Types Of Topological States Research Articles

Transformation of topological states in three-dimensional photonic crystal under magnetic field modulation

The realization and modulation of high-order topological states in three-dimensional (3D) photonic crystals (PCs) hold great significance for practical applications in optical communication, optical information processing, and optical computing. However, they have encountered technological challenges due to the difficulty in achieving a complete bandgap in 3D PCs. To address this issue, we introduce a design for a 3D PC based on the Su–Schrieffer–Heeger (SSH) model that exhibits distinctive topological surface states, topological hinge states, and topological corner states. Furthermore, these three types of topological states can be interconverted through the modulation of magnetic field because the topological states are closed and extremely sensitive to the structure, which are modulated by the magnetic field. This intriguing capability holds potential applications in the manipulation of optical flow, optical signal storage, and magnetic field sensors.

Topological states in a dimerized square-octagon lattice with staggered magnetic fluxes

We construct a dimerized square-octagon lattice model with the staggered magnetic fluxes threading plaquettes and find two types of topological states at half filling, i.e., the Chern insulator (CI) and the zero-Chern-number topological (ZCNT) states. The CI can be characterized on the basis of the chiral edge states and nonzero Chern number. Different from the CI, this ZCNT state hosts robust edge states with zero Chern number. Its topological nature can be identified by the quantized bulk polarization. More intriguingly, corner states emerge in this ZCNT state, even though its edge state is gapless. In addition, we present a phase diagram by tuning the dimerized hopping potential and the staggered fluxes. Our findings provide a promising approach to search more topological phases without time-reversal symmetry.

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.

Generalized Aubry-André-Harper model with modulated hopping and p -wave pairing

We study an extended Aubry-Andr{\'e}-Harper model with simultaneous modulation of hopping, on-site potential, and $p$-wave superconducting pairing. For the case of commensurate modulation of $\beta = 1/2$ it is shown that the model hosts four different types of topological states: adiabatic cycles can be defined which pump particles, two types of Majorana fermions, or Cooper pairs. In the incommensurate case we calculate the phase diagram of the model in several regions. We characterize the phases by calculating the mean inverse participation ratio and perform multi-fractal analysis. In addition, we characterize whether the phases found are topologically trivial or not. We find an interesting critical extended phase when incommensurate hopping modulation is present. The rise between the inverse participation ratio in regions separating localized and extended states is gradual, rather than sharp. When, in addition, the on-site potential modulation is incommensurate, we find several sharp rises and falls in the inverse participation ratio. In these two cases all different phases exhibit topological edge states. For the commensurate case we calculate the evolution of the Hofstadter butterfly and the band Chern numbers upon variation of the pairing parameter for zero and finite on-site potential. For zero on-site potential the butterflies are triangular-like near zero pairing, when gap-closure occurs, they are square-like, and hexagonal-like for larger pairing, but with the Chern numbers switched compared to the triangular case. For the finite case gaps at quarter and three-quarters filling close and lead to a switch in Chern numbers.

Observation of Double Weyl Phonons in Parity-Breaking FeSi

Condensed matter systems have now become a fertile ground to discover emerging topological quasiparticles with symmetry protected modes. While many studies have focused on fermionic excitations, the same conceptual framework can also be applied to bosons yielding new types of topological states. Motivated by Zhang etal.'s recent theoretical prediction of double Weyl phonons in transition metal monosilicides [Phys. Rev. Lett. 120, 016401 (2018)PRLTAO0031-900710.1103/PhysRevLett.120.016401], we directly measure the phonon dispersion in parity-breaking FeSi using inelastic x-ray scattering. By comparing the experimental data with theoretical calculations, we make the first observation of double Weyl points in FeSi, which will be an ideal material to explore emerging bosonic excitations and its topologically nontrivial properties.

Layer Construction of 3D Topological States and String Braiding Statistics

While the topological order in two dimensions has been studied extensively since the discover of the integer and fractional quantum Hall systems, topological states in 3 spatial dimensions are much less understood. In this paper, we propose a general formalism for constructing a large class of three-dimensional topological states by stacking layers of 2D topological states and introducing coupling between them. Using this construction, different types of topological states can be obtained, including those with only surface topological order and no bulk topological quasiparticles, and those with topological order both in the bulk and at the surface. For both classes of states we study its generic properties and present several explicit examples. As an interesting consequence of this construction, we obtain example systems with nontrivial braiding statistics between string excitations. In addition to studying the string-string braiding in the example system, we propose a generic topological field theory description which can capture both string-particle and string-string braiding statistics. Lastly, we provide a proof of a general identity for Abelian string statistics, and discuss an example system with non-Abelian strings.