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Lecture XXIII 67
Lecture XXIII: Field Theory of Superconductivity
Following our discussion of the field theory of BEC and
superfluidity in the weakly
interacting Bose gas, we turn now to condensation phenomena in
Fermi systems
Starting point is BCS Hamiltonian for local pairing interaction:
g V
Ld
H =
ddr
c(r)p2
2mc(r) g c(r)c(r)c(r)c(r)
Quantum partition function: Z= tr e(HN)
Z=()=(0)
D(, )exp
x(,r)Rdx
0
d ddr
[G(p)0 ]
1 +
p2
2m
g
where (r, ) denote Grassmann (anticommuting) fields
Analysis of interacting QFT?
Perturbative expansion in g?Transition to condensate
non-perturbative in g
Mean-field analysis:condensation of pair wavefunction signalled
by anomalous average cc
Hubbard-Stratonovich decoupling: introducing complex commuting
field (r, )
egRdx =
D(, ) exp
dx
1g
(+g)(+g)g 1g|(r, )|2 + ( + )
Z=
D(, )
D(, )e
Rdx
||2
g exp
dx
Nambu spinor ( )
Gorkov Ham. G1 [G
(p)0 ]
1
[G(h)0 ]
1
free particle/hole Hamiltonian: [G
(p/h)0 ]
1 = +/(
p2
2m
)
N.B. dx = dx () = dx Using Gaussian field integral:
D(, ) exp[ A] = det A = exp[ln det A]
Z=
D(, ) exp
dx1
g||2 + ln det G1[]
i.e. Zexpressed as functional field integral over single complex
scalar field (x)
To proceed, we must invoke some approximation:
Lecture Notes October 2005
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Lecture XXIII 68
Mean-field theory: far below transition (T Tc, c) fluctuations
small saddle-point approximation in constant Gap equation
Ginzburg-Landau theory: since transition is continuous, close to
Tc,we may develop perturbative expansion in (small)
Noting : G1 = G10
1 + G0
0 0
, G0 G( = 0)
ln det G1 = trln G1 = trln G10 1
2tr
G0
0 0
2+
N.B. ln(1 + z) = n=1(z)n/n
Zeroth order term free particle contribution, viz.
Z0 = det
G10
Using id. =
kk,n
|kk|, k = 1Ld
dx
einikreikx (x)
Second order term
tr G(p)0 G
(h)0 =
kk
G(p)0 (k)
kk/
Ld k||k G(h)0 (k)k||k
q=kk=
q
qq
pairing susceptibility (q) 1
Ld
k
G(p)0 (k)G
(h)0 (k + q)
Combined with bare term, one obtains
Z=
D[, ]eS[,], S =n,q
1
g+ (n, q)
|n,q|2 + O(4)
In principle, one can evaluate (q, n) explicitly;however we can
proceed more simply by considering...
Gradient expansion: (q, n) = (0, 0) +q2
22|q|(0, 0) + O(in, q
4)
Ginzburg-Landau theory
S[] =
ddr
t
2||2 + K
2||2 + u||4 +
t2
= 1g
+ (0, 0), K = 2|q|(0, 0) > 0 and u > 0
Lecture Notes October 2005
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Lecture XXIII 69
Note structural similarity to weakly interacting Bose gas
Landau Theory: If we assume that dominant contribution to Z= eF
arises fromminumum action, i.e. spatially homogeneous that
minimises
S[]Ld = t2 ||2 + u||4
i.e. || t + 4u||2 = 0, || = 0 t > 0t/4u t < 0i.e. for t
< 0, spontaneous breaking of continuous U(1) symmetry
associated
with phase gapless fluctuations Goldstone modes
Transition Temperature: Using identity 1Ld
k =
ddk
(2)d=
()d
(0, 0) = 1Ld
n,k
12n + (k
2/2m )2 1
n
d(+ )2n +
2 ()
n
1|n|
Introducing energy cut-off at Debye frequency D = (2nmax +
1)/
(0, 0) ()nmax
n=nmax
1
2n + 1 2()
nmax0
dn
2n + 1 () ln
D
Transition when t/2 1/g + (0, 0) = 0, i.e. kBT < kBTc = D exp
1
()g
Near Tct
2=
g
2 () ln
D
= 0 (g () ln
cD
)
= ()ln(T /Tc) = () ln(1 + (T Tc)/Tc) ()
T TcTc
i.e. physically t is reduced temperature
Lecture Notes October 2005
