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We construct the general extension of the four-dimensional Jackiw-Rossi-Dirac Hamiltonian that preserves the antilinear reflection symmetry between the positive and negative energy eigenstates. Among other systems, the resulting Hamiltonian…
We consider a particle moving on a cone and bound to its tip by $1/r$ or harmonic oscillator potentials. When the deficit angle of the cone divided by $2 \pi$ is a rational number, all bound classical orbits are closed. Correspondingly, the…
We present the electronic properties of massless Dirac fermions characterized by geometry and topology on a graphene sheet in this chapter. Topological effects can be elegantly illuminated by the Atiyah-Singer index theorem. It leads to a…
We consider the combined effect of a gap and a Zeeman interaction on the helical Dirac fermions which exist on the surface of a topological insulator. Magneto-optical properties, the magnetization, Hall effect and the density of states are…
We use supersymmetry transformations to obtain new one parameter family of inhomogeneous magnetic fields $\mathbf{B} = \widetilde{\mathcal{B}}(x,\lambda) \hat{e}_z$ for which the massless Dirac electron possesses exact solution. The…
In the mixed state of an extreme type-II d-wave superconductor and within a broad regime of weak magnetic fields H_c1 << H << H_c2, the low energy Bogoliubov-deGennes quasiparticles can be effectively described as Dirac fermions moving in…
Topological defects, such as quantum vortices, determine the properties of quantum fluids. The study of their properties has been at the center of activity in solid state and BEC communities. On the other hand, the non-trivial behavior of…
Graphene-based superlattice (SL) formed by a periodic gap modulation is studied theoretically using a Dirac-type Hamiltonian. Analyzing the dispersion relation we have found that new Dirac points arise in the electronic spectrum under…
Itinerant electrons in a two-dimensional Kagome lattice form a Dirac semi-metal, similar to graphene. When lattice and spin symmetries are broken by various periodic perturbations this semi-metal is shown to spawn interesting non-magnetic…
In view of the many quantum field theoretical descriptions of graphene in $2+1$ dimensions, we present another field theoretical feature of graphene, in the presence of defects. Particularly, we shall be interested in gapped graphene in the…
Extended defects in graphene, such as linear edges, break the translational invariance and can also have an impact on the symmetries specific to massless Dirac-like quasiparticles in this material. The paper examines the consequences of a…
Graphene on hexagonal boron nitride (hBN) can exhibit a topological phase via mutual crystallographic alignment. Recent measurements of nonlocal resistance ($R_{nl}$) near the secondary Dirac point (SDP) in ballistic graphene/hBN…
Recent studies show that periodic potentials can generate superlattice Dirac points at energies in graphene (is the Fermi velocity of graphene and G is the reciprocal superlattice vector). Here, we perform scanning tunneling microscopy and…
Formation and evolution of topological defects in course of non-equilibrium symmetry breaking phase transitions is of wide interest in many areas of physics, from cosmology through condensed matter to low temperature physics. Its study in…
A general and beautiful picture for the realization of topological insulators is that the mass term of the Dirac model has a nodal surface wrapping one Dirac point. We show that this geometric picture based on Dirac points can be…
We present a unified description of the quantum Hall effect in graphene on the basis of the 8-component Dirac Hamiltonian and the supersymmetric (SUSY) quantum mechanics. It is remarkable that the zero-energy state emerges because the…
This paper provides a pedagogical introduction to recent developments in geometrical and topological band theory following the discovery of graphene and topological insulators. Amusingly, many of these developments have a connection to…
We present full description of spectra for a Hamiltonian defined on periodic hexagonal elastic lattices. These continua are constructed out of Euler-Bernoulli beams, each governed by a scalar-valued self-adjoint operator, which is also…
Gapless Dirac surface states are protected at the interface of topological and normal band insulators. In a binary superlattice bearing such interfaces, we establish that valley-dependent dimerization of symmetry-unrelated Dirac surface…
The Dirac cone underlies many unique electronic properties of graphene and topological insulators, and its band structure--two conical bands touching at a single point--has also been realized for photons in waveguide arrays, atoms in…