Distinct Topological Phases Research Articles

Dirac fermions and topological phases in magnetic topological insulator films

We develop a Dirac fermion theory for topological phases in magnetic topological insulator films. The theory is based on exact solutions of the energies and the wave functions for an effective model of the three-dimensional topological insulator (TI) film. It is found that the TI film consists of a pair of massless or massive Dirac fermions for the surface states, and a series of massive Dirac fermions for the bulk states. The massive Dirac fermion always carries zero or integer quantum Hall conductance when the valence band is fully occupied while the massless Dirac fermion carries a one-half quantum Hall conductance when the chemical potential is located around the Dirac point for a finite range. The magnetic exchange interaction in the magnetic layers in the film can be used to manipulate either the masses or chirality of the Dirac fermions and gives rise to distinct topological phases, which cover the known topological insulating phases, such as the quantum anomalous Hall effect, quantum spin Hall effect and axion effect, and also the novel topological metallic phases, such as the half-quantized Hall effect, half quantum mirror Hall effect, and metallic quantum anomalous Hall effect.

Ideal spin-orbit-free Dirac semimetal and diverse topological transitions in Y8CoIn3 family

Topological semimetals, known for their intriguing properties arising from band degeneracies, have garnered significant attention. However, the discovery of a material realization and the detailed characterization of spinless Dirac semimetals have not yet been accomplished. Here, we propose from first-principles calculations that the RE8CoX3 group (RE = rare earth elements, X = Al, Ga, or In) contains ideal spinless Dirac semimetals whose Fermi surfaces are fourfold degenerate band-crossing points (without including spin degeneracy). Despite the lack of space inversion symmetry in these materials, Dirac points are formed on the rotation-symmetry axis due to accidental degeneracies of two bands corresponding to different 2-dimensional irreducible representations of the C6v group. We also investigate, through first-principles calculations and effective model analysis, various phase transitions caused by lattice distortion or elemental substitutions from the Dirac semimetal phase to distinct topological semimetallic phases such as nonmagnetic linked-nodal-line and Weyl semimetals (characterized by the second Stiefel–Whitney class) and ferromagnetic Weyl semimetals.

Phase diagrams and topological mixed edge states in silicene with intrinsic and extrinsic Rashba effects

We investigate the topological phases and corresponding edge states in silicene with fixed intrinsic and extrinsic Rashba spin-orbit couplings. The results show that silicene can exhibit four distinct topological phases, such as two types of valley-polarized quantum anomalous Hall effect (QAHE) and two types of QAHE that can be effectively controlled by the exchange field and electric field. In addition, these topological phases only correspond to the valley edge states distributed at the boundary of silicene. In particular, when the off-resonant circularly polarized (ORCP) light is further considered, silicene can host richer topological phases that are characterized by the Chern numbers from C=−3 to C=3, where the corresponding mixed edge states—where the valley and nonvalley edge states could coexist—can remarkably appear. More interestingly, the special higher Chern numbers C=±3 determine two valley edge states and one nonvalley edge state, with the right-moving or left-moving direction. These peculiar edge states are attributed to the ORCP light destroying the collapse of the Dirac cones that the valley edge states bridge. Additionally, silicene can experience multiple edge-state transitions for identical Chern numbers C=±2 between the valley edge state, nonvalley edge state, and two types of mixed edge state, under the ORCP light. This work may provide a platform and point of view for investigating other forms of topological edge state and is beneficial for designing multifunctional topological devices. Published by the American Physical Society 2024

Reconfigurable topological gradient metamaterials and potential applications

Elastic topological metamaterials are innovative fields of the fusion of physics and mechanics. Through novel microstructural designs, elastic topological metamaterials provide a pioneering approach to precisely manipulate elastic waves, maintaining robustness and efficiency in wave propagation even within complex environments. Recent advancements in reconfigurability and gradient features have significantly enriched the mechanical phenomena of wave manipulation, broadening the application potential of elastic topological metamaterials. In this study, we design the local resonant phononic crystal, employing a spiral support structure to ensure low-frequency characteristics and adjustable mass for flexible configuration. By utilizing the designed lattices with distinct topological phases, the topological valley edge states are constructed. The frequency and group velocity variation of the topological edge states under different stiffness and mass parameters are analyzed. Furthermore, we further constructed the mass and stiffness gradient metamaterial to analyze the transmission law of elastic waves in topological metamaterials. Additionally, the fluctuation features of the wave transmittance under disordered and impurity defects were discussed, confirming the robustness of waveguides within low-frequency topological metamaterials. Finally, we explored the potential applications of the designed metamaterials in energy localization and low-frequency reconfigurable waveguides. Numerical analysis showed that the topological gradient metamaterials can enhance the vibration energy at a specific interface, and low-frequency bend waveguide paths can be adjusted flexibly by configurable mass. In summary, this paper focuses on the low-frequency and gradient features of elastic topological metamaterials, aiming to unlock their application potential in vibrational energy harvesting, vibration suppression, and information transmission, which accelerate the practical implementation of topological metamaterials.

Broadband acoustic pseudospin topological states based on the reverse spin-orbit coupling in generalized insulators

Acoustic topological insulators have the excellent characteristic of the pseudospin-dependent one-way transmission of sound edge states immune to backscattering. We realize the broadband acoustic pseudospin topological edge states with subwavelength generalized topological insulators, which is achieved by reverse pseudospin-orbit coupling. The subwavelength band and broadband nontrivial bandgap can be achieved by adjusting the topological structure of the scatterers and introducing resonators. The results demonstrate that the resonator can significantly reduce the frequencies of p-states and d-states by introducing resonance scattering; the scattering size and rotation angles change the frequencies of p-states and d-states in opposite directions by adjusting the distribution of the sound field. Then, we experimentally realize the pseudospin-dependent one-way transmission of sound edge states along the interface separating phononic crystals with distinct topological phases. Our research provides a systematic scheme for the design of acoustic topological insulators with versatile applications.

Homotopy, symmetry, and non-Hermitian band topology

Non-Hermitian matrices are ubiquitous in the description of nature ranging from classical dissipative systems, including optical, electrical, and mechanical metamaterials, to scattering of waves and open quantum many-body systems. Seminal line-gap and point-gap classifications of non-Hermitian systems using K-theory have deepened the understanding of many physical phenomena. However, ample systems remain beyond this description; reference points and lines do not in general distinguish whether multiple non-Hermitian bands exhibit intriguing exceptional points, spectral braids and crossings. To address this we consider two different notions: non-Hermitian band gaps and separation gaps that crucially encompass a broad class of multi-band scenarios, enabling the description of generic band structures with symmetries. With these concepts, we provide a unified and comprehensive classification of both gapped and nodal systems in the presence of physically relevant parity-time ( PT ) and pseudo-Hermitian symmetries using homotopy theory. This uncovers new stable topology stemming from both eigenvalues and wave functions, and remarkably also implies distinct fragile topological phases. In particular, we reveal different Abelian and non-Abelian phases in PT -symmetric systems, described by frame and braid topology. The corresponding invariants are robust to symmetry-preserving perturbations that do not induce (exceptional) degeneracy, and they also predict the deformation rules of nodal phases. We further demonstrate that spontaneous PT symmetry breaking is captured by Chern–Euler and Chern–Stiefel–Whitney descriptions, a fingerprint of unprecedented non-Hermitian topology previously overlooked. These results open the door for theoretical and experimental exploration of a rich variety of novel topological phenomena in a wide range of physical platforms.

Machine learning of knot topology in non-Hermitian band braids

The deep connection among braids, knots and topological physics has provided valuable insights into studying topological states in various physical systems. However, identifying distinct braid groups and knot topology embedded in non-Hermitian systems is challenging and requires significant efforts. Here, we demonstrate that an unsupervised learning with the representation basis of su(n) Lie algebra on n-fold extended non-Hermitian bands can fully classify braid group and knot topology therein, without requiring any prior mathematical knowledge or any pre-defined topological invariants. We demonstrate that the approach successfully identifies different topological elements, such as unlink, unknot, Hopf link, Solomon ring, trefoil, and so on, by employing generalized Gell-Mann matrices in non-Hermitian models with n=2 and n=3 energy bands. Moreover, since eigenstate information of non-Hermitian bands is incorporated in addition to eigenvalues, the approach distinguishes the different parity-time symmetry and breaking phases, recognizes the opposite chirality of braids and knots, and identifies out distinct topological phases that were overlooked before. Our study shows significant potential of machine learning in classification of knots, braid groups, and non-Hermitian topological phases.

Coupling of photonic topological states and their dynamical control based on liquid crystal.

Optical field manipulation inspired by topology theory has recently drawn great research attention in nanophotonic. For flexible and programmable light management, the capacity to dynamically regulate the photonic topological states in fixed optical artificial microstructures is essential. Here, we propose a dynamic light manipulation of a two-dimensional (2D) photonic lattice aided by liquid crystals, which is composed of all-dielectric photonic crystals with distinct topological phases. In brief, by submerging the well-designed photonic lattice into a liquid crystal (LC), the topological edge and corner states can be actively modulated by applying external bias voltage, which offers an electrically switchable tuning capability, enabling the coupling between higher-order topological states in a structurally deterministic photonic structure. As a proof-of-principle, we use the 1D topological edge states and 0D topological corner states in one sample, respectively, to mimic line-waveguides and corner-cavities, and demonstrate their selective couplings with Fano-like profile driven by electric bias. Our work offers an effective and flexible way for light control in the potential active topological photonic devices.

Evolution of an Overlapped Bandgap and Topological Flat Bands in a Higher‐Order Valley Photonic Insulator Based on Dendritic Structure

AbstractRobust edge states and corner states in photonic topological insulators provide effective ways to manipulate the propagation of electromagnetic waves. Bandgaps of the previously reported photonic topological insulators are independent of each other. In this paper, a higher‐order valley photonic insulator composed of arrays of dendritic structure is designed. The band structure shows an overlapped bandgap is observed, and the overlapped bandgap divides the band structure into three bandgaps. The evolution of the overlapped bandgap is investigated by changing the geometric parameters and increasing the fractal to break the C3v symmetry drastically. Besides, it is notable that the band structure of the proposed valley photonic insulator is flat bands. It is demonstrated that undistorted transmission can be observed as a plane electromagnetic wave transmits through the proposed valley photonic insulator. The interface consisting of two valley photonic structures with distinct topological nontrivial phases is constructed. Edge states and corner states with strong energy localization are obtained in multi‐band frequencies. Remarkably, two triangular structures with different Wannier center configurations both have corner states in the same bandgap, which does not obey the valley selectivity. The phenomenon is caused by the weak valley locking property due to the overlapped bandgap. The proposed valley photonic insulators are expected to benefit applications in optical devices such as topological lasers.

Spin and valley-polarized Faraday rotation in irradiated buckled Xene materials

This study delves into the theoretical exploration of Faraday rotation and ellipticity in light beams transmitted through buckled Xene materials. These materials undergo topological phase transitions (TPTs), shifting from topologically non-trivial to trivial systems under the influence of an off-resonance irradiated laser field or a staggered electric potential. Specifically, we investigate the manifestation of these phenomena when the buckled Xene material is exposed to an off-resonant laser and staggered sublattice potential. Using the Kubo formula, we derive the optical conductivities of the buckled Xene material to analyze transmission spectra through Fresnel’s transmission coefficients. Additionally, we calculate the spin and valley-dependent Faraday rotation angles and ellipticities of the buckled Xene by selecting suitable parameters for the circularly polarized off-resonant laser field and staggered electric potential across distinct topological quantum phases. Our findings reveal a high sensitivity of Faraday rotation and ellipticity to the topological invariants. In particular, we find that due to the broken time-reversal symmetry (TRS), the amount of the maximum spin and valley polarized Faraday rotation angle and ellipticity in irradiated buckled Xene material without magnetic field are ≊±0.53∘ and ≊±0.2∘, respectively. Our results suggest possible techniques for probing topological numbers and topological phase transitions in buckled Xene materials by Faraday rotation.

Emergence of two-fold non-Hermitian spectral topology through synthetic spin engineering

The union of topology and non-Hermiticity has led to the unveiling of many intriguing phenomena. We introduce a synthetic spin-engineered model belonging to symmetry class AI, which is a rare occurrence, and demonstrate the emergence of a multi-fold spectral topology. As an example of our proposal, we engineer non-Hermiticity in the paradigmatic Su–Schrieffer–Heeger (SSH) model by introducing a generalized synthetic spin, leading to an emergent two-fold spectral topology that governs the decoupled behaviour of the corresponding non-Hermitian skin effect. As a consequence of the spin choice, our model exhibits a rich phase diagram consisting of distinct topological phases, which we characterize by introducing the notion of paired winding numbers, which, in turn, predict the direction of skin localization under open boundaries. We demonstrate that the choice of spin parameters enables control over the directionality of the skin effect, allowing for it to be unilateral or bilateral. Furthermore, we discover non-dispersive flat bands emerging within the inherent SSH model framework, arising from the spin-engineering approach. We also introduce a simplified toy model to capture the underlying physics of the emergent flat bands and direction-selective skin effect. As an illustration of experimental feasibility, we present a topoelectric circuit that faithfully emulates the underlying spin-engineered Hamiltonian, providing a viable platform for realizing our predicted effects. Our findings pave way for the exploration of unconventional spectral topology through spin-designed models.

Multiple topological phase transitions unveiling gapless topological superconductivity in magnet/unconventional superconductor hybrid platform

We propose a theoretical framework for generating gapless topological superconductivity (GTSC) hosting Majorana flat edge modes (MFEMs) in the presence of a two-dimensional (2D) array of magnetic adatoms with noncollinear spin texture deposited on top of an unconventional superconductor. Our observations reveal two distinct topological phase transitions within the emergent Shiba band depending on the exchange coupling strength (J) between magnetic adatom spins and superconducting electrons: the first one designates transition from gapless non-topological to gapless topological phase at lower J, while the second one denotes transition from gapless topological to a trivial gapped superconducting phase at higher J. The gapless topological superconducting phase survives at intermediate values of J, hosting MFEMs. Furthermore, we investigate the nature of the bulk effective pairings, which indicate that GTSC appears due to the interplay between pseudo “s-wave” and pseudo “px+py” types of pairing. Consequently, our study opens a promising avenue for the experimental realization of GTSC in 2D Shiba lattice based on d-wave superconductors as a high-temperature platform.

Robust multi-band acoustic router by hybridizing distinct topological phases

The acoustic router, capable of guiding sound waves along specific paths, holds a significant value in both science and engineering. Compared to traditional methods of implementing acoustic routing, the recently developed concept of topological acoustics, with its nontrivial topological phases, offers the potential to achieve a robust acoustic routing device. However, current investigations primarily focus on individual topological phases within a single bandgap, thereby limiting the exploration of diverse topological phases in multiple bandgaps and their hybridizations. In this study, we utilize topological acoustics to construct a robust dual-band acoustic router, which is challenging to achieve with traditional acoustics. By calculating Chern and valley topological phases in different bands, we reveal the competitive relations between different topological phases in a specific bandgap. Furthermore, by modifying the boundary meta-atoms, we have increased the operational frequency bands and proposed a triple-band acoustic router.

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.

Topological phases in the dynamics of the simple exclusion process.

We study the dynamical large deviations of the classical stochastic symmetric simple exclusion process (SSEP) by means of numerical matrix product states. We show that for half filling, long-time trajectories with a large enough imbalance between the number hops in even and odd bonds of the lattice belong to distinct symmetry-protected topological (SPT) phases. Using tensor network techniques, we obtain the large deviation (LD) phase diagram in terms of counting fields conjugate to the dynamical activity and the total hop imbalance. We show the existence of high activity trivial and nontrivial SPT phases (classified according to string order parameters) separated by either a critical phase or a critical point. Using the leading eigenstate of the tilted generator, obtained from infinite-system density-matrix renormalization group simulations, we construct a near-optimal dynamics for sampling the LDs, and show that the SPT phases manifest at the level of rare stochastic trajectories. We also show how to extend these results to other filling fractions, and discuss generalizations to asymmetric SEPs.

Unveiling the Anomalous Hall Response of the Magnetic Structure Changes in the Epitaxial MnBi2Te4 Films.

Recently discovered as an intrinsic antiferromagnetic topological insulator, MnBi2Te4 has attracted tremendous research interest, as it provides an ideal platform to explore the interplay between topological and magnetic orders. MnBi2Te4 displays distinct exotic topological phases that are inextricably linked to the different magnetic structures of the material. In this study, we conducted electrical transport measurements and systematically investigated the anomalous Hall response of epitaxial MnBi2Te4 films when subjected to an external magnetic field sweep, revealing the different magnetic structures stemming from the interplay of applied fields and the material's intrinsic antiferromagnetic (AFM) ordering. Our results demonstrate that the nonsquare anomalous Hall loop is a consequence of the distinct reversal processes within individual septuple layers. These findings shed light on the intricate magnetic structures in MnBi2Te4 and related materials, offering insights into understanding their transport properties and facilitating the implementation of AFM topological electronics.

One-dimensional synthetic waterborne phononic crystals

Underwater acoustics has wide applications in underwater communication, underwater positioning, underwater navigation, and so on. Inspired by the concept of topological physics, the study of topological states in waterborne phononic crystals provides a brand-new way for innovatively controlling underwater waves, which has both basic research value and important application prospects. In this work, we design a one-dimensional bilayer iron grid waterborne phononic crystal to realize a synthetic two-dimensional Dirac point by considering the relative lateral translation between the two layers as a synthetic dimension. Through changing the relative lateral translation, the double degenerate band opens a gap, which is characterized by the valley Chern number. As the band gap opens, closes and reopens, the bulk band undergoes a band inversion, that is, a topological phase transition from one valley topological phase to another. At the interface formed by two phononic crystals with distinct valley topological phases, the valley Chen number ensures the deterministic existence of the interface state. Experimental measurements are in good agreement with numerical simulations, both showing the bulk bands of waterborne phononic crystals at different valley topological phases and the interface state dispersion between them. The waterborne phononic crystal proposed in this work has a simple structure. With the help of the concept of synthetic dimension, it provides an effective way to study the topological properties of high-dimensional systems in low-dimensional real space systems, and gives new ideas for designing topological functional underwater acoustic devices. In addition, we can expand the real space system to two or even three dimensions, and introduce more synthetic dimensions to study the topological states and associated transport characteristics of higher-dimensional systems.

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.

Topological phases in coupled polyyne chains

We study the electronic properties of coupled parallel polyyne chains in a couple of symmetric stacking arrangements, namely the AA stacking and the AB stacking, with the single and triple carbon bonds of one chain aligned (AA) and anti-aligned (AB) with those of the other chain. Both these arrangements described by tight-binding Hamiltonians, whose parameters are calibrated by matching low energy dispersion provided by first principle calculations, fall in the BDI class of topological classification scheme. We calculate the topological invariants for all three topological phases of the system: one for the AA stacking and 2 for the AB one. In AA stacking, both the insulating and the metallic phase belongs to the same topological phase. Whereas, the model exhibits two different values of the topological invariant in the two different insulating phases (structurally differentiated by transverse strain). In this later stacking though the transition between two distinct topological phases with the closure of the gap is practically unachievable due to the requirement of the high transverse strain. We also show the existence of four non-zero energy edge modes in the AA stacking and that of two zero energy edge modes in one of the topological phases for the AB stacking.

Double resonance between corner states in distinct higher-order topological phases

Recent studies have shown that higher-order topologies in photonic systems lead to a robust enhancement of light–matter interactions. Moreover, higher-order topological phases have been extended to systems even without a band gap, as in Dirac semimetals. In this work, we propose a procedure to simultaneously generate two distinctive higher-order topological phases with corner states that allow a double resonant effect. This double resonance effect between the higher-order topological phases, was obtained from the design of a photonic structure with the ability to generate a higher-order topological (HOTI) insulator phase in the first bands and a higher-order Dirac half-metal phase (HODSM). Subsequently, using the corner states in both topological phases, we tuned the frequencies of both corner states such that they were separated in frequency by a second harmonic. This idea allowed us to obtain a double resonance effect with ultra-high overlap factors, and a considerable improvement in the nonlinear conversion efficiency. These results show the possibility of producing a second-harmonic generation with unprecedented conversion efficiencies in topological systems with simultaneous HOTI and HODSM phases. Furthermore, since the corner state in the HODSM phase presents an algebraic 1/r decay, our topological system can be helpful in experiments about the generation of nonlinear Dirac-ligh–matter interactions.