Tangent fermions: Dirac or Majorana fermions on a lattice without fermion doubling

I. Introduction II. Two-dimensional lattice fermions III. Methods to avoid fermion doubling (sine dispersion, sine plus cosine dispersion, staggered lattice dispersion, linear sawtooth dispersion, tangent dispersion) IV. Topologically protected Dirac cone V. Application: Klein tunneling (tangent fermions on a space-time lattice, wave packet propagation) VI. Application: Strong antilocalization (transfer matrix of tangent fermions, topological insulator versus graphene) VII. Application: Anomalous quantum Hall effect (gauge invariant tangent fermions, topologically protected zeroth Landau level) VIII. Application: Majorana metal (Dirac versus Majorana fermions, phase diagram) IX. OutlookComment: review article, 26 pages, 13 figures; V2: added three appendices, and provided code for the various implementation

Magnetic breakdown spectrum of a Kramers-Weyl semimetal

We calculate the Landau levels of a Kramers-Weyl semimetal thin slab in a perpendicular magnetic field $B$. The coupling of Fermi arcs on opposite surfaces broadens the Landau levels with a band width that oscillates periodically in $1/B$. We interpret the spectrum in terms of a one-dimensional superlattice induced by magnetic breakdown at Weyl points. The band width oscillations may be observed as $1/B$-periodic magnetoconductance oscillations, at weaker fields and higher temperatures than the Shubnikov-de Haas oscillations due to Landau level quantization. No such spectrum appears in a generic Weyl semimetal, the Kramers degeneracy at time-reversally invariant momenta is essential.Comment: 13 pages, 18 figure

Localization landscape for Dirac fermions

In the theory of Anderson localization, a landscape function predicts where wave functions localize in a disordered medium, without requiring the solution of an eigenvalue problem. It is known how to construct the localization landscape for the scalar wave equation in a random potential, or equivalently for the Schr\"{o}dinger equation of spinless electrons. Here we generalize the concept to the Dirac equation, which includes the effects of spin-orbit coupling and allows to study quantum localization in graphene or in topological insulators and superconductors. The landscape function $u(r)$ is defined on a lattice as a solution of the differential equation $\overline{{H}}u(r)=1$, where $\overline{{H}}$ is the Ostrowsky comparison matrix of the Dirac Hamiltonian. Random Hamiltonians with the same (positive definite) comparison matrix have localized states at the same positions, defining an equivalence class for Anderson localization. This provides for a mapping between the Hermitian and non-Hermitian Anderson model.Comment: 6 pages, 6 figure

Tractography-guided surgery of brain tumours: what is the best method to outline the corticospinal tract?

Background: Diffusion tensor imaging (DTI) is the imaging technique used in vivo to visualise white matter pathways. The cortico-spinal tract (CST) belongs to one of the most often delineated tracts preoperatively, although the optimal DTI method has not been established yet. Considering that various regions of interests (ROIs) could be selected, the reproducibility of CST tracking among different centres is low. We aimed to select the most reliable tractography method for outlining the CST for neurosurgeons. Materials and methods: Our prospective study consisted of 32 patients (11 males, 21 females) with a brain tumour of various locations. DTI and T1-weighed image series were acquired prior to the surgery. To draw the CST, the posterior limb of the internal capsule (PLIC) and the cerebral peduncle (CP) were defined as two main ROIs. Together with these main ROIs, another four cortical endpoints were selected: the frontal lobe (FL), the supplementary motor area (SMA), the precentral gyrus (PCG) and the postcentral gyrus (POCG). Based on these ROIs, we composed ten virtual CSTs in DSI Studio. The fractional anisotropy, the mean diffusivity, the tracts’ volume, the length and the number were compared between all the CSTs. The degree of the CST infiltration, tumour size, the patients’ sex and age were examined. Results: Significant differences in the number of tracts and their volume were observed when the PLIC or the CP stood as a single ROI comparing with the two- ROI method (all p < 0.05). The mean CST volume was 40054U (SD ± 12874) and the number of fibres was 259.3 (SD ± 87.3) when the PLIC was a single ROI. When the CP was a single ROI, almost a half of fibres (147.6; SD ± 64.0) and half of the CST volume (26664U; SD ± 10059U) was obtained (all p < 0.05). There were no differences between the various CSTs in terms of fractional anisotropy, mean diffusivity, the apparent diffusion coefficient, radial diffusivity and the tract length (p > 0.05). The CST was infiltrated by a growing tumour or oedema in 17 of 32 patients; in these cases, the mean and apparent diffusion of the infiltrated CST was significantly higher than in uncompromised CSTs (p = 0.04). CST infiltration did not alter the other analysed parameters (all p > 0.05). Conclusions: A universal method of DTI of the CST was not developed. However, we found that the CP or the PLIC (with or without FL as the second ROI) should be used to outline the CST

Chirality inversion of Majorana edge modes in a Fu-Kane heterostructure

Fu and Kane have discovered that a topological insulator with induced s-wave superconductivity (gap Delta(0), Fermi velocity v (F), Fermi energy mu) supports chiral Majorana modes propagating on the surface along the edge with a magnetic insulator. We show that the direction of motion of the Majorana fermions can be inverted by the counterflow of supercurrent, when the Cooper pair momentum along the boundary exceeds Delta(2)(0)/mu v(F) . The chirality inversion is signaled by a doubling of the thermal conductance of a channel parallel to the supercurrent. Moreover, the inverted edge can transport a nonzero electrical current, carried by a Dirac mode that appears when the Majorana mode switches chirality. The chirality inversion is a unique signature of Majorana fermions in a spinful topological superconductor: it does not exist for spinless chiral p-wave pairing.Theoretical Physic

Massless dirac fermions on a space‐time lattice with a topologically protected dirac cone

The symmetries that protect massless Dirac fermions from a gap opening may become ineffective if the Dirac equation is discretized in space and time, either because of scattering between multiple Dirac cones in the Brillouin zone (fermion doubling) or because of singularities at zone boundaries. Here an implementation of Dirac fermions on a space-time lattice that removes both obstructions is introduced. The quasi-energy band structure has a tangent dispersion with a single Dirac cone that cannot be gapped without breaking both time-reversal and chiral symmetries. It is shown that this topological protection is absent in the familiar single-cone discretization with a linear sawtooth dispersion, as a consequence of the fact that there the time-evolution operator is discontinuous at Brillouin zone boundaries.Theoretical Physic

Experimental and theoretical investigation of ligand effects on the synthesis of ZnO nanoparticles

ZnO nanoparticles with highly controllable particle sizes(less than 10 nm) were synthesized using organic capping ligands in Zn(Ac)2 ethanolic solution. The molecular structure of the ligands was found to have significant influence on the particle size. The multi-functional molecule tris(hydroxymethyl)-aminomethane (THMA) favoured smaller particle distributions compared with ligands possessing long hydrocarbon chains that are more frequently employed. The adsorption of capping ligands on ZnnOn crystal nuclei (where n = 4 or 18 molecular clusters of(0001) ZnO surfaces) was modelled by ab initio methods at the density functional theory (DFT) level. For the molecules examined, chemisorption proceeded via the formation of Zn...O, Zn...N, or Zn...S chemical bonds between the ligands and active Zn2+ sites on ZnO surfaces. The DFT results indicated that THMA binds more strongly to the ZnO surface than other ligands, suggesting that this molecule is very effective at stabilizing ZnO nanoparticle surfaces. This study, therefore, provides new insight into the correlation between the molecular structure of capping ligands and the morphology of metal oxide nanostructures formed in their presence

Surfactant-Assisted in situ Chemical Etching for the General Synthesis of ZnO Nanotubes Array

In this paper, a general low-cost and substrate-independent chemical etching strategy is demonstrated for the synthesis of ZnO nanotubes array. During the chemical etching, the nanotubes array inherits many features from the preformed nanorods array, such as the diameter, size distribution, and alignment. The preferential etching along c axis and the surfactant protection to the lateral surfaces are considered responsible for the formation of ZnO nanotubes. This surfactant-assisted chemical etching strategy is highly expected to advance the research in the ZnO nanotube-based technology