Existence Of Dirac Points Research Articles

Two-dimensional Dirac phonons without/with inversion symmetry

In a recent study, two-dimensional Dirac phonons that are protected by nonsymmorphic symmetries in spinless systems were systematically investigated. However, the focus of this study was on the classification of Dirac phonons. To address the gap in the research on the topological features of 2D Dirac phonons based on their effective models, we classified the 2D Dirac phonons into two classes: without or with inversion symmetry, thereby clarifying the minimal symmetry requirements for enforcing 2D Dirac points. Based on symmetry analysis, we discovered that screw symmetries, together with time-reversal symmetry, play an essential role in the existence of Dirac points. To validate this result, we constructed the k·p model to describe the Dirac phonons and discussed their topological features accordingly. We found that a 2D Dirac point could be considered as a composition of two 2D Weyl points with opposite chirality. Furthermore, we provided two concrete materials to demonstrate our findings. Overall, our work provides a more detailed study of 2D Dirac points in spinless systems and clarifies their topological features.

All dielectric zero-index metamaterial for TE/TM polarization

In this article, a novel design of a dielectric vein type zero-index metamaterial (ZIM) has been proposed, which can be made to operate for any of the two polarisations (transverse electric (TE) or transverse magnetic (TM)) by specifically choosing the geometrical parameters. The zero-index behaviour of the metamaterial is due the linear dispersion and existence of Dirac point at the center of the Brillouin zone. Although, ZIM sturctures comprising periodic arrays of rods-in-air and holes-in-dielectric have already been proposed, their zero-index property is restricted to TM and TE polarization, respectively. However, the structure proposed in this articles exhibits effectively zero refractive index in the TM as well as the TE mode, and hence, has an upper hand over existing dielectric ZIM structures.

Hidden symmetry and protection of Dirac points on the honeycomb lattice.

The honeycomb lattice possesses a novel energy band structure, which is characterized by two distinct Dirac points in the Brillouin zone, dominating most of the physical properties of the honeycomb structure materials. However, up till now, the origin of the Dirac points is unclear yet. Here, we discover a hidden symmetry on the honeycomb lattice and prove that the existence of Dirac points is exactly protected by such hidden symmetry. Furthermore, the moving and merging of the Dirac points and a quantum phase transition, which have been theoretically predicted and experimentally observed on the honeycomb lattice, can also be perfectly explained by the parameter dependent evolution of the hidden symmetry.

Evolution of the Hofstadter butterfly in a tunable optical lattice

Recent advances in realizing artificial gauge fields on optical lattices promise experimental detection of topologically non-trivial energy spectra. Self-similar fractal energy structures generally known as Hofstadter butterflies depend sensitively on the geometry of the underlying lattice, as well as the applied magnetic field. The recent demonstration of an adjustable lattice geometry [L. Tarruell \textit{et al.}, Nature 483, 302--305 (2012)] presents a unique opportunity to study this dependence. In this paper, we calculate the Hofstadter butterflies that can be obtained in such an adjustable lattice and find three qualitatively different regimes. We show that the existence of Dirac points at zero magnetic field does not imply the topological equivalence of spectra at finite field. As the real-space structure evolves from the checkerboard lattice to the honeycomb lattice, two square lattice Hofstadter butterflies merge to form a honeycomb lattice butterfly. This merging is topologically non-trivial, as it is accomplished by sequential closings of gaps. Ensuing Chern number transfer between the bands can be probed with the adjustable lattice experiments. We also calculate the Chern numbers of the gaps for qualitatively different spectra and discuss the evolution of topological properties with underlying lattice geometry.

Construction of Dirac points using triangular supercrystals

We show a methodology for how to construct Dirac points that occur at the corners of Brillouin zone as the Photonic counterparts of graphene. We use a triangular lattice with circular holes on a silicon substrate to create a Coupled Photonic Crystal Resonator Array (CPCRA) which its cavity resonators play the role of carbon atoms in graphene. At first we draw the band structure of our CPCRA using the tight-binding method. For this purpose we first designed a cavity which its resonant frequency is approximately at the middle of the first H-polarization band gap of the basis triangular lattice. Then we obtained dipole modes and magnetic field distribution of this cavity using the Finite Element Method (FEM). Finally we drew the two bands that construct the Dirac points together with the frequency contour plots for both bands and compared with the Plane Wave Expansion (PWE) and FEM results to prove the existence of Dirac point in the H-polarization band structure of lattices with air holes.

Two-dimensional materials with Dirac cones: Graphynes containing heteroatoms

Two graphynes, 6(H${}_{2}$),14,18 graphyne and 6${}_{\text{BN}}$,6,12 graphyne, that contain the heteroatoms hydrogen or boron and nitrogen, respectively, are shown to feature Dirac points in their band structure according to first-principles electronic structure calculations. This shows that the existence of Dirac points in the band structure of two-dimensional materials is neither restricted to graphene nor to other two-dimensional all-carbon materials. 6(H${}_{2}$),14,18 graphyne and 6${}_{\text{BN}}$,6,12 graphyne belong to the rectangular two-dimensional space groups $pmm$ and $pm$, respectively, and thus exhibit completely different symmetries than graphene. 6${}_{\text{BN}}$,6,12 graphyne features a Dirac cone with a band gap originating from an avoided crossing of its valence and conduction band at the Dirac point due to missing reflection symmetry. The examples of 6(H${}_{2}$),14,18 graphyne and 6${}_{\text{BN}}$,6,12 graphyne suggest that a wealth of two-dimensional materials with various chemical compositions and with electronic properties equally amazing as those of graphene is awaiting discovery.

Waves in Honeycomb Structures

We review recent work of the authors on the non-relativistic Schrodinger equation with a honeycomb lattice potential, $V$. In particular, we summarize results on (i) the existence of Dirac points, conical singularities in dispersion surfaces of $H_V=-\Delta+V$ and (ii) the two-dimensional Dirac equations, as a large, but finite time, effective description of $e^{-iH_Vt}\psi_0$, for data $\psi_0$, which is spectrally localized at a Dirac point. We conclude with a formal derivation and discussion of the effective large time evolution for the nonlinear Schrodinger - Gross Pitaevskii equation for small amplitude initial conditions, $\psi_0$. The effective dynamics are governed by a nonlinear Dirac system.