Carsten Timm: Theory of superconductivity

From this, we obtain the energy of a single vortex,
∫
E 1 = E core + d 2 r 1 2 E · E = E core − 1 2 2γ α ∫
β
d 2 r 1 r 2 = E core − 2πγ α β
∫
dr 1 r . (7.32)
Now we note that the derivation does not hold for small distances from the vortex center since there |ψ| 2 is not
close to −α/β. Thus we cut off the radial integral at the lower end at some “vortex core radius” r 0 and put all
energy contributions from the core into E core . r 0 is on the order of the coherence length ξ. But the integral still
diverges; if our system has a characteristic size of L, we obtain
E 1 = E core − 2πγ α β
∫ L
r 0
dr
r = E core − 2πγ α ln L . (7.33)
β r
} {{ } 0
> 0
Thus the energy of a single, isolated vortex diverges logarithmically with the system size. This suggests that
isolated vortices will never be present as thermal fluctuations as long as α < 0. This is not true, though. The
probability of such vortices should be
( 2π
e −Ecore/kBT exp
k B T γ α β ln L )
r 0
p 1 ∝ 1 r0
2 e −E1/kBT = 1 r0
2
= 1 r 2 0
e −E core/k B T
( ) 2π
L
k B T γ α β
1 =
r 0 r0
2 e −E core/k B T
The typical area per vortex is 1/p 1 and the total number of vortices will be, on average,
N v =
( L
r 0
) − 1
2η
. (7.34)
( ) 2− 1
L2
= L 2 p 1 = e −Ecore/k BT L
2η
. (7.35)
1/p 1 r 0
For η > 1/4, N v diverges in the thermodynamic limit so that infinitely many vortices are present. For η < 1/4,
N v → 0 for L → ∞ and according to our argument, which is essentially due to Kosterlitz and Thouless, there are
no vortices. It is plausible and indeed true that free vortices destroy quasi-long-range order and in this sense also
superfluidity. Note that
η = − 1
4π
k B T β
γ α(T )
!
= 1 4
(7.36)
is an equation for a critical temperature for the appearance of free vortices. We thus find that the critical
temperature in superfluid films should be reduced from the point where α = 0 (η = ∞) to the one where η = 1/4
due to vortices appearing as fluctuations of the order parameter. While qualitatively true, our description is still
incomplete, though, since we have so far neglected interactions between vortices.
Vortex interaction
The energy of two vortices with vorticities ±1 can easily be obtained from the electrostatic analogy. We assume
that core regions do not overlap, i.e., the separation is R ≥ 2r 0 . The pseudo-electric field of the two vortices,
assumed to be located at ±R/2 = ±R ˆx/2, is
E(r) =
=
√
−2γ α β
√
−2γ α β
r − R/2
√−2γ
|r − R/2| 2 − α β
r + R/2
|r + R/2| 2
|r + R/2| 2 (r − R/2) − |r − R/2| 2 (r + R/2)
|r − R/2| 2 |r + R/2| 2 . (7.37)
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