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A topological concern is addressed in view of the extensively and intensively studied topological phases of condensed matter. In this realm, the phases with topological order cannot be characterized by symmetry alone. Moreover, the relevant phase transitions do occur without spontaneous symmetry breaking, beyond the scope of Landaus theory. The first realization of such phases is the discovery of the integer quantum Hall effect (QHE), which was followed soon by a topological interpretation. Later on, a distinct, half-integer QHE was also found from graphene, which has almost spin degeneracy described by $SU(2)$ symmetry. The previous theoretical predictions were realized in this finding. It has been well understood that the anomaly of this half-integer QHE originates from the presence of $2$D massless Dirac fermions around the zero energy with respect to the original Dirac points (DPs). The very characteristic lies in that there exists a topologically robust zero-mode LL that is a constant function of the perpendicular magnetic field. More deeply, this zero-mode LL is protected by the local chiral symmetry (CS), against CS preserving perturbations provided that intervalley scattering between the double DPs is inhibited, where the CS arises from the global sublattice symmetry in spinless graphene. Since massless Dirac particles are broadly present in condensed matter with various symmetries, not to mention Dirac bosonic systems, it is of interest to see how about the situations in other systems with $2$D massless Dirac fermions. We address several notes in a topological viewpoint on the presence of $2$D massless Dirac fermions in $3$D layered systems.In particular, we focus on the zero-mode LL since this LL signifies $2$D massless Dirac fermions.
The recent theoretical prediction and experimental realization of topological insulators (TI) has generated intense interest in this new state of quantum matter. The surface states of a three-dimensional (3D) TI such as Bi_2Te_3, Bi_2Se_3 and Sb_2Te_
The effect of disorder on the Landau levels of massless Dirac fermions is examined for the cases with and without the fermion doubling. To tune the doubling a tight-binding model having a complex transfer integral is adopted to shift the energies of
We present a magneto-infrared spectroscopy study on a newly identified three-dimensional (3D) Dirac semimetal ZrTe$_5$. We observe clear transitions between Landau levels and their further splitting under magnetic field. Both the sequence of transiti
In this article we discuss generalized harmonic confinement of massless Dirac fermions in (2+1) dimensions using smooth finite magnetic fields. It is shown that these types of magnetic fields lead to conditional confinement, that is confinement is po
The intense search for topological superconductivity is inspired by the prospect that it hosts Majorana quasiparticles. We explore in this work the optimal design for producing topological superconductivity by combining a quantum Hall state with an o