Carsten Timm: Theory of superconductivity

Next, we have to express Z ′ in terms of the new length scale r ′ := r + dr. This is only relevant in expressions not
already linear in dr. This applies to, on the one hand,
1
r 2 = 1
(r ′ − dr) 2 = 1 + 2 dr
r ′
(r ′ ) 2 , (7.62)
and, on the other,
⎛
⎞ ⎛
⎞
exp ⎝− 1 ∑
2πK N i N j ln r⎠ = exp ⎝− 1 ∑
2πK N i N j ln(r ′ − dr) ⎠
2
2
i≠j
i≠j
⎛
⎞ ⎛
= exp ⎝− 1 ∑
2πK N i N j ln r ′ ⎠ exp ⎝ 1 ∑
2
2
i≠j
i≠j
⎛
⎞
= exp ⎝− 1 ∑
(
2πK N i N j ln r ′ ⎠
2
neglecting terms of order N 0 compared to N. The renormalized partition function is finally
[ ( ) ]
2 ( ) 2N [
y
Z ′ = exp 2π
(r ′ ) 2 r ′ 1 y
dr V
(N!) 2 (r ′ ) 2 1 + (2 − πK) dr ] 2N
r ′ N
} {{ }
∫
×
D ′ 1
d 2 r 1 · · ·
=: Z pair
∑
∫
D ′ 2N
i≠j
⎡
d 2 r 2N exp ⎣− 1 ∑
2
i≠j
(
−2πK + 8π 4 y 2 K 2 dr
r ′ )
⎞
dr
2πK N i N j
⎠
r ′
1 − πK dr
r ′ ) 2N
, (7.63)
⎤
N i N j ln |r i − r j |
⎦
r ′ . (7.64)
Here, Z pair is the partition function of the small pair we have integrated out. It is irrelevant for the renormalization
of y and K. Apart from this factor, Z ′ is identical to Z if we set
[
y(r ′ ) = 1 + (2 − πK(r)) dr ]
r ′ y(r) (7.65)
Introducing the logarithmic length scale
we obtain the Kosterlitz RG flow equations
K(r ′ ) = K(r) − 4π 3 y 2 K 2 (r) dr
r ′ . (7.66)
l := ln r r 0
⇒ dl = dr
r , (7.67)
The initial conditions are
dy
dl
dK
dl
= (2 − πK) y, (7.68)
= −4π 3 y 2 K 2 . (7.69)
i.e., the parameters assume their “bare” values at r = r 0 (l = 0).
y(l = 0) = y 0 = e −E core/k B T , (7.70)
K(l = 0) = K 0 = − 1
k B T 2γ α β , (7.71)
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