Chemistry Reference
In-Depth Information
m
±
Note that Ψ
B
(0) receives contribution from
=
1 channel while for
Ψ
A
(0), the
m
= 0 channel contributes. The Kondo coupling of the electrons
with the impurity spin is given by
N
c
−
1
N
f
N
s
N
c
2
Ψ
s †
lα
a
αβ
Ψ
lβ
(0)
a
H
K
τ
S
,
=
(0)
(6)
k
c
s
=1
l
=1
α,β
=1
a
=1
where
is the effective Kondo coupling for energy scales up to the cutoff Λ,
S
denotes the spin at the impurity site,
g
N
c
)
spin group, and we have now generalized the fermions, in the spirit of large
N
τ
are the generators of the SU(
analysis, to have
N
s
flavors (valley indices)
N
f
colors (sublattice indices)
and
N
c
spin. For realistic systems
N
f
=
N
c
=
N
s
=2. Herewehave
chosen Kondo coupling
to be independent of sublattice and valley indices.
This is not a necessary assumption. However, we shall avoid extension of
our analysis to flavor and/or color dependent coupling term for simplicity.
Also, the Dirac nature of the graphene conduction electrons necessitates
the Kondo Hamiltonian to mix
g
= 0 channels (Eqs. (5) and
(6)). This is in complete contrast to the conventional Kondo systems where
the Kondo coupling involves single angular momentum channel.
The kinetic energy of the Dirac electrons can also be expressed in terms
of the
m
=
±
1and
m
ξ
fields:
H
0
=
∞
0
∞
kdk
2
ms †
+
α
ξ
ms
E
+
(
k
)
ξ
+
α
π
m
=
−∞
s,α
ms †
−α
ξ
ms
+
E
−
(
k
)
ξ
.
(7)
−α
Typically such a term involves all angular momenta channels. For our
purpose here, it will be enough to consider the contribution from electrons
in the
1 channels which contribute to scattering from the impurity
(Eqs. (5) and (6)). To make further analytical progress, we now unfold the
range of momenta
m
=0
, ±
c
1(2)
α
k
from (0
,∞
)to(
−∞,∞
) by defining the fields
)=
0(
−
sgn(
s
))
s
+
α
s
1(2)
α
c
k
|k|ξ
|k|
,
k>
,
(
(
)
0
)
|k|ξ
0(
−
sgn(
s
))
s
−α
c
1(2)
α
(
k
)=+(
−
(
|k|
)
,
k<
0
,
(8)
so that one can express the Ψ fields as Ψ
s
A
(
B
)
α
(0) =
∞
dk
2
π
|k|c
1(2)
α
(
k
).
−∞
s
1(2)
α
In terms of the
c
fields, the kinetic energy (in the
m
=0
, ±
1 channels)
