Interplay of Coulomb Repulsion and Spin–Orbit Coupling in Superconducting. . .
365
these combinations, for L = 0, 1, are listed in [15, 16]. We introduce the projectors
ˆ
P ± in Eq. (12) to decompose the gap equation on the eigenstates of ˆ
H 0 on the upper
(+) and lower (−) bands, that we, respectively, associate to eigenstates ±3/2 and
±1/2 of the helicity operator ˆ
λ = ˆ
k · ˆ
J. Note that, for a given value of J , there
is a finite number of components V that contribute in the summation in Eq. (10).
For example, for s−wave J = L = S = 0 only = 0 and 2 contribute. In the
following, we further simplify Eq. (10) by considering a gap function in a unique
(J, L, S) sector, Δ J,LS , and write A
J,LSLS
1 σ 2
= A
J,LS
1 σ 2
. A more refined analysis
would allow for mixing between different values of (L, S) for a fixed J .
It is expected that the amplitude of the pairing potential depends on the largest
combination of the coefficients V and A J,LS close the Fermi surface, where
σ 1 = σ 2 = − and k 1 = k 2 = k F . To be more accurate, one should consider the full
k−dependence but let us work in this simpler limit. It was shown that this amplitude
is the strongest for J = L = S = 0 [15], which is precisely the order parameter
we consider in our work (see Sect. 3). This logic of maximizing the product
V A
J,LS
applies well for superconductiviy from an attractive potential, like the
electron-phonon coupling, where the eigenvalue of Eq. (10) with the largest absolute
value, λ 1 (T ), is already positive. However, it is not straightforward to extend
to superconductivity from a repulsive potential, such as the Coulomb repulsion
between electrons, where in the s−wave channel the eigenvalue with the largest
absolute value is negative, λ 1 (T ) < 0, because of the overall repulsive nature
of the Coulomb potential. Then, the s−wave solution to Eq. (10) comes from the
second largest-in-absolute-value eigenvalue, λ 2 (T ) > 0, which corresponds to the
first electronic configuration where the Coulomb potential is attractive [9].
In a multi-orbital system, the coefficient A
J,LS
is a matrix that can have positive
or negative eigenvalues. Then, the superconducting pairing can be favored not only
by increasing the product V A
J,LS
but also by changing its sign. In the case of
the repulsive Coulomb potential in Eq. (10), the eigenvalue λ 1 (T ) with the largest
magnitude is negative for J = L = S = 0 and it will not be responsible for the
critical temperature defined by λ(T c ) = 1 > 0. Yet, the sign of λ 1 can be positive for
other order parameters Δ J,LS if A J LS
has a negative eigenvalue, which we obtain
for J = L = S = 1 = i.e. ˆ
N 111 =
√
3(−k z ( ˆ
J x +i ˆ
J y )+(k x +ik y )J z )/(
√
5k) [16],
and which decomposes as a matrix on the bands with helicity ±1/2 and ±3/2
A
111
1 =
2/5 3/10
3/10 0
.
(13)
This matrix has eigenvalues (2 ±
√
13)/10, with one negative (≈ −0.16). For this
channel, the sign of λ 1 can be positive and eventually be a solution to λ 1 (T c ) = 1.
This may have a larger critical temperature since we now consider the largest
eigenvalue in magnitude, instead of the second largest. However, it is difficult to
speculate on the resulting critical temperature and a refined study is needed to
evaluate the corresponding critical temperature.
365
these combinations, for L = 0, 1, are listed in [15, 16]. We introduce the projectors
ˆ
P ± in Eq. (12) to decompose the gap equation on the eigenstates of ˆ
H 0 on the upper
(+) and lower (−) bands, that we, respectively, associate to eigenstates ±3/2 and
±1/2 of the helicity operator ˆ
λ = ˆ
k · ˆ
J. Note that, for a given value of J , there
is a finite number of components V that contribute in the summation in Eq. (10).
For example, for s−wave J = L = S = 0 only = 0 and 2 contribute. In the
following, we further simplify Eq. (10) by considering a gap function in a unique
(J, L, S) sector, Δ J,LS , and write A
J,LSLS
1 σ 2
= A
J,LS
1 σ 2
. A more refined analysis
would allow for mixing between different values of (L, S) for a fixed J .
It is expected that the amplitude of the pairing potential depends on the largest
combination of the coefficients V and A J,LS close the Fermi surface, where
σ 1 = σ 2 = − and k 1 = k 2 = k F . To be more accurate, one should consider the full
k−dependence but let us work in this simpler limit. It was shown that this amplitude
is the strongest for J = L = S = 0 [15], which is precisely the order parameter
we consider in our work (see Sect. 3). This logic of maximizing the product
V A
J,LS
applies well for superconductiviy from an attractive potential, like the
electron-phonon coupling, where the eigenvalue of Eq. (10) with the largest absolute
value, λ 1 (T ), is already positive. However, it is not straightforward to extend
to superconductivity from a repulsive potential, such as the Coulomb repulsion
between electrons, where in the s−wave channel the eigenvalue with the largest
absolute value is negative, λ 1 (T ) < 0, because of the overall repulsive nature
of the Coulomb potential. Then, the s−wave solution to Eq. (10) comes from the
second largest-in-absolute-value eigenvalue, λ 2 (T ) > 0, which corresponds to the
first electronic configuration where the Coulomb potential is attractive [9].
In a multi-orbital system, the coefficient A
J,LS
is a matrix that can have positive
or negative eigenvalues. Then, the superconducting pairing can be favored not only
by increasing the product V A
J,LS
but also by changing its sign. In the case of
the repulsive Coulomb potential in Eq. (10), the eigenvalue λ 1 (T ) with the largest
magnitude is negative for J = L = S = 0 and it will not be responsible for the
critical temperature defined by λ(T c ) = 1 > 0. Yet, the sign of λ 1 can be positive for
other order parameters Δ J,LS if A J LS
has a negative eigenvalue, which we obtain
for J = L = S = 1 = i.e. ˆ
N 111 =
√
3(−k z ( ˆ
J x +i ˆ
J y )+(k x +ik y )J z )/(
√
5k) [16],
and which decomposes as a matrix on the bands with helicity ±1/2 and ±3/2
A
111
1 =
2/5 3/10
3/10 0
.
(13)
This matrix has eigenvalues (2 ±
√
13)/10, with one negative (≈ −0.16). For this
channel, the sign of λ 1 can be positive and eventually be a solution to λ 1 (T c ) = 1.
This may have a larger critical temperature since we now consider the largest
eigenvalue in magnitude, instead of the second largest. However, it is difficult to
speculate on the resulting critical temperature and a refined study is needed to
evaluate the corresponding critical temperature.