306
Appendix
0 | ˆ
C
†
I
ˆ
O
(n)
J | 0 = 0
(A.6.22)
if the rank r of the operator ˆ
O
(n)
J is lower than the rank of C
†
I , that is, if r < [I ]. Thus,
the intended proof can be carried out by inspecting the rank and order of successive
contributions to the operator ˆ
O J .
The two parts ˆ
K
(0)
J and ˆ
K
(1)
J of the commutator (A.6.18) have the ranks [J ] and
[J ] + 1, respectively. More general, the commutator [ ˆ
A, ˆ
B] of two operators ˆ
A and ˆ
B
with definite ranks, r and r
, respectively, is of rank r + r
− 1. Another commutator
feature relates to the number of unphysical fermion operators in an operator product:
A commutator [ ˆ
A, ˆ
B] of a general product operator ˆ
A and a physical operator ˆ
B, as
in ˆ
K
(1)
J , reduces the number of unphysical fermion operators in ˆ
A by 1. Accordingly,
the terms of ˆ
K
(1)
J exhibit at most three unphysical c-operators compared to the four
unphysical operators in ˆ
H I . As a particular case, the zeroth-order commutator ˆ
K
(0)
J
consists only of physical c-operators, reflecting the fact that ˆ
H 0 is a diagonal oneparticle operator. As a consequence, ˆ
K
(0)
J commutes with the CC operator ˆ
T so that
e
− ˆ
T ˆ
K
(0)
J e
ˆ
T
= ˆ
K
(0)
J
(A.6.23)
Accordingly, the matrix element (A.6.17) can be written as the sum
M
cc
I J = = 0 | ˆ
C
†
I
ˆ
K
(0)
J | 0 + + 0 | ˆ
C
†
I e
− ˆ
T ˆ
K
(1)
J e
ˆ
T
| 0
(A.6.24)
of a zeroth-order contribution and a contribution involving ˆ
K
(1)
J , being at least of first
order. Recalling that the rank of ˆ
K
(0)
J is [J ], the zeroth-order term, M
cc(0)
I J , necessarily
vanishes unless [I ] = [J ]; that is, both I and J belong to the same excitation class;
to put it differently, zeroth-order contributions arise only in the diagonal blocks of
M
cc .
To analyze the higher-order contribution, we make use of the Baker–Hausdorff
(BH) expansion according to
e
− ˆ
T ˆ
K
(1)
J e
ˆ
T
= ˆ
K
(1)
J + [ ˆ
K
(1)
J , ˆ
T ] +
1
2
[[ ˆ
K
(1)
J , ˆ
T ], ˆ
T ] +
1
6
[[[ ˆ
K
(1)
J , ˆ
T ], ˆ
T ], ˆ
T ]
(A.6.25)
which here terminates after the threefold commutator since each successive commutator eliminates one of the original three unphysical c-operators in ˆ
K
(1)
J . The BH
expansion (A.6.25) begins with ˆ
K
(1)
J , being of the rank [J ] + 1 and representing
the only first-order contribution in the operator expansion (A.6.25). This means that
there is a non-vanishing first-order contribution if (and only if) [I ] = [J ] + 1. Stated
differently, the sub-diagonal blocks of M
cc obey the order relation M
cc
μ+1μ ∼ O(1);
all other LL blocks are at least of second order.
Now consider matrix elements M
cc
I J where I and J differ by more than one excitation class: [I ] = [J ] + μ, μ ≥ 2. Thus, the minimal rank in the operator (A.6.25)
required to yield a non-vanishing matrix element is r = [J ] + μ. In the first commu-
Appendix
0 | ˆ
C
†
I
ˆ
O
(n)
J | 0 = 0
(A.6.22)
if the rank r of the operator ˆ
O
(n)
J is lower than the rank of C
†
I , that is, if r < [I ]. Thus,
the intended proof can be carried out by inspecting the rank and order of successive
contributions to the operator ˆ
O J .
The two parts ˆ
K
(0)
J and ˆ
K
(1)
J of the commutator (A.6.18) have the ranks [J ] and
[J ] + 1, respectively. More general, the commutator [ ˆ
A, ˆ
B] of two operators ˆ
A and ˆ
B
with definite ranks, r and r
, respectively, is of rank r + r
− 1. Another commutator
feature relates to the number of unphysical fermion operators in an operator product:
A commutator [ ˆ
A, ˆ
B] of a general product operator ˆ
A and a physical operator ˆ
B, as
in ˆ
K
(1)
J , reduces the number of unphysical fermion operators in ˆ
A by 1. Accordingly,
the terms of ˆ
K
(1)
J exhibit at most three unphysical c-operators compared to the four
unphysical operators in ˆ
H I . As a particular case, the zeroth-order commutator ˆ
K
(0)
J
consists only of physical c-operators, reflecting the fact that ˆ
H 0 is a diagonal oneparticle operator. As a consequence, ˆ
K
(0)
J commutes with the CC operator ˆ
T so that
e
− ˆ
T ˆ
K
(0)
J e
ˆ
T
= ˆ
K
(0)
J
(A.6.23)
Accordingly, the matrix element (A.6.17) can be written as the sum
M
cc
I J = = 0 | ˆ
C
†
I
ˆ
K
(0)
J | 0 + + 0 | ˆ
C
†
I e
− ˆ
T ˆ
K
(1)
J e
ˆ
T
| 0
(A.6.24)
of a zeroth-order contribution and a contribution involving ˆ
K
(1)
J , being at least of first
order. Recalling that the rank of ˆ
K
(0)
J is [J ], the zeroth-order term, M
cc(0)
I J , necessarily
vanishes unless [I ] = [J ]; that is, both I and J belong to the same excitation class;
to put it differently, zeroth-order contributions arise only in the diagonal blocks of
M
cc .
To analyze the higher-order contribution, we make use of the Baker–Hausdorff
(BH) expansion according to
e
− ˆ
T ˆ
K
(1)
J e
ˆ
T
= ˆ
K
(1)
J + [ ˆ
K
(1)
J , ˆ
T ] +
1
2
[[ ˆ
K
(1)
J , ˆ
T ], ˆ
T ] +
1
6
[[[ ˆ
K
(1)
J , ˆ
T ], ˆ
T ], ˆ
T ]
(A.6.25)
which here terminates after the threefold commutator since each successive commutator eliminates one of the original three unphysical c-operators in ˆ
K
(1)
J . The BH
expansion (A.6.25) begins with ˆ
K
(1)
J , being of the rank [J ] + 1 and representing
the only first-order contribution in the operator expansion (A.6.25). This means that
there is a non-vanishing first-order contribution if (and only if) [I ] = [J ] + 1. Stated
differently, the sub-diagonal blocks of M
cc obey the order relation M
cc
μ+1μ ∼ O(1);
all other LL blocks are at least of second order.
Now consider matrix elements M
cc
I J where I and J differ by more than one excitation class: [I ] = [J ] + μ, μ ≥ 2. Thus, the minimal rank in the operator (A.6.25)
required to yield a non-vanishing matrix element is r = [J ] + μ. In the first commu-