Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Topological characterization of Landau levels for $2$D massless Dirac
  fermions in $3$D layered systems   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 Landau's 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.

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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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The categories of this paper are: ao-sci
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The categories of this paper are: eess
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The categories of this paper are: nlin
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The categories of this paper are: hep-ex
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The categories of this paper are: math
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex