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b
ij
defined earlier with a minor change of notation). The su-
perconducting order parameter, off-diagonal long range order (ODLRO), is
a
α
(this is just
|
2
=
|
Φ
|
R
i
−
R
j
|−→∞
F
lim
(
R
i
−
R
j
)
.
(23)
R
j
)=
α
F
αα
(
R
i
−
where
R
j
). All results we show in this paper
are performed on lattices with 13
2
unit cells.
The superconducting order parameter
F
(
R
i
−
|
2
as a function of doping, cal-
culated for physical parameters corresponding to graphene, obtained using
the optimized wavefunction is shown in Fig. 8. Remarkably, a “supercon-
ducting dome”, reminiscent of cuprates,
[
49
]
is obtained and is consistent
with the RVB physics. The result indicates that undoped graphene had no
long range superconducting order consistent with physical arguments and
mean-field theory
[
40
]
of the phenomenological GB Hamiltonian. Interest-
ingly, the present calculation suggests an “optimal doping”
|
Φ
x
of about 0.2
at which the the ODLRO attains a maximum. These calculations strongly
suggest a superconducting ground state in doped graphene.
We now further investigate the system near optimal doping in order to
estimate
T
c
. Figure 9 shows a plot of the (singlet)pair correlation function
F
. The function has oscillations up to
about six to seven lattice spacings and then attains a nearly constant value.
From an exponential fit one can infer that the coherence length
(
r
) as function of the separation
r
of the
superconductor is about six to seven lattice spacings. A crude estimate of
an upper bound of transition temperature can then be obtained by using
results from weak coupling BCS theory, using
ξ
1
1
.
764
v
F
πξ
k
b
T
c
=
.
Conserva-
k
b
T
c
=
t
50
tive estimates give us
T
c
is about twice room temperature.
Evidently, this is an upper bound, and an order of magnitude lower than
the mean-field theory estimates of Black-Schaffer and Doniach
[
40
]
.Further
improvement of our estimate of
, i.e.,
T
c
becomes technically dicult. It is inter-
esting to compare these results with those obtained in a Hubbard model on
a square lattice that captures cuprate physics. In this latter case, a similar
estimate of the coherence length
is about two to three lattice spacings
[
49
]
; however, the hopping scale is nearly an order of magnitude lower and
the estimate of
ξ
T
Room
. Again, this provides further support
for the possibility of high temperature superconductivity in graphene.
It is important to ask about the possibility of competing orders that
could overshadow superconductivity at optimal doping that we have found.
Honerkamp [51] has addressed this issue by means of a functional renormal-
ization group study of a general Hamiltonian on the honeycomb lattice. He
T
c
is about 2
