interest as a fascinating quantum spin Hall (QSH) insulator. However, there are two critical aspects impeding its exploration and potential applications of QSH effects. One is its semimetallic feature with a negative band gap, leading to nontrivial edge channels annihilated by the bulk states. The other is its fabrication always accompanied by a mixed phase of 1T′ and 2H. Based on first‐principles calculations, it is shown that the large work‐function difference results in strong interlayer interactions and proximity effects in ML 1T′‐MoTe2 via interfacing a 3D topological insulator Bi2Te3, facilitating the realization of pure 1T′ phase and even the band gap opening. It is further verified that the epi‐grown ML 1T′‐MoTe2 on Bi2Te3 is nearly in single phase. Furthermore, the measurements of angle resolved photoemission spectroscopy and scanning tunneling spectroscopy confirm the obvious separated‐tendency of conduction and valence bands as well as the strong metallic edge states in ML 1T′‐MoTe2. The results also reveal the nontrivial band topology in ML 1T′‐MoTe2 is preserved in 1T′‐MoTe2/Bi2Te3 heterostructure. This work offers a promising candidate to realize QSH effects and provides guidance for controlling the nontrivial band gap opening by proximity effects in van der Waals engineering.

Topological Insulating Phase in Single-Layer Pentagonal Covalent Organic Frameworks: A Reticular Design Using Metal Phthalocyanine

Organic topological insulators have received tremendous attention lately due to their designability and highly chemical diversity. In this work, we propose an isoreticular series of single-layer π-conjugated covalent organic frameworks with the Cairo pentagonal tiling, termed mcm-ZnPc-1, mcm-ZnPc-2, mcm-ZnPc-3, and mcm-ZnPc-3N, where mcm is the crystal net’s symbol and ZnPc stands for zinc phthalocyanine. First-principles calculations using density functional theory show that mcm-ZnPc-1 and mcm-ZnPc-2 are trivial insulators with band gaps of 0.96 and 1.18 eV, respectively. Interestingly, expanding the π-conjugated system of mcm-ZnPc-2, followed by a chemical substitution, results in mcm-ZnPc-3 and mcm-ZnPc-3N with a distorted Dirac point and a quadratic band-crossing point in their band structures, respectively. The topological analyses indicate that mcm-ZnPc-3N is a Z2 topological insulator with a nontrivial gapless edge state. Our tight-binding model for the pentagonal lattice suggests that the topologically trivial-to-nontrivial transition can be attributed to the sufficient electronic coupling between two adjacent linkers and to the fact that the band-crossing point locates at the Fermi level. Remarkably, we show that replacing Zn in mcm-ZnPc-3N with Cd and Hg can substantially enlarge the nontrivial band gap from 0.3 meV up to 10 meV due to strong spin–orbit coupling strengths although a biaxial strain is necessary to tune the Fermi level of mcm-HgPc-3N in order to enter its topological insulating phase. Our proposed materials are predicted to be dynamically, thermally, and mechanically stable, thus paving the way for their experimental realization. This work not only introduces the pentagonal lattice as a building motif for framework structures but also demonstrates a strategy to engineer the exotic band structure of organic materials.

Optical switching of topological phase in a perovskite polariton lattice.

Strong light-matter interaction enriches topological photonics by dressing light with matter, which provides the possibility to realize active nonlinear topological devices with immunity to defects. Topological exciton polaritons-half-light, half-matter quasiparticles with giant optical nonlinearity-represent a unique platform for active topological photonics. Previous demonstrations of exciton polariton topological insulators demand cryogenic temperatures, and their topological properties are usually fixed. Here, we experimentally demonstrate a room temperature exciton polariton topological insulator in a perovskite zigzag lattice. Polarization serves as a degree of freedom to switch between distinct topological phases, and the topologically nontrivial polariton edge states persist in the presence of onsite energy perturbations, showing strong immunity to disorder. We further demonstrate exciton polariton condensation into the topological edge states under optical pumping. These results provide an ideal platform for realizing active topological polaritonic devices working at ambient conditions, which can find important applications in topological lasers, optical modulation, and switching.