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effectively one-dimensional
.Tothatextent,
in a final theory
, we may expect
our spin-1 excitation to be a triplet bound state of 'two neutral spin
−
2
e
+
e
−
electron-hole pairs'. Further, as the energy of
the spin-1 quantum approaches zero the binding energy also approaches
zero and the electron-hole bound state wave function becomes elliptical,
with diverging size. We may then view the low energy spin-1 quanta as a
'critically (loosely) bound' two spinon state, very much like the quantum
number fractionization of the des Cloizeaux-Pearson spin-1 excitation in
the 1d spin-
2
spinons' rather than '
antiferromagnetic Heisenberg model. Our result also suggest
a non-linear sigma model and novel 2 + 1 dimensional bosonization scheme
for graphite
[
26
]
.
We find
[
27
]
that our spin-1 collective mode survives in carbon nano-
tubes in a modified fashion. Preliminary study shows that three dimen-
sional semimetals Bi, HgTe and
-Sn do not have spin-1 collective modes
at low energies, because of quadratic dispersion at the zero gap.
α
3. Relativistic Type Effects
Phenomena unique to relativistic quantum field theories, such as Klein tun-
nelling and Zitterbewegung, have been predicted
[
3
]
to occur in graphene,
because of the mathematical similarity of the graphene electron dynamics
to dynamics of massless Dirac electronsin2+1dimensions,withfermi
velocity playing the role of light velocity. Partly keeping the above in mind,
along with Lukose and Shankar
[
28
]
we investigated the effect of a uniform
electric field, applied along the graphene sheet, on its already anomalous
Landau level spectrum. We find that, within the low energy approximation
near the Fermi surface (Dirac points), the problem can be exactly solved.
We find strikingly new effects of electric field on the Landau levels which is
different from the Landau levels of standard
2
electron gas
. What we find
can be termed as analogue of Lorenz boost, Lorenz contraction and time
dilation that one is familiar in the context of special theory of relativity.
We find that the Landau spectrum gets scaled, for a given
d
k
y
quantum
number, by an electric field dependent dimensionless parameter(
E
v
F
B
).
As the value of this parameter is increased, spacings between the Landau
levels decreases. This Landau level contraction is consequence of electric
field induced quantum mechanical mixing of Landau levels. The entire Lan-
dau level structure collapses at a critical value of this parameter. Further,
the 'relativistic' character of the spectrum (with Fermi velocity replacing
the velocity of light), leads to a novel interpretation of our result in terms of
β
=
