Bulk Topology Research Articles

Electronic manipulation of magnon topology by chirality injection from boundaries

Magnon bands are known to exhibit nontrivial topology in ordered magnets under suitable conditions, engendering topological phases referred to as magnonic topological insulators. Conventional methods to drive a magnonic topological phase transition are bulk magnetic or thermal operations such as changing the direction of an external magnetic field or varying the temperature of the system, which are undesired in device applications of magnon topology. In this work, we lift the limitation of the magnon topology control on the bulk nonelectronic manipulation by proposing a scheme to manipulate magnonic topological phases by electronic boundary operations of spin chirality injection. More specifically, we consider a ferromagnetic honeycomb lattice and show that a finite spin chirality injected from the boundary of the system via the spin Hall effects introduces a tunable sublattice-symmetry-breaking mass term to the bosonic counterpart of the Haldane model for the Chern insulators and thereby allows us to electronically manipulate the bulk topology of magnons from the boundary. The ``shoulder'' in the thermal Hall conductivity profile is proposed as an experimental probe of the chirality-induced topological phase transition. The scheme for the boundary manipulation of the magnon topology is shown to work for a honeycomb antiferromagnet as well. We envisage that the interfacial chirality injection may offer a nonintrusive electronic means to tune the static and the dynamical bulk properties of general magnetic systems.

Vacancy-superlattice–induced higher-order topological corner states in a Chern insulator

Recent studies have discovered local potentials can induce nontrivial eigenmodes responding to the bulk topology for the system. While previous studies focused on conventional first-order topological states emerging from vacancy superlattice, here we study higher-order topological properties of the vacancy superlattice on a two-dimensional Chern insulator with particle-hole symmetry. The vacancy superlattice with alternate lattice spacings exhibits an emergent second-order topological phase characterized by the nontrivial edge polarization. This topological phase is robust against particle-hole symmetry-preserved perturbations as long as the energy gap remains open for the mid-gap states. Our work generalizes the nontrivial higher-order topological properties to a Chern insulator with local defect vacancies and provides a controllable platform for engineering higher-order topological corner states.

Higher-order topological phases emerging from Su-Schrieffer-Heeger stacking

In this work, we develop a systematic approach of constructing and classifying the model Hamiltonians for two-dimensional (2D) higher-order topological phase with corner zero-energy states (CZESs). Our approach is based on the direct construction of an analytical solution of the CZESs in a series of 2D systems stacking the 1D extended Su-Schrieffer-Heeger (SSH) model, two copies of the two-band SSH model, along with two orthogonal directions. Fascinatingly, our approach not only gives the celebrated Benalcazar-Bernevig-Hughes and 2D SSH models but also reveals a novel model and we name it 2D crossed SSH model. Although these three models exhibit completely different bulk topology, we find that the CZESs can be universally characterized by edge winding number for 1D edge states, attributing to their unified Hamiltonian construction form and edge topology. Remarkably, our principle of constructing CZESs can be readily generalized to 3D and superconducting systems. Our work sheds new light on the theoretical understanding of the higher-order topological phases and paves the way to looking for higher-order topological insulators and superconductors.

Higher-order topological corner states induced solely by on-site potentials with mirror symmetry

Higher-order topological insulators have triggered a great deal of interest because they exhibit nontrivial bulk topology on lower-dimensional boundaries such as corners and hinges. While such interesting phases have been investigated in a plethora of systems by tuning staggered tunneling strength or manipulating existing topological phases, here we show that a higher-order topological phase can be driven solely by mirror-symmetric on-site potentials. We start with a chain model, and we show that it has nonzero edge states due to the lack of internal symmetries such as chiral and particle-hole symmetries. We then generalize the model to two dimensions, and we demonstrate the emergence of topological corner modes. These corner modes are an intrinsic manifestation of nontrivial bulk band topology protected by mirror symmetry, and thus they are robust against symmetry-preserved perturbations. Our model only requires the tuning of on-site energies, and it provides a realistic platform for studying a class of higher-order topological insulators using state-of-the-art experiment techniques.

Spectroscopic ellipsometry-based investigations into the scattering characteristics of topologically distinct photonic stopbands

Topological band theory provides a framework to establish the equivalence/inequivalence of bandgaps in photonic topological insulators. However, experimental discernment of bandgap topological characteristics encounters inherent measurement complexities, particularly beyond the terahertz frequencies. To surmount this difficulty, we resort to the prolific optical technique of spectroscopic ellipsometry and carry out detailed experimental examination of attributes of one-dimensional photonic crystal stopbands and, in consequence, identify an appropriate classifier of the implicit topological characteristics. It is found that governed by the bulk topology, the band edge locations in the dispersion diagram provide a conditional site for the appearance of zeros of a complex reflection ratio. This leads to a selective appearance of topologically robust phase singularities with integer (unity positive) topological charge. We demonstrate that the presence of these phase singularities on either the blue or the red band edges of the stopbands provides us with an experimental marker of their distinctive topological characteristics.

A topological nonlinear parametric amplifier

Topological boundary states are well localized eigenstates at the boundary between two different bulk topologies. As long as bulk topology is preserved, the topological boundary mode will endure. Here, we report topological nonlinear parametric amplification of light in a dimerized coupled waveguide system based on the Su-Schrieffer-Heeger model with a domain wall. The good linear transmission properties of the topological waveguide arising from the strong localization of light to the topological boundary is demonstrated through successful high-speed transmission of 30 Gb/s non-return-to-zero and 56 Gb/s pulse amplitude 4-level data. The strong localization of a co-propagating pump and probe to the boundary waveguide is harnessed for efficient, low power optical parametric amplification and wavelength conversion. A nonlinear tuning mechanism is shown to induce chiral symmetry breaking in the topological waveguide, demonstrating a pathway in which Kerr nonlinearities may be applied to tune the topological boundary mode and control the transition to bulk states.

Topological corner states in graphene by bulk and edge engineering

Two-dimensional higher-order topology is usually studied in (nearly) particle-hole symmetric models, so that an edge gap can be opened within the bulk one. But more often the edge anticrossing deviates even into the bulk, where corner states are difficult to pinpoint. We address this problem in a graphene-based ${\mathbb{Z}}_{2}$ topological insulator with spin-orbit coupling and in-plane magnetization both originating from substrates through a Slater-Koster multiorbital model. The gapless helical edge modes cross inside the bulk, where the magnetization-induced edge gap is also located. After demonstrating its second-order nontriviality in bulk topology by a series of evidence, we show that a difference in bulk-edge on-site energy can adiabatically tune the position of the crossing/anticrossing of the edge modes to be inside the bulk gap. This can help unambiguously identify two pairs of topological corner states with nonvanishing energy degeneracy for a rhombic flake. We further find that the obtuse-angle pair is more stable than the acute-angle one.

Pressure-Tuning Superconductivity in Noncentrosymmetric Topological Materials ZrRuAs.

Recently, the hexagonal phase of ternary transition metal pnictides TT'X (T = Zr, Hf; T' = Ru; X = P, As), which are well-known noncentrosymmetric superconductors, were predicted to host nontrivial bulk topology. In this work, we systematically investigate the electronic responses of ZrRuAs to external pressure. At ambient pressure, ZrRuAs show superconductivity with Tc ~ 7.74 K, while a large upper critical field ~ 13.03 T is obtained for ZrRuAs, which is comparable to the weak-coupling Pauli limit. The resistivity of ZrRuAs exhibits a non-monotonic evolution with increasing pressure. The superconducting transition temperature Tc increases with applied pressure and reaches a maximum value of 7.93 K at 2.1 GPa, followed by a decrease. The nontrivial topology is robust and persists up to the high-pressure regime. Considering both robust superconductivity and intriguing topology in this material, our results could contribute to studies of the interplay between topological electronic states and superconductivity.

Nonsymmorphic chiral symmetry and solitons in the Rice-Mele model

The Rice-Mele model has two topological and spatially-inversion symmetric phases, namely the Su-Schrieffer-Heeger (SSH) phase with alternating hopping only, and the charge-density-wave (CDW) phase with alternating energies only. The chiral symmetry of the SSH phase is robust in position space, so that it is preserved in the presence of the ends of a finite system and of textures in the alternating hopping. However, the chiral symmetry of the CDW wave phase is nonsymmorphic, resulting in a breaking of the bulk topology by an end or a texture in the alternating energies. We consider the presence of solitons (textures in position space separating two degenerate ground states) in finite systems with open boundary conditions. We identify the parameter range under which an atomically-sharp soliton in the CDW phase supports a localized state which lies within the band gap, and we calculate the expectation value $p_y$ of the nonsymmorphic chiral operator for this state, and the soliton electric charge. As the spatial extent of the soliton increases beyond the atomic limit, the energy level approaches zero exponentially quickly or inversely proportionally to the width, depending on microscopic details of the soliton texture. In both cases, the difference of $p_y$ from one is inversely proportional to the soliton width, while the charge is independent of the width. We investigate the robustness of the soliton level in the presence of disorder and sample-to-sample parameter variations, comparing with a single soliton level in the SSH phase with an odd number of sites.

Generic theory of characterizing topological phases under quantum slow dynamics

Dynamical characterization of equilibrium topological phases has attracted considerable attention in recent years. In this paper, we make a thorough exploration of the non-adiabatic characterization of topological phases under slow quench protocol. We first propose an exactly solvable multi-state Landau-Zener model that can be directly applied to the non-adiabatic slow quench dynamics of topological systems. Then we present two different schemes to characterize the bulk topology of the system based on the so called spin inversion surface. The first one needs least number of quenching processes, but requires to measure the gradients of time-averaged spin-polarization on the SIS. The second one only needs to measure the value of time-averaged spin-polarization on the SIS, thus makes it possible to directly characterize the topological phases by introducing an extra quenching process. Moreover, high-order SIS or band inversion surface (BIS) relying on the dimension reduction approach, is also generalized to the above two different characterization schemes. One can extract the topological invariant from pairs of points with opposite signs both on the 0D highest order BIS and on the 0D highest order SIS, which greatly simplifies the measurement strategy and characterization process. In a word, superior to the sudden quench protocol, we demonstrate that the topological invariant can be captured not only by the topological information on BIS, but also on the SIS. In particular, direct characterization of topological phases based on BIS and SIS can be realized.

Higher-order topological semimetals and nodal superconductors with an order-two crystalline symmetry

Using a systematic relation between topological gapless phases in three dimensions and topological gapped phases in two dimensions, we identify four types of higher-order topological semimetals or nodal superconductors (HOTS), hosting (i) flat zero-energy "Fermi arcs" at crystal hinges, (ii) flat zero-energy hinge arcs coexisting with surface Dirac cones, (iii) chiral or helical hinge modes, or (iv) flat zero-energy hinge arcs connecting nodes only at finite momentum. Bulk-boundary correspondence relates the hinge states to the bulk topology protecting the nodal point or loop. We classify all HOTS for all tenfold-way classes with an order-two crystalline (anti-)symmetry, such as mirror, twofold rotation, or inversion.

Second-order topological insulator in periodically driven optical lattices.

The higher-order topological insulator (HOTI) is a new type of topological system which has special bulk-edge correspondence compared with conventional topological insulators. In this work, we propose a scheme to realize Floquet HOTI with ultracold atoms in optical lattices. With the combination of periodically spin-dependent driving of the superlattices and a long-range coupling term, a Floquet second-order topological insulator with four zero-energy corner states emerges, whose Wannier bands are gapless and exhibit interesting bulk topology. Furthermore, the nearest-neighbor anisotropic coupling term also induced other intriguing topological phenomena, e.g. non-topologically protected corner states and topological semimetal for two different types of lattice structures respectively. Our scheme may give insight into the construction of different types of higher-order topological insulators in synthetic systems. It also provides an experimentally feasible platform to research the relations between different types of topological states and may have a wide range of applications in future.

Using Nanosphere Embedding to Probe the Surface and Bulk Relaxation in Vitrimers.

The relaxation behavior of vitrimers that is dominated by chemical exchange reactions plays a critical role in vitrimer processing and applications such as self-healing, welding, and others involving the dynamic nature of the vitrimers. Here, we use atomic force microscopy to image embedding of gold nanospheres into epoxy-based vitrimers to assess the surface and bulk relaxation in this material. The results show that even at temperatures well below the bulk topology freezing transition temperature, the nanospheres embed into the vitrimers. The activation energy for the relaxation at the surface and in bulk were estimated in the single measurement, and the former is found to be much lower than the latter. The increase in the surface relaxation is attributed to a combination of an acceleration effect on relaxation by network defects and a decrease in the number of intermolecular exchanges at the surface.

Role of topology and symmetry for the edge currents of a two-dimensional superconductor

The bulk-boundary correspondence guarantees topologically protected edge states in a two-dimensional topological superconductor. Unlike in topological insulators, these edge states are, however, not connected to a quantized (spin) current as the electron number is not conserved in a Bogolyubov--de Gennes Hamiltonian. Still, edge currents are in general present. Here we use the two-dimensional Rashba system as an example to systematically analyze the effect symmetry reductions have on the order-parameter mixing and the edge properties in a superconductor of Altland-Zirnbauer class DIII (time-reversal-symmetry preserving) and D (time-reversal-symmetry breaking). In particular, we employ both Ginzburg-Landau and microscopic modeling to analyze the bulk superconducting properties and edge currents appearing in a strip geometry. We find edge (spin) currents independent of bulk topology and associated topological edge states which evolve continuously even when going through a phase transition into a topological state. Our findings emphasize the importance of symmetry over topology for the understanding of the nonquantized edge currents.

Extrinsic topology of Floquet anomalous boundary states in quantum walks

Bulk-boundary correspondence is a fundamental principle for topological phases where bulk topology determines gapless boundary states. On the other hand, it has been known that corner or hinge modes in higher order topological insulators may appear due to "extrinsic" topology of the boundaries even when the bulk topological numbers are trivial. In this paper, we find that Floquet anomalous boundary states in quantum walks have similar extrinsic topological natures. In contrast to higher order topological insulators, the extrinsic topology in quantum walks is manifest even for first-order topological phases. We present the topological table for extrinsic topology in quantum walks and illustrate extrinsic natures of Floquet anomalous boundary states in concrete examples.

Observation of symmetry-protected corner states in breathing honeycomb topolectrical circuits

We report the experimental observation of second-order corner states in a two-dimensional breathing honeycomb topolectrical circuit with sixfold rotational symmetry C6 through voltage measurements. The topological corner states originate from the nontrivial bulk topology, which can be characterized by the topological invariant associated with the rotation eigenspectrum. We confirm two types of corner states, both originate from the C6 symmetry, while one of them is specially pinned to zero admittance because of the emerging chiral symmetry protection. We then examine the robustness of zero modes in the presence of next-nearest-neighbor hopping terms that destroy chiral symmetry but still preserve C6 symmetry. Our work provides a paradigm in circuit systems to study the exotic topological physics.

Dynamics of the Entanglement Zero Modes in the Haldane Model under a Quantum Quench

We investigate evolution of entanglement spectra of the Haldane model for Chern insulators upon a sudden quench within the same topological phase. In particular, we focus on the location of the entanglement spectrum crossing, which signifies the bulk topology. It is shown that the coplanarity condition for the pseudomagnetic field of the model, which can be used to determine the crossing in the equilibrium case, needs to be relaxed. We analytically derive the non-equilibrium condition with the help of an edge-state wave function ansatz and a dynamically induced length-scale cutoff. With spectral analyses, it is realized that the oscillatory behavior of the crossing is dominated by the interband excitations at the van Hove singularities.

Quantum emulation of topological magneto-optical effects using ultracold atoms

Magneto-optical effect is a fundamental but broad concept in magnetic mediums. Here we propose a scheme for its quantum emulation using ultracold atoms. By representing the light-medium interaction in the quantum-emulation manner, the artificial magneto-optical effect emerges under an entirely different mechanism from the conventional picture. The underlying polarization state extracted in the synthetic dimension displays a different response to various experimental setups. Notably, the magneto-optical rotation is related to the bulk topology in synthetic dimensions, and thus provides an unambiguous evidence for the desired topological magneto-optical effect, which has not been developed hitherto in ultracold atoms. This scheme is simple and feasible, and can be realized by current experimental techniques. The implementation of the scheme is able to offer an intriguing platform for exploring topological magneto-optical effects and associated physics.

Topological triple phase transition in non-Hermitian Floquet quasicrystals

Phase transitions connect different states of matter and are often concomitant with the spontaneous breaking of symmetries. An important category of phase transitions is mobility transitions, among which is the well known Anderson localization1, where increasing the randomness induces a metal–insulator transition. The introduction of topology in condensed-matter physics2–4 lead to the discovery of topological phase transitions and materials as topological insulators5. Phase transitions in the symmetry of non-Hermitian systems describe the transition to on-average conserved energy6 and new topological phases7–9. Bulk conductivity, topology and non-Hermitian symmetry breaking seemingly emerge from different physics and, thus, may appear as separable phenomena. However, in non-Hermitian quasicrystals, such transitions can be mutually interlinked by forming a triple phase transition10. Here we report the experimental observation of a triple phase transition, where changing a single parameter simultaneously gives rise to a localization (metal–insulator), a topological and parity–time symmetry-breaking (energy) phase transition. The physics is manifested in a temporally driven (Floquet) dissipative quasicrystal. We implement our ideas via photonic quantum walks in coupled optical fibre loops11. Our study highlights the intertwinement of topology, symmetry breaking and mobility phase transitions in non-Hermitian quasicrystalline synthetic matter. Our results may be applied in phase-change devices, in which the bulk and edge transport and the energy or particle exchange with the environment can be predicted and controlled.

Topology by Dissipation: Majorana Bosons in Metastable Quadratic Markovian Dynamics.

Majorana bosons, that is, tight bosonic analogs of the Majorana fermionic quasiparticles of condensed-matter physics, are forbidden for gapped free bosonic matter within a standard Hamiltonian scenario. We show how the interplay between dynamical metastability and nontrivial bulk topology makes their emergence possible in noninteracting bosonic chains undergoing Markovian dissipation. This leads to a distinctive form of topological metastability, whereby a conserved Majorana boson localized on one edge is paired, in general, with a symmetry generator localized on the opposite edge. We argue that Majorana bosons are robust against disorder and identifiable by signatures in the zero-frequency steady-state power spectrum. Our results suggest that symmetry-protected topological phases for free bosons may arise in transient metastable regimes, which persist over practical timescales.