17.7. The BCS superconducting ground state
221
c ˆ
† c ˆ is the number operator for the electrons, which because of the Pauli
k k
Principle has eigenvalues 0 or 1; this term is of course completely analogous
to (7.55), and sums the single particle energies ∈ k for each occupied level.
ˆ
We immediately note that H BCS is invariant under the global U(1) transformation
c ˆ → c ˆ
′ = e
−iα c ˆ
(17.110)
k
k
k
ˆ
ˆ
for all k, which is equivalent to ψ
′ (x) = e
−iα ψ(x) for the electron field operator
ˆ
at x. Thus fermion number is conserved by H BCS . However, just as for
the superfluid, we shall see that the BCS ground state does not respect the
symmetry.
We follow Bogoliubov (1958) and Bogoliubov et al. (1959) (see also Valatin
1958), and make a canonical transformation on the operators c ˆ , ˆ
c
†
similar
k −k
to the one employed for the superfluid problem in (17.38), as motivated by
the ‘pair condensate’ picture. We set
ˆ
β
= u k c ˆ − v k c ˆ
† ,
β
† = u k c ˆ
† − v k c ˆ
k
k
−k
k
k
−k
ˆ
β
= u
−k
k c ˆ + v
−k
k c ˆ
† ,
β
†
= u
v
111)
k
k c ˆ
†
+ k c ˆ
(17.
k
−k
−k
where u k and v k are real, depend only on k = |k|, and are chosen so as to
preserve anticommutation relations for the β’s. This last condition implies
(problem 17.7)
u
2
2
k + v k = 1
(17.112)
so that we may conveniently set
u k = cos θ k , v k = sin θ k .
(17.113)
Just as in the superfluid case, the transformations (17.111) only make sense in
the context of a number non-conserving ground state, since they do not respect
ˆ
the symmetry (17.110). Although H BCS of (17.109) is number conserving, we
shall shortly make a crucial number non-conserving approximation.
We seek a diagonalization of (17.109), analogous to (17.40), in terms of
ˆ
ˆ
the mode operators β and β
† :
∑
ˆ
ˆ
H BCS =
ω k (β
† ˆ
ˆ
β
† ˆ
+ β β ) + γ.
(17.114)
k k
−k −k
k
It is easy to check (problem 17.8) that the form (17.114) implies
ˆ
ˆ
[H BCS , β
†
ˆ
] = ω β
l
l
†
(17.115)
l
as in (17.41), despite the fact that the operators obey anticommutation relations. Equation (17.115) then implies that the ω k are the energies of states
ˆ
ˆ
created by the quasiparticle operators β
† and β
† , the ground state being
k
−k
defined by
ˆ
β k |ground›
ˆ
BCS = β −k |ground› BCS = 0.
(17.116)
221
c ˆ
† c ˆ is the number operator for the electrons, which because of the Pauli
k k
Principle has eigenvalues 0 or 1; this term is of course completely analogous
to (7.55), and sums the single particle energies ∈ k for each occupied level.
ˆ
We immediately note that H BCS is invariant under the global U(1) transformation
c ˆ → c ˆ
′ = e
−iα c ˆ
(17.110)
k
k
k
ˆ
ˆ
for all k, which is equivalent to ψ
′ (x) = e
−iα ψ(x) for the electron field operator
ˆ
at x. Thus fermion number is conserved by H BCS . However, just as for
the superfluid, we shall see that the BCS ground state does not respect the
symmetry.
We follow Bogoliubov (1958) and Bogoliubov et al. (1959) (see also Valatin
1958), and make a canonical transformation on the operators c ˆ , ˆ
c
†
similar
k −k
to the one employed for the superfluid problem in (17.38), as motivated by
the ‘pair condensate’ picture. We set
ˆ
β
= u k c ˆ − v k c ˆ
† ,
β
† = u k c ˆ
† − v k c ˆ
k
k
−k
k
k
−k
ˆ
β
= u
−k
k c ˆ + v
−k
k c ˆ
† ,
β
†
= u
v
111)
k
k c ˆ
†
+ k c ˆ
(17.
k
−k
−k
where u k and v k are real, depend only on k = |k|, and are chosen so as to
preserve anticommutation relations for the β’s. This last condition implies
(problem 17.7)
u
2
2
k + v k = 1
(17.112)
so that we may conveniently set
u k = cos θ k , v k = sin θ k .
(17.113)
Just as in the superfluid case, the transformations (17.111) only make sense in
the context of a number non-conserving ground state, since they do not respect
ˆ
the symmetry (17.110). Although H BCS of (17.109) is number conserving, we
shall shortly make a crucial number non-conserving approximation.
We seek a diagonalization of (17.109), analogous to (17.40), in terms of
ˆ
ˆ
the mode operators β and β
† :
∑
ˆ
ˆ
H BCS =
ω k (β
† ˆ
ˆ
β
† ˆ
+ β β ) + γ.
(17.114)
k k
−k −k
k
It is easy to check (problem 17.8) that the form (17.114) implies
ˆ
ˆ
[H BCS , β
†
ˆ
] = ω β
l
l
†
(17.115)
l
as in (17.41), despite the fact that the operators obey anticommutation relations. Equation (17.115) then implies that the ω k are the energies of states
ˆ
ˆ
created by the quasiparticle operators β
† and β
† , the ground state being
k
−k
defined by
ˆ
β k |ground›
ˆ
BCS = β −k |ground› BCS = 0.
(17.116)