Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Intrinsic Zeeman Effect in Graphene   The intrinsic Zeeman energy is precisely one half of the cyclotron energy for
electrons in graphene. As a result a Landau-level mixing occurs to create the
energy spectrum comprised of the $4j$-fold degenerated zero-energy level and
4-fold degenerated nonzero-energy levels in the $j$-layer graphene, where
$j=1,2,3$ for monolayer, bilayer and trilayer, respectively. The degeneracy
manifests itself in the quantum Hall (QH) effect. We study how the degeneracy
is removed by the Coulomb interactions. With respect to the zero-energy level,
an excitonic gap opens by making a BCS-type condensation of electron-hole pairs
at the filling factor $\nu =0$. It gives birth to the Ising QH ferromagnet at
$\nu =\pm 1$ for monolayer, $\nu =\pm 1,\pm 3$ for bilayer, and $\nu =\pm 1,\pm
3,\pm 5$ for trilayer graphene from the zero-energy degeneracy. With respect to
the nonzero-energy level, a remarkable consequence is derived that the
effective Coulomb potential depends on spins, since a single energy level
contains up-spin and down-spin electrons belonging to different Landau levels.
The spin-dependent Coulomb interaction leads to the valley polarization at $\nu
=\pm 4, \pm 8, \pm 12, ...$ for monolayer, $\nu =\pm 2, \pm 6, \pm 10, >...$
for bilayer, and $\nu =\pm 2,\pm 4, \pm 8, \pm 12, ...$ for trilayer graphene.

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The categories of this paper are: hep-ph
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
2
The categories of this paper are: math
25
The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph