We study theoretically two-dimensional single-crystalline sheets of semiconductors that form a honeycomb lattice with a period below 10 nm. These systems could combine the usual semiconductor properties with Dirac bands. Using atomistic tight-binding calculations, we show that both the atomic lattice and the overall geometry influence the band structure, revealing materials with unusual electronic properties. In rocksalt Pb chalcogenides, the expected Dirac-type features are clouded by a complex band structure. However, in the case of zinc-blende Cd-chalcogenide semiconductors, the honeycomb nanogeometry leads to rich band structures, including, in the conduction band, Dirac cones at two distinct energies and nontrivial flat bands and, in the valence band, topological edge states. These edge states are present in several electronic gaps opened in the valence band by the spin-orbit coupling and the quantum confinement in the honeycomb geometry. The lowest Dirac conduction band has S-orbital character and is equivalent to the pi-pi* band of graphene but with renormalized couplings. The conduction bands higher in energy have no counterpart in graphene; they combine a Dirac cone and flat bands because of their P-orbital character. We show that the width of the Dirac bands varies between tens and hundreds of meV. These systems emerge as remarkable platforms for studying complex electronic phases starting from conventional semiconductors. Recent advancements in colloidal chemistry indicate that these materials can be synthesized from semiconductor nanocrystals.
Honeycomb lattice can support electronic states exhibiting Dirac energy dispersion, with graphene as the icon. We propose to derive nontrivial topology by grouping six neighboring sites of honeycomb lattice into hexagons and enhancing the inter-hexagon hopping energies over the intra-hexagon ones. We reveal that this manipulation opens a gap in the energy dispersion and drives the system into a topological state. The nontrivial topology is characterized by the $\mathbb{Z}_2$ index associated with a pseudo time-reversal symmetry emerging from the $C_6$ symmetry of the Kekulé hopping texture, wh…
Carbon allotropes such as diamond, nano-tube, Fullerene, and Graphene, have unique lattice symmetries of crystal lattice, but these are topologically trivial. We have proposed a topologically-nontrivial allotrope, named Hopfene, which has three-dimensional (3D) arrays of Hopf-links to bind 2D Graphene sheets both vertically and horizontally. Here, we describe the electronic structures of Hopfene by simple tight-binding calculations. We confirmed the original Dirac points of 2D Graphene were topologically protected upon the introduction of the Hopf links, and low-energy excitations are describe…
We use the methods of topological quantum chemistry to explore the topology of phonons on time-reversal symmetric crystals with the structure of the planar honeycomb (layer group $p6/mmm$). This approach is not tied to a particular model of atomic vibrations, but is applied to the most general dynamical matrix constrained only by the symmetries of the system. We show that four distinct fragile topological phonon phases are generically possible. Truncating the dynamical matrix to third nearest neighbors yields a model that realizes the different phonon topologies, characterized by the existence…
We revisit the spectral problem for Bloch electrons in a two-dimensional bipartite honeycomb lattice under a uniform magnetic field. It is well-known that such a honeycomb structure is realized in graphene. We present a systematic framework to compute the perturbative magnetic flux expansions near two distinct band edges. We then analyze the nonperturbative bandwidth of the spectrum. It turns out that there is a novel similarity between the spectrum near the Dirac point in the honeycomb lattice and the spectrum in the supersymmetric sine-Gordon quantum mechanics. We finally confirm a nontrivia…
We theoretically study a spin Hamiltonian for spin-orbit-coupled ferromagnets on the honeycomb lattice. We find that the effective Hamiltonian for magnons, a quanta of spin-wave excitations from ordered states, is equivalent to the Haldane model for electrons in the honeycomb lattice, which indicates the nontrivial topology of the magnon bands and the existence of the associated edge state. We also take the Schwinger boson (or bosonic spinon) representations of spins that are applicable both to ordered and disordered phases. We show that the spinon bands within the mean-field treatment are top…
We present an exactly solvable spin-orbital model based on the Gamma-matrix generalization of a Kitaev-type Hamiltonian. In the presence of small magnetic fields, the model exhibits a critical phase with a spectrum characterized by topologically protected Fermi points. Upon increasing the magnetic field, Fermi points carrying opposite topological charges move toward each other and annihilate at a critical field, signaling a phase transition into a gapped phase with trivial topology in three dimensions. On the other hand, by subjecting the system to a staggered magnetic field, an effective time…
The discovery of monolayer graphene has initiated two fertile fields in modern condensed matter physics, Dirac semimetals and atomically-thin layered materials. When these trends meet again in transition metal compounds, which possess spin and orbital degrees of freedom and strong electron correlations, more exotic phenomena are expected to emerge in the cross section of topological states of matter and Mott physics. Here, we show by using ab initio calculations that a monolayer form of transition metal trichalcogenides (TMTs), which has a honeycomb network of transition metal cations, may exh…
We examine the combined effects of a Kekule coupling texture (KC) and a Dzyaloshinskii-Moriya interaction (DMI) in a two-dimensional ferromagnetic honeycomb lattice. By analyzing the gap closing conditions and the inversions of the bulk bands, we identify the parameter range in which the system behaves as a trivial or a nontrivial topological magnon insulator. We find four topological phases in terms of the KC parameter and the DMI strength. We present the bulk-edge correspondence for the magnons in a honeycomb lattice with an armchair or a zigzag boundary. Furthermore, we find Tamm-like edge …
Square-root topological insulators are recently-proposed intriguing topological insulators, where the topologically nontrivial nature of Bloch wave functions is inherited from the square of the Hamiltonian. In this paper, we propose that higher-order topological insulators can also have their square-root descendants, which we term square-root higher-order topological insulators. There, emergence of in-gap corner states is inherited from the squared Hamiltonian which hosts higher-order topology. As an example of such systems, we investigate the tight-binding model on a decorated honeycomb latti…
We report on the theoretical prediction of asymmetric edge spin waves, propagating in opposite directions on the edges of honeycomb nanoribbons with zigzag and bearded edges boundaries. The simultaneous propagation of edge spin waves in the same direction on both edges of these nanoribbons is hence predicted to be forbidden. These asymmetric edge exchange spin waves are analogous to the nonreciprocal surface spin waves reported in magnetic thin films. Their existence is related to the nontrivial symmetry underlying the nanoribbons under study. The edge and discretized bulk exchange spin waves …
In crystalline systems, charge polarization is related to Zak's phase determined by bulk band topology. Nontrivial charge polarization induces robust edge states accompanied with fractional charge. In Su-Schrieffer-Heeger (SSH) model, it is known that the strong modulation of electron hopping causes nontrivial charge polarization even in the presence of inversion symmetry. Here, we consider a bi-atomic honeycomb lattice to introduce such strong modulation, i.e. A$_3$B sheet. By tuning hopping ratio and onsite potential difference between A and B atoms, we show that topological phase transition…
Periodic Behavior of Topology in Graphene with Nanohole Array
Institute for Quantum Science and Technology, and Department of Physics, Shanghai University（上海大学量子科学与技术研究院及物理系）；School of Engineering, Institute of Science Tokyo（东京科学大学工程学院）
The search for topological semimetals is mainly focused on heavy-element compounds as following the footsteps of previous research on topological insulators, with less attention on light-element materials. However, the negligible spin orbit coupling with light elements may turn out to be beneficial for realizing topological band features.Here, using first-principles calculations, we propose a new two-dimensional light-element material-the honeycomb borophene oxide (h-B2O), which has nontrivial topological properties.The proposed structure is based on the recently synthesized honeycomb borophen…
A novel gapped metallic state coined orthogonal Dirac semimetal is proposed in the honeycomb lattice in terms of $Z_{2}$ slave-spin representation of Hubbard model. This state corresponds to the disordered phase of slave-spin and has the same thermaldynamical and transport properties as usual Dirac semimetal but its singe-particle excitation is gapped and has nontrivial topological order due to the $Z_{2}$ gauge structure. The quantum phase transition from this orthogonal Dirac semimetal to usual Dirac semimetal is described by a mean-field decoupling with complementary fluctuation analysis an…
The doubly-connected polygonal geometry of planar graphene rings is found to bring forth topological configurations for accessing nontrivial relativistic quantum field (RQF) theory models that carry beyond the constant-mass Dirac-fermion theory. These include generation of sign-alternating masses, solitonic excitations, and charge fractionization. The work integrates a RQF Lagrangian formulation with numerical tight-binding Aharonov-Bohm electronic spectra and the generalized position-dependent-mass Dirac equation. In contrast to armchair graphene rings (aGRGs) with pure metallic arms, certain…
In the field of topological insulators, the topological properties of quantum states in samples with simple geometries, such as a cylinder or a ribbon, have been classified and understood during the last decade. Here, we extend these studies to a Möbius band, and argue that its lack of orientability prevents a smooth global definition of parity-odd quantities such as pseudovectors. In particular, the Chern number, the topological invariant for the quantum Hall effect, lies in this class. The definition of spin on the Möbius band translates into the idea of the orientable double cover, an analo…
We propose that a quantum anomalous Hall insulator (QAHI) can be realized in a nanopatterned two-dimensional electron gas (2DEG) with a small in-plane magnetic field and a high carrier density. The Berry curvatures originating from the in-plane magnetic field and Rashba and Dresselhaus spin-orbit coupling, in combination with a nanoscale honeycomb lattice potential modulation, lead to topologically nontrivial insulating states in the 2DEG without Landau levels. In the bulk insulating gaps, the anomalous Hall conductivity is quantized $-e^{2}/h$, corresponding to a finite Chern number -1. There…
A unified form of time-reversal symmetry (TRS) breaking terms in phononic crystals, leading to nontrivial phononic topology, has been proposed recently, but is contradicted by some other works which introduce gyroscopic effect as TRS-breaking. We re-study gyroscopic phononic crystals using Newtonian mechanical method, and find the correct TRS-breaking term in consistent with the unified form. Applying this term we calculate the basic topological phononics in a honeycomb lattice. Furthermore, we study nonlinear phonon-phonon scattering effect on topological phononic edge states by molecular dyn…
We consider two-dimensional Fermi systems with quadratic band touching and $C_3$ symmetry, as realizable in Bernal-stacked honeycomb bilayers. Within a renormalization-group analysis, we demonstrate the existence of a quantum critical point at a finite value of the density-density interactions, separating a semimetallic disordered phase at weak coupling from a gapped ordered phase at strong coupling. The latter may be characterized by, for instance, antiferromagnetic, quantum anomalous Hall, or charge density wave order. In the semimetallic phase, each point of quadratic band touching splits i…
Due to nonzero intrinsic spin-orbit interaction in buckled honeycomb crystal structures, silicene and germanene exhibit interesting topological properties, and are therefore candidates for the realization of the quantum spin Hall effect. We employ the Kane-Mele model to investigate the electron states in hexagonal silicene and germanene nanorings having either zigzag or armchair edges in the presence of a perpendicular magnetic field. We present results for the energy spectra as function of magnetic field, the electron density of the spin-up and spin-down states in the ring plane, and the calc…
We theoretically study the topological properties of the tight-binding model on the breathing kagome lattice with antisymmetric spin-orbit coupling (SOC) between nearest neighbors. We show that the system hosts nontrivial topological phases even without second-nearest-neighbor hopping, and that the weakly dispersing band of the kagome lattice can become topological. The main results are presented in the form of phase diagrams, where the $\mathbb{Z}_2$ topological index is shown as a function of SOC (intrinsically allowed and Rashba) and lattice trimerization. In addition, exact diagonalization…
We demonstrate the possibility of topological states for non-Dirac electrons. Specifically it is shown that, because of the $C_{\rm 3}$ crystal symmetry and time reversal symmetry, $p_x$ and $p_y$ orbits accommodated on triangular lattice exhibit a quadratic band touching at $Γ$ point at the Fermi level. When the atomic spin-orbit coupling (SOC) is taken into account, a gap is opened resulting in a quantum spin Hall effect state. As revealed explicitly by a $k\cdot p$ model, the topology is associated with a meron structure in the pseudo spin texture with vorticity two, a mechanism different f…
The existence of robust chiral edge states in a finite topologically nontrivial chern insulator is a consequence of the bulk-boundary correspondence. In this paper, we present a theoretical framework based on lattice Green's function to study the scattering of such chiral edge electrons by a single localized impurity. To this end, in the first step, we consider the standard topological Haldane model on a honeycomb lattice with strip geometry. We obtain analytical expressions for the wave functions and their corresponding energy dispersion of the low-energy chiral states localized at the edge o…
Topological flat bands at the Fermi level offer a promising platform to study a variety of intriguing correlated phase of matter. Here we present band engineering in the twisted orbital-active bilayers with spin-orbit coupling. The symmetry constraints on the interlayer coupling that determines the effective potential for low-energy physics of moiré electrons are exhaustively derived for two-dimensional point groups. We find the line graph or biparticle sublattice of moiré pattern emerge with a minimal $C_3$ symmetry, which exhibit isolated electronic flat bands with nontrivial topology. The b…
