Second-order Topological Insulator Research Articles

Topological quantum phase transitions on the breathing kagomé lattice

We explore extensively topological quantum phase transitions (TQPTs) of the breathing kagomé lattice model in the presence of staggered fluxes. We obtain rich topological phases, including the Chern insulator (CI) and the second-order topological insulator (SOTI) phases, by tuning the dimerized hopping parameter t 1′ and the staggered-flux parameter ϕ. The CI phases can be identified on the basis of the chiral edge states and the non-zero Chern numbers. However, in sharp contrast to the CI phases, the SOTI phases are characterized by the robust corner states and the quantized polarizations. In addition, we explore the TQPTs considering the next-nearest-neighbor hopping parameter t 2. We demonstrate the existence of two-dimensional SOTIs with broken time-reversal symmetry and reveal the TQPTs between the CIs and the SOTIs.

Four dimensional second-order topological insulator based on a synthetic plasmonic metasurface.

It is possible to explore higher dimensional topological properties in lower dimensional structures by introducing additional synthetic dimensions. In this Letter, we construct a four-dimensional (4D) second-order topological insulator using gradient nanoparticle arrays arranged in a periodic lattice. The nanoparticle array has spatially varying inter-particle distance along x and y directions, which can be regarded as two synthetic dimensions. Different from higher-order topological insulators in classical wave systems, the higher-order topological states in this 4D system are protected by a pair of first Chern numbers in two-dimensional (2D) subspaces instead of by the quantized 2D Zak phases. It is shown that there exist (4-1)- and (4-2)-dimensional boundary states for both transverse and longitudinal collective resonant modes, which provides new, to the best of our knowledge, mechanisms for light confinement and control in such a plasmonic superlattice.

Surface theory of a second-order topological insulator beyond the Dirac approximation

We study the surface states and chiral hinge states of a 3D second-order topological insulator in the presence of an external magnetic gauge field. Surfaces pierced by flux host Landau levels, while surfaces parallel to the applied field are not significantly affected. The chiral hinge modes mediate spectral flow between neighbouring surfaces. As the magnetic field strength is increased, the surface Landau quantization deviates from that of a massive Dirac cone. Quantitatively, the $n = 0$ Landau level falls inside the surface Dirac gap, and not at the gap edge. The $n \ne 0$ levels exhibit a further, qualitative discrepancy: while the massive Dirac cone is expected to produce pairs of levels ($\pm n$) which are symmetric around zero energy, the $n$ and $-n$ levels become asymmetric in our lattice model -- one of the pair may even be absent from the spectrum, or hybridized with the continuum. In order to resolve the issue, we extend the standard 2D massive Dirac surface theory, by including additional Hamiltonian terms at $\mathcal{O} (k^2)$. While these terms do not break particle-hole symmetry in the absence of magnetic field, they lead to the aforementioned Landau level asymmetry once the magnetic field is applied. We argue that similar $\mathcal{O}(k^2)$ correction terms are generically expected in lattice models containing gapped Dirac fermions, using the BHZ model of a 2D topological insulator as an example.

Room-Temperature Topological Phase Transition in Quasi-One-Dimensional Material Bi4I4

Quasi-one-dimensional (1D) materials provide a superior platform for characterizing and tuning topological phases for two reasons: i) existence for multiple cleavable surfaces that enables better experimental identification of topological classification, and ii) stronger response to perturbations such as strain for tuning topological phases compared to higher dimensional crystal structures. In this paper, we present experimental evidence for a room-temperature topological phase transition in the quasi-1D material Bi$_4$I$_4$, mediated via a first order structural transition between two distinct stacking orders of the weakly-coupled chains. Using high resolution angle-resolved photoemission spectroscopy on the two natural cleavable surfaces, we identify the high temperature $\beta$ phase to be the first weak topological insulator with gapless Dirac cones on the (100) surface and no Dirac crossing on the (001) surface, while in the low temperature $\alpha$ phase, the topological surface state on the (100) surface opens a gap, consistent with a recent theoretical prediction of a higher-order topological insulator beyond the scope of the established topological materials databases that hosts gapless hinge states. Our results not only identify a rare topological phase transition between first-order and second-order topological insulators but also establish a novel quasi-1D material platform for exploring unprecedented physics.

Graphyne as a second-order and real Chern topological insulator in two dimensions

Higher-order topological phases and real topological phases are two emerging topics in topological states of matter, which have been attracting considerable research interest. However, it remains a challenge to find realistic materials that can realize these exotic phases. Here, based on first-principles calculations and theoretical analysis, we identify graphyne, the representative of the graphyne-family carbon allotropes, as a two-dimensional (2D) second-order topological insulator and a real Chern insulator. We show that graphyne has a direct bulk band gap at the three $M$ points, forming three valleys. The bulk bands feature a double band inversion, which is characterized by the real Chern number enabled by the spacetime-inversion symmetry. The real Chern number is explicitly evaluated by both the Wilson loop method and the parity approach. We demonstrate that a nontrivial real Chern number for a 2D system dictates the existence of Dirac-type edge bands and the topological corner states. Furthermore, we find that the topological phase transition in graphyne from the real Chern insulator to a trivial insulator is mediated by a 2D Weyl semimetal phase. The robustness of the corner states against symmetry breaking and possible experimental detection methods are discussed.

Bulk-hinge correspondence and three-dimensional quantum anomalous Hall effect in second-order topological insulators

The chiral hinge modes are the key feature of a second order topological insulator in three dimensions. Here we propose a quadrupole index in combination of a slab Chern number in the bulk to characterize the flowing pattern of chiral hinge modes along the hinges at the intersection of the surfaces of a sample. We further utilize the topological field theory to demonstrate the correspondent connection of the chiral hinge modes to the quadrupole index and the slab Chern number, and present a picture of three-dimensional quantum anomalous Hall effect as a consequence of chiral hinge modes. The two bulk topological invariants can be measured in electric transport and magneto-optical experiments. In this way we establish the bulk-hinge correspondence in a three-dimensional second order topological insulator.

Lattice topological edge and corner modes of photonic crystal slabs

Surface lattice modes, generated by the evanescent coupling between localized modes of optical resonators arranged in a two-dimensional (2D) array, generally exhibit remarkable optical response beyond the single photonic particle. Here, by employing the lattice mode concept, we demonstrate that lattice topological edge and corner modes can be achieved in properly designed photonic crystal (PhC) slabs. Such slabs consist of an array of finite-sized second-order topological insulators mimicking the 2D Su-Schrieffer-Heeger model. The proposed lattice edge and corner modes emerge within the topological band gap of the PhC slab, which dictates their topological nature. In particular, the band diagram of the lattice corner modes shows that they possess non-degenerate eigenfrequencies and dispersive bands. In addition, we show that the eigenfrequency of the lattice topological modes can be shifted by tuning the intercell and/or intracell optical coupling. Finally, by finely tuning the geometric parameters of the slab, we realize a lattice corner mode possessing flatband dispersion characteristics. Our study can find applications to topological lasing, nonlinearity enhancement, and slow-light effects in topological photonic systems.

Higher-Order Fabry-Pérot Interferometer from Topological Hinge States.

We propose an intrinsic three-dimensional Fabry-Pérot type interferometer, coined "higher-order interferometer," that is based on the chiral hinge states of second-order topological insulators and cannot be mapped to an equivalent two-dimensional setting because of higher-order topological obstructions. Quantum interference patterns in the two-terminal conductance of this interferometer are controllable not only by tuning the strength but also, particularly, by rotating the direction of the magnetic field applied perpendicularly to the transport direction. Remarkably, the conductance exhibits a characteristic beating pattern with multiple frequencies depending on the field strength and direction in a unique fashion. Our novel interferometer thus provides feasible and robust magnetotransport signatures for hinge states of higher-order topological insulators.

Chiral hinge magnons in second-order topological magnon insulators

When interacting spins in condensed matter order ferromagnetically, their ground-state wave function is topologically trivial. Nonetheless, in two dimensions, ferromagnets can support spin excitations with nontrivial topology, an exotic state known as topological magnon insulator (TMI). Here, we theoretically unveil and numerically confirm a ferromagnetic state in three dimensions dubbed second-order TMI, whose hallmarks are excitations at its hinges, where facets intersect. Since ferromagnetism naturally comes with broken time-reversal symmetry, the hinge magnons are chiral, rendering backscattering impossible. Hence, they trace out three-dimensional paths about the sample unimpeded by defects and are topologically protected by the spectral gap. They are remarkably robust against disorder and highly tunable by atomic-level engineering of the sample termination. We predict that a van der Waals heterostructure built from chromium trihalide and transition metal dichalcogenide monolayers exhibits second-order magnon topology. Our findings empower magnonics, the harnessing of spin waves as information carriers, with the tools of higher-order topology, a promising route to combine low-energy information transfer free of Joule heating with three-dimensional vertical integration.

Corner states in a second-order mechanical topological insulator

Demonstration of topological boundary modes in elastic systems has attracted a great deal of attention over the past few years due to its unique protection characteristic. Recently, second-order topological insulators have been proposed in manipulating the topologically protected localized states emerging only at corners. Here, we numerically and experimentally study corner states in a two-dimensional phononic crystal, namely a continuous elastic plate with embedded bolts in a hexagonal pattern. We create interfacial corners by adjoining trivial and non-trivial topological configurations. Due to the rich interaction between the bolts and the continuous elastic plate, we find a variety of corner states of and devoid of topological origin. Strikingly, some of the corner states are not only highly-localized but also tunable. Taking advantage of this property, we experimentally demonstrate asymmetric corner localization in a Z-shaped domain wall. This finding could create interest in exploration of tunable corner states for the use of advanced control of wave localization.

Structural-Disorder-Induced Second-Order Topological Insulators in Three Dimensions.

Higher-order topological insulators are established as topological crystalline insulators protected by crystalline symmetries. One celebrated example is the second-order topological insulator in three dimensions that hosts chiral hinge modes protected by crystalline symmetries. Since amorphous solids are ubiquitous, it is important to ask whether such a second-order topological insulator can exist in an amorphous system without any spatial order. Here, we predict the existence of a second-order topological insulating phase in an amorphous system without any crystalline symmetry. Such a topological phase manifests in the winding number of the quadrupole moment, the quantized longitudinal conductance, and the hinge states. Furthermore, in stark contrast to the viewpoint that structural disorder should be detrimental to the higher-order topological phase, we remarkably find that structural disorder can induce a second-order topological insulator from a topologically trivial phase in a regular geometry. We finally demonstrate the existence of a second-order topological phase in amorphous systems with time-reversal symmetry.

Experimental demonstration of a reconfigurable acoustic second-order topological insulator using condensed soda cans array

Traditional topological insulators support the topologically protected boundary states that are one dimension lower than the system itself. Recently, higher-order topological insulators have received increasing attention in the field of acoustic wave manipulation due to their unique bulk-boundary correspondence principle, hosting both gapped edge states and in-gap corner states simultaneously. However, for most of the topological acoustic systems, the lack of reconfigurability and the inevitable outer trivial regions with considerable thickness restrict the potential applications of acoustic topological insulators. Here, we experimentally demonstrate a reconfigurable condensed acoustic second-order topological insulator in free space by using subwavelength soda cans whose side length is significantly reduced to 1.89 times of the corresponding wavelength. The topological nontrivial phase is introduced through tunably modulating the interval between cans. Without the typically required outer trivial regions, we observe the topological corner states at the corner of the finite structures in both simulations and experiments. Furthermore, the robustness against the defects induced by dislocations and deformations is discussed. We foresee that the proposal may facilitate the application potentials of topological acoustics in low-frequency sound manipulations.

Robustness of helical hinge states of weak second-order topological insulators

Robustness of helical hinge states of three-dimensional weak second-order topological insulators (WSOTIs) against disorders is studied. The pure WSOTI is obtained from a weak ${\mathbb{Z}}_{2}$ first-order topological insulator through a surface band inversion. Both bulk states and surface states in the WSOTI are gapped, and in-gap valley-momentum locked helical hinge states are topologically protected by the surface valley-Chern number. In the presence of weak disorders, helical hinge states are robust against disorders while the quantized conductance of the states is fragile due to the intervalley scattering. As disorder increases, the system undergoes a series of quantum phase transitions: from the WSOTI to the weak first-order topological insulator, then to a diffusive metal and finally to an Anderson insulator. Our results thus fully establish the WSOTI phase as a genuine state of matters and open a door for the second-order valleytronics that allows one to control the valley degree of freedom through helical hinge states.

Transport induced dimer state from topological corner states

Recently, a new type of second-order topological insulator has been theoretically proposed by introducing an in-plane Zeeman field into the Kane-Mele model in the two-dimensional honeycomb lattice. A pair of topological corner states arise at the corners with obtuse angles of an isolated diamond-shaped flake. To probe the corner states, we study their transport properties by attaching two leads to the system. Dressed by incoming electrons, the dynamic corner state is very different from its static counterpart. Resonant tunneling through the dressed corner state can occur by tuning the in-plane Zeeman field. At the resonance, the pair of spatially well separated and highly localized corner states can form a dimer state, whose wavefunction extends almost the entire bulk of the diamond-shaped flake. By varying the Zeeman field strength, multiple resonant tunneling events are mediated by the same dimer state. This re-entrance effect can be understood by a simple model. These findings extend our understanding of dynamic aspects of the second-order topological corner states.

Higher-order topological semimetal in acoustic crystals.

The notion of higher-order topological insulators has endowed materials with topological states beyond the first order. Particularly, a three-dimensional (3D) higher-order topological insulator can host topologically protected 1D hinge states, referred to as the second-order topological insulator, or 0D corner states, referred to as the third-order topological insulator. Similarly, a 3D higher-order topological semimetal can be envisaged if it hosts states on the 1D hinges. Here we report the realization of a second-order topological Weyl semimetal in a 3D-printed acoustic crystal, which possesses Weyl points in 3D momentum space, 2D Fermi arc states on surfaces and 1D gapless states on hinges. Like the arc surface states, the hinge states also connect the projections of the Weyl points. Our experimental results evidence the existence of the higher-order topological semimetal, which may pave the way towards innovative acoustic devices.

Higher-Order Band Topology in Twisted Moiré Superlattice.

The two-dimensional (2D) twisted bilayer materials with van der Waals coupling have ignited great research interests, paving a new way to explore the emergent quantum phenomena by twist degree of freedom. Generally, with the decreasing of twist angle, the enhanced interlayer coupling will gradually flatten the low-energy bands and isolate them by two high-energy gaps at zero and full filling, respectively. Although the correlation and topological physics in the low-energy flat bands have been intensively studied, little information is available for these two emerging gaps. In this Letter, we predict a 2D second-order topological insulator (SOTI) for twisted bilayer graphene and twisted bilayer boron nitride in both zero and full filling gaps. Employing a tight-binding Hamiltonian based on first-principles calculations, three unique fingerprints of 2D SOTI are identified, that is, nonzero bulk topological index, gapped topological edge state, and in-gap topological corner state. Most remarkably, the 2D SOTI exists in a wide range of commensurate twist angles, which is robust to microscopic structure disorder and twist center, greatly facilitating the possible experimental measurement. Our results not only extend the higher-order band topology to massless and massive twisted moiré superlattice, but also demonstrate the importance of high-energy bands for fully understanding the nontrivial electronics.

Floquet second-order topological insulators in non-Hermitian systems

Second-order topological insulator (SOTI) is featured with the presence of $(d-2)$-dimensional boundary states in $d$-dimension systems. The non-Hermiticity induced breakdown of bulk-boundary correspondence (BBC) and the periodic driving on systems generally obscure the description of non-Hermitian SOTI. To prompt the applications of SOTIs, we explore the role of periodic driving in controllably creating exotic non-Hermitian SOTIs both for 2D and 3D systems. A scheme to retrieve the BBC and a complete description to SOTIs via the bulk topology of such nonequilibrium systems are proposed. It is found that rich exotic non-Hermitian SOTIs with a widely tunable number of 2D corner states and 3D hinge states and a coexistence of the first- and second-order topological insulators are induced by the periodic driving. Enriching the family of topological phases, our result may inspire the exploration to apply SOTIs via tuning the number of corner/hinge states by the periodic driving.

Real-space observation of topological invariants in 2D photonic systems.

Topological materials are capable of inherently robust transport and propagation of physical fields against disorder and perturbations, holding the promise of revolutionary technologies in a wide spectrum. Higher-order topological insulators are recently predicted as topological phases beyond the standard bulk-edge correspondence principle, however, their topological invariants have been proven very challenging to observe, even not possible yet by indirect ways. Here, we demonstrate theoretically and experimentally that the topological invariants in two-dimensional systems can be directly revealed in real space by measuring single-photon bulk dynamics. By freely writing photonic lattices with femtosecond laser, we construct and identify the predicted second-order topological insulators, as well as first-order topological insulators with fractional topological winding number. Furthermore, we show that the accumulation and statistics on individual single-particle registrations can eventually lead to the same results of light waves, despite the fact that the development of topological physics was originally based on wave theories, sharing the same spirit of wave-particle nature in quantum mechanics. Our results offer a direct fashion of observing topological phases in two-dimensional systems and may inspire topologically protected artificial devices in high-order topology, high-dimension and quantum regime.

A proposal of strong and weak phases in second-order topological insulators

Recently, it has been discovered theoretically that there exist novel types of 3-dimensional topological insulators (TIs) whose gapless states manifest themselves at 1-dimensional hinges. They are called second-order topological insulators (SOTIs). Most mathematical models of SOTIs compose of conventional strong topological insulators (STIs) and additional mass terms In this paper, we investigate whether the models made based on weak topological insulators (WTIs) can have hinge states as the same as those made based on the STIs. We found that the models based on WTIs have only trivial index determined by the Wilson loop formalism unlike the models based on STIs. However, they can be regarded as stacking of 2-dimensional SOTIs along the z direction. Thus, we propose that there are topologically three different phases in our considering system: ordinary insulator, strong SOTI, and weak SOTI phases. This classification suggests the existence of other topological invariants besides the index. Finally, we propose new indices which can distinguish weak SOTIs from ordinary insulators.

Higher-order non-Hermitian skin effect

The non-Hermitian skin effect is a unique feature of non-Hermitian systems, in which an extensive number of boundary modes appear under the open boundary conditions. Here, we discover higher-order counterparts of the non-Hermitian skin effect that exhibit new boundary physics. In two-dimensional systems with the system size $L \times L$, while the conventional (first-order) skin effect accompanies $O\,( L^{2} )$ skin modes, the second-order skin effect accompanies $O\,( L )$ corner skin modes. This also contrasts with Hermitian second-order topological insulators, in which only $O\,( 1 )$ corner zero modes appear. Moreover, for the third-order skin effect in three dimensions, $O\,( L )$ corner skin modes appear from all $O\,( L^{3} )$ modes. We demonstrate that the higher-order skin effect originates from intrinsic non-Hermitian topology protected by spatial symmetry. We also show that it accompanies the modification of the non-Bloch band theory in higher dimensions.