Interplay between Topological States and Rashba States as Manifested on Surface Steps at Room Temperature.

The unique spin texture of quantum states in topological materials underpins many proposed spintronic applications. However, realizations of such great potential are stymied by perturbations, such as temperature and local fields imposed by impurities and defects, that can render a promising quantum state uncontrollable. Here, we report room-temperature scanning tunneling microscopy/spectroscopy observation of interaction between Rashba states and topological surface states, which manifests local electronic structure along step edges controllable by the layer thickness of thin films. The first-principles theoretical calculation elucidates the robust Rashba states coexisting with topological surface states along the surface steps with characteristic spin textures in momentum space. Furthermore, the Rashba edge states can be switched off by reducing the thickness of a topological insulator Bi2Se3 to bolster their interaction with the hybridized topological surface states. The study unveils a manipulating mechanism of the spin textures at room temperature, reinforcing the necessity of thin film technology in controlling the quantum states.

Achievement splitting for topological states with pseudospin in phase modulation by using gyromagnetic photonic crystals.

As we know, valley-Hall kink states or pseudospin helical edge states are excited by polarized-momentum-locking [left-handed circular polarization (LCP) and right-handed circular polarization (RCP)] because the valley-Hall kink modes or pseudospin polarized modes have intrinsic and local chirality, which is difficult for these states to achieve phase modulation. Here we theoretically design and study a compatible topological photonic system with coexistence of photonic quantum Hall phase and pseudospin Hall phase, which is composed of gyromagnetic photonic crystals with a deformed honeycomb lattice containing six cylinders. A typical kind of hybrid topological waveguide states with pseudospin-characteristic, magnetic field-dependent, and strong robustness against backscattering and perfect electric conductor (PEC) is realized in the present system. Furthermore, we re-design a structure with intersection-liked, achieve splitting for one-way pseudospin quantum Hall edge states by using phase modulation. Robustness of the one-way pseudospin-quantum Hall edge states in splitting has been demonstrated as well. Additionally, PEC inserted in transport channel brings optical path difference in waveguide transmission, which would influence splitting for hybrid topological waveguide states in phase difference modulation. This work not only provides a new way for manipulation (i.e., phase modulation) of hybrid topological waveguide states in compatible topological photonic system from distinct topological classes but also has potential in various applications, such as sensing, signal processing, and on-chip communications.

Topological tight binding models on some non-trivial lattices: Union of geometry, flat bands and topology.

We introduce a topological tight binding model based on certain rules that we have formulated to study systems with certain non-trivial bulks (abbreviated as SAB). These rules allows us to study bulks that have twists and branching. We discuss certain cases in the SAB model with different number of bands, exhibiting several interesting physical properties. For every bulk there can be two sets of configurations: the orientable and the non-orientable configuration. The later exhibits several non-trivial physical properties like exact flat bands (exactly at particle hole symmetry level), zero energy states localised in the bulk, topological edge states etc. We then discuss a three band non-orientable SAB model which is easy to visualise and hence can be realized in experiments first. We also investigate the effects of disorder (both chiral symmetry preserving and breaking) in the non-orientable configurations hosting flat bands. We find for chiral symmetry preserving disorders, some of them (non-degenerate flat band) are robust to large disorders while others (degenerate flat band) exhibit an insulator to metal transition beyond certain disorder strength due to band gap closing as a result of the broadening of the zero energy states. For chiral symmetry breaking disorders, in both the cases the zero energy bulk states broaden and close the gap beyond certain critical disorder strength.

Observation of 3D acoustic quantum Hall states.

Quantum Hall effect, the quantized transport phenomenon of electrons under strong magnetic fields, remains one of the hottest research topics in condensed matter physics since its discovery in 2D electronic systems. Recently, as a great advance in the research of quantum Hall effects, the quantum Hall effect in 3D systems, despite its big challenge, has been achieved in the bulk ZrTe5 and Cd3As2 materials. Interestingly, Cd3As2 is a Weyl semimetal, and quantum Hall effect is hosted by the Fermi arc states on opposite surfaces via the Weyl nodes of the bulk, and induced by the unique edge states on the boundaries of the opposite surfaces. However, such intriguing edge state distribution has not yet been experimentally observed. Here, we aim to reveal experimentally the unusual edge states of Fermi arcs in acoustic Weyl system with the aid of pseudo-magnetic field. Benefiting from the macroscopic nature of acoustic crystals, the pseudo-magnetic field is introduced by elaborately designed the gradient on-site energy, and the edge states of Fermi arcs on the boundaries of the opposite surfaces are unambiguously demonstrated in experiments. Our system serves as an ideal and highly tunable platform to explore the Hall physics in 3D system, and has the potential in the application of new acoustic devices.

Anomalous-Chern Steering of Topological Nonreciprocal Guided Waves.

Nonreciprocal topological edge states based on external magnetic bias have been regarded as the last resort for genuine unidirectional wave transport, showing superior robustness over topological states with preserved time-reversal symmetry. However, fast and efficient reconfigurability of their trajectory has remained a formidable challenge due to the difficulty in controlling the spatial distribution of magnetic fields over large areas and short times. Here, this persistent issue is solved by leveraging the rich topology of unitary scattering networks, and achieve fast steering of nonreciprocal topological transport at an interface between a Chern and an anomalous topological insulator, without having to control a magnetic field. Such interface can be drawn by doping the network with scatterers located at the center of each link, whose level of reflection is electrically tuned. With experiments in the GHz range, the possibility to actively steer the way of unidirectional edge states is demonstrated, switching the transmission path thousands of times per second in a fully-robust topological heterostructure. The approach represents a significant step towards the realization of practical reconfigurable topological meta-devices with broken time-reversal symmetry, and their application to future robust communication technologies.

Phase- and temperature-driven chiral topological superfluids on a honeycomb lattice.

The correlated spinful Haldane model exhibits rich topological phases consisting of chiral topological superfluids (TSFs) and topological spin density waves. However, most of previous studies mainly focus on the case with the fixed hopping phase or at zero temperature. In this paper, we study the attractive spinful Haldane model with arbitrary phase at finite temperature. The chiral TSFs with Chern numberC = 2 and 4 emerge driven by the phase and temperature. In particular, the temperature can drive aC = 2 topological superfluid from a trivial normal insulator phase at an appropriate interaction. The bulk topology of all TSFs is uncovered by the Wilson loop method, and confirmed by the responses of edge dislocations.

Valley edge states as bound states in the continuum.

Bound states in the continuum (BICs) are spatially localized states with energy embedded in the continuum spectrum of extended states. The combination of BICs physics and nontrivial band topology theory givs rise to topological BICs, which are robust against disorders and meanwhile, the merit of conventional BICs is attracting wide attention recently. Here, we report valley edge states as topological BICs, which appear at the domain wall between two distinct valley topological phases. The robustness of such BICs is demonstrated. The simulations and experiments show great agreement. Our findings of valley related topological BICs shed light on both BICs and valley physics, and may foster innovative applications of topological acoustic devices.

Entanglement of edge modes in (very) strongly correlated topological insulators.

Identifying topological phases for a strongly correlated theory remains a non-trivial task, as defining order parameters, such as Berry phases, is not straightforward. Quantum information theory is capable of identifying topological phases for a theory that exhibits quantum phase transition with a suitable definition of order parameters that are related to different entanglement measures for the system. In this work, we study entanglement entropy for a coupled SSH model, both in the presence and absence of Hubbard interaction and at varying interaction strengths. For the free theory, edge entanglement acts as an order parameter, which is supported by analytic calculations and numerical (DMRG) studies. We calculate the symmetry-resolved entanglement and demonstrate the equipartition of entanglement for this model which itself acts as an order parameter when calculated for the edge modes. As the DMRG calculation allows one to go beyond the free theory, we study the entanglement structure of the edge modes in the presence of on-site Hubbard interaction for the same model. A sudden reduction of edge entanglement is obtained as interaction is switched on. The explanation for this lies in the change in the size of the degenerate subspaces in the presence and absence of interaction. We also study the signature of entanglement when the interaction strength becomes extremely strong and demonstrate that the edge entanglement remains protected. In this limit, the energy eigenstates essentially become a tensor product state, implying zero entanglement. However, a remnant entropy survives in the non-trivial topological phase, which is exactly due to the entanglement of the edge modes.

Broadband Emission Induced by Band-Edge Carrier Reconfiguration in 2D Hybrid Lead Halide Perovskites.

Broadband emission in hybrid lead halide perovskites (LHPs) has gained significant attention due to its potential applications in optoelectronic devices. The origin of this broadband emission is primarily attributed to the interactions between electrons and phonons. Most investigations have focused on the impact of structural characteristics of LHPs on broadband emission, while neglecting the role of electronic mobility. In this work, the study investigates the electronic origins of broadband emission in a family of 2D LHPs. Through spectroscopic experiments and density functional theory calculations, the study unveils that the electronic states of the organic ligands with conjugate effect in LHPs can extend to the band edges. These band-edge carriers are no longer localized only within the inorganic layers, leading to electronic coupling with molecular states in the barrier and giving rise to additional interactions with phonon modes, thereby resulting in broadband emission. The high-pressure photoluminescence measurements and theoretical calculations reveal that hydrostatic pressure can induce the reconfiguration of band-edge states of charge carriers, leading to different types of band alignment and achieving macroscopic control of carrier dynamics. The findings can provide valuable guidance for targeted synthesis of LHPs with broadband emission and corresponding design of state-of-the-art optoelectronic devices.

The type-I, III nodal ring, type-I, III quadratic nodal point, and Dirac valley phonons in 2D kagome lattices M2C3(M = As, Bi, Cd, Hg, P, Sb, Zn).

Topological phases in kagome systems have garnered considerable interest since the introduction of the colloidal kagome lattice. Our study employs first-principle calculations and symmetry analysis to predict the existence of ideal type-I, III nodal rings (NRs), type-I, III quadratic nodal points (QNPs), and Dirac valley phonons (DVPs) in a collection of two-dimensional (2D) kagome lattices M2C3(M = As, Bi, Cd, Hg, P, Sb, Zn). Specifically, the Dirac valley points (DVPs) can be observed at two inequivalent valleys with Berry phases of +πand-π, connected by edge arcs along the zigzag and armchair directions. Additionally, the QNP is pinned at the Γ point, and two edge states emerge from its projections. Notably, these kagome lattices also exhibit ideal type-I and III nodal rings protected by time inversion and spatial inversion symmetries. Our work examines the various categories of nodal points and nodal ring phonons within the 2D kagome systems and presents a selection of ideal candidates for investigating topological phonons in bosonic systems.

Photonic Molecule Approach to Multiorbital Topology.

The concepts of topology provide a powerful tool to tailor the propagation and localization of the waves. While electrons have only two available spin states, engineered degeneracies of photonic modes provide novel opportunities resembling spin degrees of freedom in condensed matter. Here, we tailor such degeneracies for the array of femtosecond laser written waveguides in the optical range exploiting the idea of photonic molecules: clusters of strongly coupled waveguides. In our experiments, we observe unconventional topological modes protected by the Z3 invariant arising due to the interplay of interorbital coupling and geometric dimerization mechanism. We track multiple topological transitions in the system with the change in the lattice spacings and excitation wavelength. This strategy opens an avenue for designing novel types of photonic topological phases and states.

A brief review of hybrid skin-topological effect.

The finding of non-Hermitian skin effect (NHSE) has revolutionized our understanding of non-Hermitian topological phases, where the usual bulk-boundary correspondence is broken and new topological phases specific to non-Hermitian system are uncovered. Hybrid skin-topological effect (HSTE) is a class of newly discovered non-Hermitian topological states that simultaneously supports skin-localized topological edge states and extended bulk states. Here we provide a brief review of HSTE, starting from different mechanisms that have been used to realize HSTE, including non-reciprocal couplings, onsite gain/loss, and non-Euclidean lattice geometries. We also review some theoretical developments closely related to the HSTE, including the concept of higher-order NHSE, parity-time symmetry engineering, and non-Hermitian chiral skin effect. Finally, we summarize recent experimental exploration of HSTE, including its realization in electric circuits systems, non-Hermitian photonic crystals, and active matter systems. We hope this review can make the concept of hybrid-skin effect clearer and inspire new finding of non-Hermitian topological states in higher dimensional systems.

Observation of D-class topology in an acoustic metamaterial.

Topological materials and metamaterials opened new paradigms to create and manipulate phases of matter with unconventional properties. Topological D-class phases (TDPs) are archetypes of the ten-fold classification of topological phases with particle-hole symmetry. In two dimensions, TDPs support propagating topological edge modes that simulate the elusive Majorana elementary particles. Furthermore, a piercing of π-flux Dirac-solenoids in TDPs stabilizes localized Majorana excitations that can be braided for the purpose of topological quantum computation. Such two-dimensional (2D) TDPs have been a focus in the research frontier, but their experimental realizations are still under debate. Here, with a novel design scheme, we realize 2D TDPs in an acoustic crystal by synthesizing both the particle-hole and fermion-like time reversal symmetries for a wide range of frequencies. The design scheme leverages an enriched unit cell structure with real-valued couplings that emulate the targeted Hamiltonian of TDPs with complex hoppings: A technique that could unlock the realization of all topological classes with passive metamaterials. In our experiments, we realize a pair of TDPs with opposite Chern numbers in two independent sectors that are connected by an intrinsic fermion-like time-reversal symmetry built in the system. We measure the acoustic Majorana-like helical edge modes and visualize their robust topological transport, thus revealing the unprecedented D and DIII class topologies with direct evidence. Our study opens up a new pathway for the experimental realization of two fundamental classes of topological phases and may offer new insights in fundamental physics, materials science, and phononic information processing.

Engineering Gapless Edge States from Antiferromagnetic Chern Homobilayer.

We put forward that stacked Chern insulators with opposite chiralities offer a strategy to achieve gapless helical edge states in two dimensions. We employ the square lattice as an example and elucidate that the gapless chiral and helical edge states emerge in the monolayer and antiferromagnetically stacked bilayer, characterized by Chern number C=-1 and spin Chern number CS=-1, respectively. Particularly, for a topological phase transition to the normal insulator in the stacked bilayer, a band gap closing and reopening procedure takes place accompanied by helical edge states disappearing, where the Chern insulating phase in the monolayer vanishes at the same time. Moreover, EuO is revealed as a suitable candidate for material realization. This work is not only valuable to the research of the quantum anomalous Hall effect but also offers a favorable platform to realize magnetic topologically insulating materials for spintronics applications.

Parity of polaritons in a molecular aggregate coupled to a single-mode cavity.

We investigated the parity of polaritons, particularly the parity of topological polariton states resulting from light fields, in a molecular aggregate with uniform and alternating excitation transfer interaction coupled to a single-mode cavity. We find that all polariton states are with even parity, in terms of parity conservation, and the even-parity edge states of the Su-Schrieffer-Heeger model with alternating excitation transfer interaction induce even-parity topological polariton states. Thus, the odd-parity edge states are almost unaffected. The original odd-parity edge state is then affected with respect to the parity non-conservation case, i.e. the occupation number of the edge states shifts from one edge to another. This result entails the preparation of edge states from the photonic excited states through an adiabatic process.

Topological phases of monolayer and bilayer depleted Lieb lattices.

Existence of nontrivial topological phases in a tight binding Haldane-like model on the depleted Lieb lattice is reported. This two-band model is formulated by considering the nearest-neighbor (NN), next-NN (NNN) and next-NNN hopping terms along with complex phase which breaks the time reversal symmetry of this semi-metallic system. Topological feature of this model is studied along with the presence of sublattice symmetry breaking staggered onsite energy. Combined effect of these two broken symmetries is found crucial for an additional transition between nontrivial and trivial phases. System exhibits two types of phase transitions, say, between two nontrivial phases and nontrivial to trivial phases. Nonzero Chern numbers, existence of Hall plateau and symmetry protected edge states confirm the presence of the nontrivial phases. This two-band system hosts four different types of phases where two are topological. Additionally topological properties of stacked bilayer of the depleted Lieb lattices is also studied with similar Haldane-like Hamiltonian. This four-band system is found to host Chern insulating phases, with higher values of Chern numbers supported by in-gap edge states.

Simultaneous pseudospin and valley topological edge states of elastic waves in phononic crystals made of distorted Kekulé lattices.

Topological metamaterials protected by the spatial inversion symmetry mainly support single type edge state, interpreted by either the quantum valley Hall effect or the quantum spin Hall effect. However, owing to the existence of the complicated couplings and waveform conversions during elastic wave propagation, realizing topologically protected edge states that support both pseudospin and valley degrees of freedom in elastic system remains a great challenge. Here, we propose a two-dimensional Kekulé phononic crystal (PC) that can simultaneously possess pseudospin- and valley-Hall edge states in different frequency bands. By inhomogeneously changing the elliptical direction in a Kekulé lattice of elliptical cylinders, three complete phononic bandgaps exhibiting distinct topological phase transitions can be obtained, one of which supports a pair of pseudospin-Hall edge states and the other hosts valley-Hall edge states in the low and high frequency regime. Furthermore, a sandwiched PC heterostructure and a four-channel cross-waveguide splitter are constructed to achieve selective excitation and topological robust propagation of pseudospin- and valley-momentum locking edge states in a single configuration. These results provide new possibilities for manipulating in-plane bulk elastic waves with both pseudospin and valley degrees of freedom in a single configuration, which has potential applications for multiband and multifunctional waveguiding.

Edge state behavior in a Su-Schrieffer-Heeger like model with periodically modulated hopping.

Su-Schrieffer-Heeger (SSH) model is one of the simplest models to show topological end/edge states and the existence of Majorana fermions. Here we consider a SSH like model both in one and two dimensions where a nearest neighbor hopping features spatially periodic modulations. In the 1D chain, we witness appearance of new in-gap end states apart from a pair of Majorana zero modes (MZMs) when the hopping periodicity go beyond two lattice spacings. The pair of MZMs, that appear in the topological regime, characterize the end modes each existing in either end of the chain. These, however, crossover to both-end end modes for small hopping modulation strength in a finite chain. Contrarily in a 2D SSH model with symmetric hopping that we consider, both non-zero and zero energy topological states appear in a finite square lattice even with a simple staggered hopping, though the zero energy modes disappear in a ribbon configuration. Apart from edge modes, the 2D system also features corner modes as well as modes with satellite peaks distributed non-randomly within the lattice. In both the dimensions, an increase in the periodicity of hopping modulation causes the zero energy Majorana modes to become available for either sign of the modulation. But interestingly with different periodicity for hopping modulations in the two directions, the zero energy modes in a 2D model become rarer and does not appear for all strength and sign of the modulation.

Modification of topological corner excitations at the interplay of boundary geometry and Coulomb interaction.

The effect of Coulomb interaction on the 2D second order topological superconductor is investigated taking into account different geometries of the boundary in the mainframe of the mean-field approximation. The spontaneous symmetry breaking, described earlier in Aksenovet al(2023Phys. Rev.B107125401), is found to be robust against the boundary deformation. Meanwhile, the details of the state with spontaneously broken symmetry is found to be dependent on the specific boundary geometry. Considering different types of the boundary of the 2D system, it is demonstrated that the deviation of the electron density in the broken symmetry state is determined by the position of the zero-dimensional (second-order) excitations with nearly zero energy. The critical value of the Coulomb interaction, at which the transition occurs, is found to be determined by the energy of these excitations, which is non-zero due to overlapping of the wave-functions at different corners.

Observation of dark edge states in parity-time-symmetric quantum dynamics.

Topological edge states arise in non-Hermitian parity-time ([Formula: see text])-symmetric systems, and manifest themselves as bright or dark edge states, depending on the imaginary components of their eigenenergies. As the spatial probabilities of dark edge states are suppressed during the non-unitary dynamics, it is a challenge to observe them experimentally. Here we report the experimental detection of dark edge states in photonic quantum walks with spontaneously broken [Formula: see text] symmetry, thus providing a complete description of the topological phenomena therein. We experimentally confirm that the global Berry phase in [Formula: see text]-symmetric quantum-walk dynamics unambiguously defines topological invariants of the system in both the [Formula: see text]-symmetry-unbroken and -broken regimes. Our results establish a unified framework for characterizing topology in [Formula: see text]-symmetric quantum-walk dynamics, and provide a useful method to observe topological phenomena in [Formula: see text]-symmetric non-Hermitian systems in general.