Nonzero Chern Number Research Articles

Photonic topological insulators in bianisotropic metamaterials.

We analyze the photonic topological phases in bianisotropic metamaterials characterized by a lossless and reciprocal magnetoelectric tensor. The underlying medium is considered a topological insulator that supports a pair of counterpropagating helical edge states. By introducing the pseudospin basis, the photonic system can be described by the spin-orbit Hamiltonians with spin 1, which result in nonzero spin Chern numbers that determine the topological properties. Surface modes at the interface between two bianisotropic media with opposite chirality exist in their common band gap, which are represented by elliptic or hyperbolic equations. In particular, two branches of hyperbolic surfaces are degenerate at the frequency where the chiral nihility occurs, which depict the helical nature of edge states between two distinct topological phases. Topological features of the bianisotropic metamaterials are further illustrated with the robust transport of surface modes at an irregular boundary.

Bound on resistivity in flat-band materials due to the quantum metric

The quantum metric is a central quantity of band theory but has so far not been related to many response coefficients due to its nonclassical origin. However, within a newly developed Kubo formalism for fast relaxation, the decomposition of the dc electrical conductivity into both classical (intraband) and quantum (interband) contributions recently revealed that the interband part is proportional to the quantum metric. Here, we show that interband effects due to the quantum metric can be significantly enhanced and even dominate the conductivity for semimetals at charge neutrality and for systems with highly quenched bandwidth. This is true in particular for topological flat-band materials of nonzero Chern number, where for intermediate relaxation rates an upper bound exists for the resistivity due to the common geometrical origin of quantum metric and Berry curvature. We suggest to search for these effects in highly tunable rhombohedral trilayer graphene flakes.

Transport properties of the electronic states in the gate voltage-modulated skyrmion crystal

As a result of the Hund's coupling, the band structure of the conducting electrons in the skyrmion crystal (SkX) shares similar topological properties with that of graphene, such as its cone-like shape, nonzero band Chern number, edge states, and etc. In this work, we rigorously demonstrate that the Klein tunneling phenomena is also shared by these two. We use the Green's function technique and calculated the transmission probability of the electrons tunneling through an electrostatic barrier in the SkX expressed by the double exchange model. Numerical results of the SkX reproduced the Dirac model obtained by linear fitting the two-dimensional band structure of the SkX.

Disordered transmission-line networks with and without parity symmetry

Topological states are useful because they are robust against disorder and imperfection. In this study, we consider the effect of disorder and the breaking of parity symmetry on a topological network system in which the edge states are protected by Chern numbers. In the absence of periodicity, the local Chern number is adopted to characterize the topological features of the network. Our numerical results show that the local Chern number and the edge states are very robust against onsite disorder as long as the gap of the bulk state continuum remains open and survives even when the bulk band gap is closed. Breaking the parity symmetry can destroy the quantization of local Chern numbers, compromising the existence of edge modes. We observed non-integer local Chern number peaks that are non-zero inside the bulk bands but these non-zero non-integral local Chern numbers are not associated with the existence of robust edge states.

Coexistence of Quantum Hall and Quantum Anomalous Hall Phases in Disordered MnBi2Te4

In most cases, to observe quantized Hall plateaux, an external magnetic field is applied in intrinsic magnetic topological insulators $\mathrm{MnBi_2Te_4}$. Nevertheless, whether the nonzero Chern number ($C\neq 0$) phase is a quantum anomalous Hall (QAH) state, or a quantum Hall (QH) state, or a mixing state of both is still a puzzle, especially for the recently observed $C=2$ phase [Deng \textit{et al}., Science \textbf{367}, 895 (2020)]. In this Letter, we propose a physical picture based on the Anderson localization to understand the observed Hall plateaux in disordered $\mathrm{MnBi_2Te_4}$. Rather good consistency between the experimental and numerical results confirms that the bulk states are localized in the absence of a magnetic field and a QAH edge state emerges with $C=1$. However, under a strong magnetic field, the lowest Landau band formed with the localized bulk states, survives disorder, together with the QAH edge state, leading to a $C=2$ phase. Eventually, we present a phase diagram of a disordered $\mathrm{MnBi_2Te_4}$ which indicates more coexistence states of QAH and QH to be verified by future experiments.

Antiferromagnetic topological crystalline insulator and mixed Weyl semimetal in two-dimensional NpAs monolayer

Magnetic topological states have attracted significant attentions due to their intriguing quantum phenomena and potential applications in topological spintronic devices. Here, we propose a two-dimensional material NpAs monolayer as a candidate for multiple topological states accompanied with the changes of magnetic structures. Under the antiferromagnetic configuration, the long-awaited topological crystalline insulator (TCI) emerges with a nonzero mirror Chern number and a giant band gap of 630 meV, and remarkably a pair of gapless edge states can be tailored by rotating the magnetization directions while the TCI phase survives. Moreover, we establish the existence of quantum anomalous Hall effect and nontrivial nodal points under the ferromagnetic configuration, thereby giving rise to the mixed Weyl semimetal after adding the magnetization direction to topological classification. Our findings not only provide an ideal candidate for uncovering exotic topological characters with magnetism but also put forward potential applications in topological spintronics.

Ferromagnetic coupling in a two-dimensional Cairo pentagonal Ni2(TCNQ)2 lattice

Magnetism has revolutionized important technologies, and continues to bring forth new phenomena in emergent materials and reduced dimensions. Here, using first-principles calculations, we demonstrate that the already-synthesized two-dimensional (2D) Ni-tetracyanoquinodimethane (Ni2(TCNQ)2) lattice is a stable ferromagnetism material with multiple spin-polarized Dirac cones. The conical bands in proximity of the Fermi level can be tuned by external tensile strain and show the fourfold degenerate electronic states at the critical tensile strain of ∼2.35 %, whose energy dispersion is consistent with 2D Cairo pentagonal lattice. In addition, spin-orbital coupling can open a band gap at the Dirac point of A, leading to topologically nontrivial electronic states characterized by the non-zero Chern number and the edge states of nanoribbon. Our results offer versatile platforms for the realization of massless spintronics with full-spin polarization in 2D Cairo pentagonal Ni2(TCNQ)2 Lattice.

Switchable valley polarization and quantum anomalous Hall state in the VN2X2Y2 nanosheets (X = group-III and Y = group-VI elements)

Motivated by the recent synthesis of MA2Z4 family materials, we perform a first-principles investigation on the structural stability and electronic properties of VN2X2Y2 nanosheets (X = B-Ga, Y = O-Te), which are the isostructural and isoelectronic analogues of the VSi2N4 system. We find that among the 16 possible III–VI combinations, there are four XY compositions, i.e., XY = BO, BS, AlO, and GaO, that can form stable VN2X2Y2 nanosheets with robust dynamic and thermal stabilities. Intrinsic ferromagnetism appears in these VN2X2Y2 nanosheets, among which the VN2B2S2 and VN2Ga2O2 ones possess an easy out-of-plane magnetization. Thus, unlike the VSi2N4 system, the VN2B2S2 and VN2Ga2O2 nanosheets exhibit spontaneous valley polarization with a sizeable polarization value in the bottom conduction band. Through the strain modulation, the valley polarization can be switched from the bottom conduction band to the top valence band, which is closely related to the strain-induced band inversion. Due to the existence of valley polarization, the band inversion occurs at different strains for the two valleys. Therefore, in the critical strain region, an intriguing quantum anomalous Hall state is present in the strained VN2X2Y2 nanosheets, which is characterized by a nonzero Chern number of C = 1. Such nontrivial topology is further confirmed by a quantized Hall conductance and a single gapless edge state in the bulk gap. Our study demonstrates as the cousin of MA2Z4 materials, the III–VI VN2X2Y2 nanosheets possess robust structural stability, peculiar electronic and topological properties, which have potential applications in electronics, spintronics, and valleytronics.

Multiple Dirac states in two-dimensional topological half-metallic CrB2C2

Recently, Dirac states in two dimensional materials have drawn great attention because of its desirable transport properties. In this paper, we predicted a two-dimensional hexagonal CrB2C2, which exhibits stable mechanical and thermal properties. Young's modulus of CrB2C2 is 485.67 N/m, which is comparable with that of graphene. In addition, CrB2C2 is half-metallic, and the ferromagnetic Curie temperature is about 1500 K, which is far above room temperature. A CrB2C2 nanosheet exhibits multiple Dirac states in the spin minority channel and forms compensated electron-hole pockets. The Fermi velocity of two different Dirac states is about 4.18×105 and 3.42×105 m/s. In the electronic band structure, the Dirac state at the K point is weakly gapped by 0.38 meV under a spin orbital interaction, while the other one could be topological nontrivially gapped by 37.4 meV, and the bottom Dirac band exhibits a nonzero Chern number at about 0.25 eV above the Fermi energy.

Chiral Topological Surface States on a Finite Square Photonic Crystal Bounded by Air

Chiral, topologically protected, photonic surface states can be found at the boundary between gyrotropic photonic crystals where a changing magnetic field induces different topology across the interface. Typically, photonic crystals with either a suitable band structure on both sides of the interface to provide a band gap and evanescent decay of the surface states away from the interface, or an outer layer with engineered material properties is required. In this paper, we show the emergence of topological, unidirectional surface states at the termination of finite gyrotropic photonic crystals with a simple square lattice and ${\mathcal{C}}_{4}$ rotational symmetry bounded by a vacuum, eliminating the need for an outside layer to enable chiral surface modes. We start from an infinite, time-reversal-symmetry-breaking photonic crystal with a band gap associated with bands with nonzero Chern numbers, different from all-zero Chern numbers in air. We then modify the photonic crystal to move this band gap below the light line, while maintaining the Chern-number discontinuities. Band-structure calculations for a supercell approximating a photonic crystal finite in the direction normal to the surface demonstrate the existence, dispersion, and chirality of the surface mode. Extensive direct scattering calculations for a point source and spatial Fourier analysis further reveal a unidirectional free-space topological surface state, which propagates counterclockwise around the surface of a finite photonic crystal, providing a nearly foolproof way to cross-check the surface-mode band structure unaffected by backscattering from local defects. Additionally, scattering simulations allow an independent characterization of the state dispersion and unveil the robustness of the topological plasmonic mode propagation around the ${90}^{\ensuremath{\circ}}$ bends of the structure, being due to only radiation leakage. In contrast to buried topological surface states, the observed surface modes at the photonic crystal--air interface have the advantage of being accessible to the outside world, allowing one to take advantage of the defect-tolerant backscattering-free surface modes to engineer emission from photonic crystal surfaces into arbitrary free-space beam shapes and directions.

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.

Chiral edge modes in evolutionary game theory: A kagome network of rock-paper-scissors cycles.

We theoretically demonstrate the realization of a chiral edge mode in a system beyond natural science. Specifically, we elucidate that a kagome network of rock-paper-scissors (K-RPS) hosts a chiral edge mode of the population density which is protected by the nontrivial topology in the bulk. The emergence of the chiral edge mode is demonstrated by numerically solving the Lotka-Volterra (LV) equation. This numerical result can be intuitively understood in terms of the cyclic motion of a single rock-paper-scissors cycle, which is analogous to the cyclotron motion of fermions. Furthermore, we point out that a linearized LV equation is mathematically equivalent to the Schrödinger equation describing quantum systems. This equivalence allows us to clarify the topological origin of the chiral edge mode in the K-RPS; a nonzero Chern number of the payoff matrix induces the chiral edge mode of the population density, which exemplifies the bulk-edge correspondence in two-dimensional systems described by evolutionary game theory.

Simulating higher-order topological insulators in density wave insulators

Since the discovery of the Harper-Hofstadter model, it has been known that condensed matter systems with periodic modulations can be promoted to non-trivial topological states with emergent gauge fields in higher dimensions. In this work, we develop a general procedure to compute the gauge fields in higher dimensions associated to low-dimensional systems with periodic (charge- and spin-) density wave modulations. We construct two-dimensional (2D) models with modulations that can be promoted to higher-order topological phases with $U(1)$ and $SU(2)$ gauge fields in 3D. Corner modes in our 2D models can be pumped by adiabatic sliding of the phase of the modulation, yielding hinge modes in the promoted models. We also examine a 3D Weyl semimetal (WSM) gapped by charge-density wave (CDW) order, possessing quantum anomalous Hall (QAH) surface states. We show that this 3D system is equivalent to a 4D nodal line system gapped by a $U(1)$ gauge field with a nonzero second Chern number. We explain the recently identified interpolation between inversion-symmetry protected phases of the 3D WSM gapped by CDWs using the corresponding 4D theory. Our results can extend the search for (higher-order) topological states in higher dimensions to density wave systems.

Magnetism-mediated transition between crystalline and higher-order topological phases in NpSb

Merging the fields of topology and magnetism expands the scope of fundamental quantum phenomena with novel functionalities for topological spintronics enormously. Here, we theoretically demonstrate that ferromagnetism provides an efficient means to achieve a topological switching between crystalline and higher-order topological insulator phases in two dimensions. Using a tight-binding model and first-principles calculations, we identify layered NpSb as a long-awaited two-dimensional topological crystalline insulator with intrinsic ferromagnetic order with a band gap which is as large as 220 meV. We show that when ${\mathcal{M}}_{z}$ symmetry is preserved for the out of plane magnetization of this material, it exhibits a pair of gapless edge states along all boundaries and carries a nonzero mirror Chern number ${\mathcal{C}}_{\mathcal{M}}=1$. Remarkably, when rotating the magnetization into the plane a higher-order topological insulator phase with a parity-based invariant ${\ensuremath{\nu}}_{2\text{D}}=1$ is achieved, and in-gap topological corner states emerge. Our results pave the way to understanding and engineering topological insulating states in two-dimensional ferromagnets.

Itinerant Topological Magnons in SU(2) Symmetric Topological Hubbard Models with Nearly Flat Electronic BandsSupported by the National Natural Science Foundation of China (Grant No. 11774152), and National Key R&D Program of China (Grant No. 2016YFA0300401).

We show that a suitable combination of flat-band ferromagnetism, geometry and nontrivial electronic band topology can give rise to itinerant topological magnons. An SU(2) symmetric topological Hubbard model with nearly flat electronic bands, on a Kagome lattice, is considered as the prototype. This model exhibits ferromagnetic order when the lowest electronic band is half-filled. Using the numerical exact diagonalization method with a projection onto this nearly flat band, we can obtain the magnonic spectra. In the flat-band limit, the spectra exhibit distinct dispersions with Dirac points, similar to those of free electrons with isotropic hoppings, or a local spin magnet with pure ferromagnetic Heisenberg exchanges on the same geometry. Significantly, the non-flatness of the electronic band may induce a topological gap at the Dirac points, leading to a magnonic band with a nonzero Chern number. More intriguingly, this magnonic Chern number changes its sign when the topological index of the electronic band is reversed, suggesting that the nontrivial topology of the magnonic band is related to its underlying electronic band. Our work suggests interesting directions for the further exploration of, and searches for, itinerant topological magnons.

Three-dimensional time reversal invariant topological superconductivity in doped chiral topological semimetals

Chiral topological semimetals host multifold degenerate band crossing points under the protection of crystalline symmetries. In this paper, we suggest that the recently discovered chiral topological semimetals in space group 198, parts of which are superconducting upon doping, can be new candidates of time reversal invariant topological superconductors. By investigating the Fermi surfaces around the band crossing points that carry nonzero Chern numbers, we clarify how the nontrivial topology of chiral topological semimetals affects their superconducting state and show the existence of topological superconductivity in $s_\pm$-wave pairing with surface Majorana fermions. We further demonstrate that the topological superconductivity is favored by the inter-unit-cell phonon-mediated electron-electron interaction.

Two-dimensional ferromagnetic Chern insulator: WSe2 monolayer

Quantum anomalous Hall (QAH) effect presents a promising platform for next generation nano-electronics due to its dissipationless edge states. Unfortunately, until now, QAH materials suitable for practical applications are rare. Based on the first-principles calculations, we found a novel two-dimensional (2D) QAH insulator: WSe2 monolayer. Both the spin-polarized electronic structure and spin-orbit coupling (SOC) of WSe2 contribute to the QAH effect. When SOC is ignored, the WSe2 monolayer is a spin-polarized Dirac half-metal (DHM). Taking SOC into account, a global band gap about 30 meV is opened and the system behaves as an intrinsic Chern insulator with a Curie temperature of 160 K. The non-trivial properties are proved by the nonzero Chern number, quantized anomalous Hall conductivity, and gapless chiral edge states. We expect the material will become a promising candidate for realizing the QAH effect with higher experimental temperature.

Exciton Band Topology in Spontaneous Quantum Anomalous Hall Insulators: Applications to Twisted Bilayer Graphene.

We uncover topological features of neutral particle-hole pair excitations of correlated quantum anomalous Hall (QAH) insulators whose approximately flat conduction and valence bands have equal and opposite nonzero Chern number. Using an exactly solvable model we show that the underlying band topology affects both the center-of-mass and relative motion of particle-hole bound states. This leads to the formation of topological exciton bands whose features are robust to nonuniformity of both the dispersion and the Berry curvature. We apply these ideas to recently reported broken-symmetry spontaneous QAH insulators in substrate aligned magic-angle twisted bilayer graphene.

Simulation of higher-order topological phases and related topological phase transitions in a superconducting qubit

Higher-order topological phases give rise to new bulk and boundary physics, as well as new classes of topological phase transitions. While the realization of higher-order topological phases has been confirmed in many platforms by detecting the existence of gapless boundary modes, a direct determination of the higher-order topology and related topological phase transitions through the bulk in experiments has still been lacking. To bridge the gap, in this work we carry out the simulation of a two-dimensional second-order topological phase in a superconducting qubit. Owing to the great flexibility and controllability of the quantum simulator, we observe the realization of higher-order topology directly through the measurement of the pseudo-spin texture in momentum space of the bulk for the first time, in sharp contrast to previous experiments based on the detection of gapless boundary modes in real space. Also through the measurement of the evolution of pseudo-spin texture with parameters, we further observe novel topological phase transitions from the second-order topological phase to the trivial phase, as well as to the first-order topological phase with nonzero Chern number. Our work sheds new light on the study of higher-order topological phases and topological phase transitions.

Fragile Topology and Flat-Band Superconductivity in the Strong-Coupling Regime.

In flat bands, superconductivity can lead to surprising transport effects. The superfluid "mobility", in the form of the superfluid weight D_{s}, does not draw from the curvature of the band but has a purely band-geometric origin. In a mean-field description, a nonzero Chern number or fragile topology sets a lower bound for D_{s}, which, via the Berezinskii-Kosterlitz-Thouless mechanism, might explain the relatively high superconducting transition temperature measured in magic-angle twisted bilayer graphene (MATBG). For fragile topology, relevant for the bilayer system, the fate of this bound for finite temperature and beyond the mean-field approximation remained, however, unclear. Here, we numerically use exact MonteCarlo simulations to study an attractive Hubbard model in flat bands with topological properties akin to those of MATBG. We find a superconducting phase transition with a critical temperature that scales linearly with the interaction strength. Then, we investigate the robustness of the superconducting state to the addition of trivial bands that may or may not trivialize the fragile topology. Our results substantiate the validity of the topological bound beyond the mean-field regime and further stress the importance of fragile topology for flat-band superconductivity.