D.4 Volume of Invariant Measure for Unitary Matrices
451
×
∞
−∞
. . .
∞
−∞
dd H 1 dh R
1,N . . .dh R
N −1,N dh I
1,N . . .dh I
N −1,N
( ¯
1 + ¯
w † · ¯
w) α−1 Det[T 2
1 + ¯
v· ¯
v † ] α
.
(D.111)
Let us now note that
Det[T
2
1 + ¯
v· ¯
v
†
] = Det[T
2
1 + T 1 · ¯
w· ¯
w
†
· ¯
T
†
1 ]
= Det[T
2
1 ] Det[ ¯
1 + ¯
w· ¯
w
†
] = Det[T
2
1 ] ( ¯
1 + ¯
w
†
· ¯
w).
(D.112)
Then Eq. (D.111) takes the form
I
α
N (2) =
√
π
−
1
2 )
(α)
2
N −1
×
∞
−∞
. . .
∞
−∞
dd H 1 dh R
1,N . . .dh R
N −1,N dh I
1,N . . .dh I
N −1,N
Det[T 2
1 ] α ( ¯
1 + ¯
w † · ¯
w) 2α−1
.
(D.113)
The next step is to integrate over the matrix elements, h 1,N , . . ., h N −1,N in
Eq. (D.113). Note that
dh
R
1,N . . .dh
R
N −1,N dh
I
1,N . . .dh
I
N −1,N = Det[T 1 ]
2 dw
R
1 . . .dw
R
N −1 dw
I
1 . . .dw
I
N −1 .
(D.114)
The Jacobian Det[T 1 ] 2 enters because Re[ ¯
v] = T 1 ·Re[ ¯
w] and Im[ ¯
v] = T 1 ·Im[ ¯
w].
We can now write Eq. (D.113) in the form
I
α
N (2) =
√
π
−
1
2 )
(α)
2
(N −1) I
α−1
N −1 (2)
×
∞
−∞
. . .
∞
−∞
dw R
1 . . .dw R
N −1 dw I
1 . . .dw I
N −1
[1 + (w R
1 ) 2 + (w I
1 ) 2 + . . . + (w R
N −1 ) 2 + (w I
N −1 ) 2 ] 2α−1 ,
(D.115)
where
I
α−1
N −1 (2) =
dd H 1 (2)
Det[ ¯
1 + ¯
H 2
1 ] N −1
.
(D.116)
But
∞
−∞
dx
[b 2 + x 2 ] 2α−1 =
√
π
(2α −
3
2 )
(2α − 1)
1
[b 2 ]
2α−
3
2
.
(D.117)