Carsten Timm: Theory of superconductivity

The constant is irrelevant for the following derivation and is omitted from now on. Since H BCS is bilinear in c, c †
it describes a non-interacting effective system. But what is unusual is that H BCS contains terms of the form cc
and c † c † , which do not conserve the electron number. We thus expect that the eigenstates of H BCS do not have a
sharp electron number. We had already seen that the BCS ground state has this property. This is a bit strange
since superpositions of states with different electron numbers are never observed. One can formulate the theory
of superconductivity in terms of states with fixed electron number, but this formulation is cumbersome and we
will not pursue it here.
To diagonalize H BCS , we introduce new fermionic operators, which are linear combinations of electron creation
and annihilation operators,
(
γk↑
γ † −k,↓
) ( u
∗
= k
−v k
v ∗ k
u k
) (
ck↑
c † −k,↓
)
. (10.8)
This mapping is called Bogoliubov (or Bogoliubov-Valatin) transformation. Again, it is not clear yet that u k , v k
are related to the previously introduced quantities denoted by the same symbols. For the γ to satisfy fermionic
anticommutation relations, we require
{ }
γ k↑ , γ † k↑
= γ k↑ γ † k↑ + γ† k↑ γ k↑
= u ∗ ku k c k↑ c † k↑ − u∗ kv ∗ k c k↑ c −k,↓ − v k u k c † −k,↓ c† k↑ + v kv ∗ k c † −k,↓ c −k,↓
+ u k u ∗ k c † k↑ c k↑ − u k v k c † k↑ c† −k,↓ − v∗ ku ∗ k c −k,↓ c k↑ + vkv ∗ k c −k,↓ c †
{ −k,↓
}
}
= |u k | 2 c k↑ , c † k↑
−u ∗ kvk ∗ {c k↑ , c −k,↓ } −v k u k
{c { }
†
−k,↓
} {{ } } {{ }
, c† k↑
+ |v k | 2 c † −k,↓ , c −k,↓
} {{ } } {{ }
= 0
= 1
= 0
= 1
= |u k | 2 + |v k | 2 ! = 1. (10.9)
Using this constraint, we find the inverse transformation,
( ) ( ) (
ck↑ uk v
=
k γk↑
c † −k,↓
−v ∗ k
u ∗ k
γ † −k,↓
)
. (10.10)
Insertion into H BCS yields
H BCS = ∑ ) (
) (
) )
{ξ k
(u ∗ kγ † k↑ + v∗ kγ −k,↓ u k γ k↑ + v k γ † −k,↓
+ ξ k −vkγ ∗ k↑ + u ∗ kγ † −k,↓
(−v k γ † k↑ + u kγ −k,↓
k
) (
) ) (
)}
−∆ ∗ k
(−v k γ † k↑ + u kγ −k,↓ u k γ k↑ + v k γ † −k,↓
− ∆ k
(u ∗ kγ † k↑ + v∗ kγ −k,↓ −vkγ ∗ k↑ + u ∗ kγ † −k,↓
= ∑ {(
)
ξ k |u k | 2 − ξ k |v k | 2 + ∆ ∗ kv k u k + ∆ k u ∗ kvk
∗ γ † k↑ γ k↑
k
(
+
(
+
+
)
−ξ k |v k | 2 + ξ k |u k | 2 + ∆ ∗ ku k v k + ∆ k vku ∗ ∗ k γ † −k,↓ γ −k,↓
ξ k u ∗ kv k + ξ k u ∗ kv k + ∆ ∗ kvk 2 − ∆ k (u ∗ k) 2) γ † k↑ γ† −k,↓
(ξ k vku ∗ k + ξ k vku ∗ k − ∆ ∗ ku 2 k + ∆ k (vk) ∗ 2) }
γ −k,↓ γ k↑ + const. (10.11)
The coefficients u k , v k should now be chosen such that the γγ and γ † γ † terms vanish. This requires
Writing
2ξ k u ∗ kv k + ∆ ∗ kv 2 k − ∆ k (u ∗ k) 2 = 0. (10.12)
∆ k = |∆ k | e iϕ k
, (10.13)
u k = |u k | e iα k
, (10.14)
v k = |v k | e iβ k
, (10.15)
90