Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors
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1502 D. Manske,I. Eremin, and K.H.Bennemann
(T)Im(k, !)=−
1
N
k
∞
−∞
d!
b(§)+f (§ − !)
×P
F
(q, !)Im
$
(k
, !
)
(
!Z
)
2
−
k
+
2
&
,
(T
c
)=1. (23.191)
Here,
F
in P
F
is set to zero identically. For decreas-
ing temperature T the eigenvalue (T) increases and
reachesunity atT = T
c
.Thespectralfunctionscalcu-
lated during the solution of (23.191) are then appro-
priatestarting values for thesolution oftheequations
below T
c
. Solving the generalized Eliashberg equa-
tions for cuprates we find that below T
c
the super-
conducting gap function has d
x
2
−y
2
-wave symmetry
for both hole-doped and electron-doped supercon-
ductors. Vertex corrections for the two-particle cor-
relation function are not included. The calculations
for Sr
2
RuO
4
are performed similarly.
As mentioned earlier there exists an important
feedback effect of the self-energy on the spec-
tral functions of the corresponding quasi-particles
(dressed holes or electrons). This is in particular
the case in the superconducting state. Below T
c
a
gap appears in the spectral density of the quasi-
particles. These not only condensed into Cooper-
pairs, but also provide the effective pairing inter-
action V
eff
(q, !). The superconducting gap func-
tion (k, !)=Z(k, !)(k, !)willbecalculated
self-consistently including all scattering and damp-
ing effects.This strong feedback effect has important
consequences: for example, the dynamical spin sus-
ceptibility Im
s
(Q, !)shows a rearrangement of its
spectral weight for small frequencies ! < 2(Q, !)
followed by a peak at approximately ! =2.Note
that for simplicity we restrict our discussion to the
antiferromagneticnesting vector q = Q =(, ).As
discussed in the Introduction this behavior provides
an explanation for the observed “resonance peak”
in inelastic neutron scattering in many cuprate su-
perconductors. Therefore, the strong feedback effect
of superconductivity on the elementary excitations
and thus on Im
s
(Q, !) can be viewed as a general
phenomenon of a strongly interacting system and
in particular as an important fingerprint of spin-
fluctuation-mediated pairing in cuprate high-T
c
su-
perconductors. The opening of the superconducting
gap also helps to obtaina kink in the antinodaldirec-
tion. To demonstrate this interdependence the self-
consistent procedure shown in Fig.23.76 is required.
Appendix C:
Derivation of the Quasiparticle Self-Energ y
(Weak-Coupling Case)
In order to discuss the anisotropic k-dependence of
the elementary excitations,
k
=
0
k
+£(k, !), it is in-
structiveto consider the weak-coupling limit (Z =1)
of the generalized Eliashberg equations.Using a one-
loop approximation the self-energy is given by:
£(k, !)=−T
2
!
n
,
m
p,q
ˆ
i
G(k − p, i!
n
− i
m
)
׈
i
U
eff
(p)U(p) (23.192)
×
1
2
Tr
ˆ
i
G(p + q, i!
n
+ i
m
)
׈
i
G(k, i!
n
)
.
Forspinfluctuationswehave
i
=
0
.Forsimplicity
we use U
eff
U = U
2
and assume that the Coulomb in-
teraction U (p) is constant. The full Green’s function
is approximated by its non-interacting part
G(k, i!
n
) G
0
(k, i!
n
)=
i!
n
ˆ
0
+
k
ˆ
3
−
k
ˆ
1
(
i!
n
)
2
−
2
k
−
2
k
. (23.193)
Using (23.193) and introducingthe quasiparticle en-
ergy dispersion in the superconducting state E
2
p
=
2
p
+
2
p
and the coherence factors
c
±
(p + q, q)=1±
p+q
q
+
p+q
q
E
p+q
E
q
, (23.194)
we can perform the sum over !
n
in (23.193).Noting
that
T
!
n
1
(i!
n
+ i
m
− y)(i!
n
− x)
=
f (x)−f (y)
i
m
+ x − y
, (23.195)
one obtains
23 Electronic Theory for Superconductivity 1503
T
!
n
1
2
Tr
[
G(p + q, i!
n
+ i
m
) G(q, i!
n
)
]
=
1
4
f (E
q
)−f (E
p+q
)
i
m
+ E
q
− E
p+q
+
f (−E
q
)−f (−E
p+q
)
i
m
− E
q
+ E
p+q
c
+
(p + q, q)
+
f (−E
q
)−f (E
p+q
)
i
m
− E
q
− E
p+q
+
f (E
q
)−f (−E
p+q
)
i
m
+ E
q
+ E
p+q
c
−
(p + q, q)
.
(23.196)
Equation (23.196) has to be multiplied by
i!
n
−i
m
(i!
n
−i
m
)
2
−E
2
k−p
for the ˆ
0
-component and by
1
(i!
n
−i
m
)
2
−E
2
k−p
for the ˆ
3
and ˆ
1
-components, respec-
tively, and summed over
m
.Using(n = Bose func-
tion) n(−x)=−(1+n(x)), f (−x)=1−f (x)and
−T
m
1
(i
m
−x)(i
m
−y )
=
n(x)−n(y)
x−y
in the form
− T
m
1
(i
m
− i!
n
− x)(i
m
− y)
=
n(i!
n
+ x)−n(y)
i!
n
+ x − y
=−
f (x)+n(y )
i!
n
+ x − y
,
after the analytic continuation i!
n
→ ! +i we ob-
tain the self-energy.For example, the imaginary part
of the self-energy is
Im £
0
(k, !)=
U
2
8
(23.197)
×
⎡
⎣
p
f (E
k−p
)+n(E
p+q
+ !)
×
q
c
+
(p + q, q)
f (E
q
)−f (E
p+q
)
×
ı(! + E
k−p
− E
q
+ E
p+q
)
− ı(! + E
k−p
+ E
q
− E
p+q
)
+ c
−
(p + q, q)
1−f (E
q
)−f (E
p+q
)
×
ı(! + E
k−p
+ E
q
+ E
p+q
)
− ı(! + E
k−p
− E
q
− E
p+q
)
+
p
f (E
k−p
)+n(E
k−p
− !)
×
q
c
+
(p + q, q)
f (E
q
)−f (E
p+q
)
×
ı(! − E
k−p
+ E
q
− E
p+q
)
− ı(! − E
k−p
− E
q
+ E
p+q
)
+ c
−
(p + q, q)
1−f (E
q
)−f (E
p+q
)
×
ı(! − E
k−p
− E
q
− E
p+q
)
− ı(! − E
k−p
+ E
q
+ E
p+q
)
.
We introduce the susceptibilityin the superconduct-
ing state
(p, i
m
)=−T
!
n
q
1
2
Tr (23.198)
×
[
G(p + q, i!
n
+ i
m
)G(q, i!
n
)
]
=−
1
4
q
c
+
(p + q, q)
f (E
q
)−f (E
p+q
)
×
1
i
m
+ E
q
− E
p+q
1
i
m
− E
q
+ E
p+q
+ c
−
(p + q, q)
1−f (E
q
)−f (E
p+q
)
×
1
i
m
− E
q
− E
p+q
1
i
m
+ E
q
+ E
p+q
.
Then,
Im£
0
(k, !)=
U
2
2
p
−
f (E
k−p
) (23.199)
+ n(E
k−p
+ !)
Im (p, ! + E
k−p
)
×
f (E
k−p
)+n(E
k−p
− !)
× Im (p, ! − E
k−p
)
.
With the helpof f (x)+n(y)=
1
2
coth
y
2T
−tanh
x
2T
onefindstheimportantresult
Im£
0
(k, !)=−
U
2
4
p
∞
−∞
d!
(23.200)
×
coth
!
2T
−tanh
!
− !
2T
× Im (k − p, !
) ı(|! − !
| − E
p
) .
1504 D. Manske,I. Eremin, and K.H.Bennemann
Therefore, the anisotropy of the spin excitations
(weighted by the density of states) is intimately re-
lated to the anisotropy of the self-energy of the ele-
mentary excitations. The strength of this anisotropy
is controlled by U
2
.Thus,it is clear that spin dynam-
icswill control the quasiparticlebehavior in cuprates.
As a result one expects an important feedback of spin
fluctuationson quasiparticle dynamics.In particular
the characteristic spin fluctuation frequencies in the
normal (!
sf
) and superconducting (!
res
) will deter-
mine the characteristic features in the energy disper-
sion.
For the ˆ
1
and ˆ
3
-components of £ one obtains
similar results. It is instructive to write them in a
unified way using the nesting approximation. In this
approximation one assumes that the main contribu-
tion to the momentum sum comes from Im at the
nesting vector Q =(, ). Then, the self-energy no
longer depends on k and one can write ( =0, 1, 3):
Im £
(!)=−
U
2
4
∞
−∞
d!
(23.201)
×
coth
!
2T
−tanh
!
− !
2T
×Im (Q, !) N
(|! − !
|) ,
where
N
0
(x)=
p
ı(|x|− E
p
) ,
N
1
(x)=−signx
p
p
E
p
ı(|x| − E
p
) ,
N
3
(x)=signx
p
p
E
p
ı(|x|− E
p
) . (23.202)
This completes the derivation of the self-energy in
the weak-couplingcase.In a strong-coupling case the
procedure described in Appendix A has to be used.
Then, it is possible to obtain the self-energy only nu-
merically.
Appendix D:
Cooper-Pair Phase Fluctuations
As suggested by the observation of T
c
∝ n
s
for the
underdoped cuprates [14] and by the small super-
fluid density, n
s
<< n for x → 0, and also by
the weakly coupled two-dimensional CuO
2
planes,
one expects that Cooper-pair phase fluctuations are
important. These fluctuations of phase-incoherent
Cooper-pairs will destroy the long-range supercon-
ducting order and the Meissner effect. In conven-
tional, bulk superconductors these phase fluctua-
tions are not relevant, since for large superfluid
density the phase fluctuations are too costly ener-
getically. Thus, in conventional superconductors the
superconducting transition is due to the destruc-
tion of the Cooper pairs and T
c
is proportional
to
0
[76]. Note that the superconducting energy
gap governs the thermal breaking of Cooper-
pairs. Clearly, the observation by Uemura et al. that
T
c
∝ n
s
in underdoped hole-doped cuprates [14] in-
dicates that Cooper-pair phase fluctuations are im-
portant [78,79]. The Cooper-pairs break up only at
a crossover temperature around T
∗
c
> T
c
where T
∗
c
is approximately given by the transition tempera-
ture one would obtain without phase fluctuations
and where at T
c
Cooper-pairs become phase coher-
ent. Between T
c
and T
∗
c
local Cooper-pairs without
long-range phase coherence are present.
Inthefollowing,wediscussphasefluctuationsand
their influence on elementary excitations, Meissner
effect, interplay of phase and amplitude fluctuations,
and superconducting transition temperature T
c
.
Role of Phase and Amplitude Fluctuations for
Determining T
c
We start from the expression for the free energy of a
superconductor:
F ≡ F
S
− F
N
= (23.203)
−
1
ˇ
ln
D
†
De
−
1
S
eff
[
†
,]
,
where S
eff
[
†
, ] is the effective action obtained af-
ter integrating out the fermion variables. This is a
23 Electronic Theory for Superconductivity 1505
functional of the Cooper-pair field (t, r). Note that
(23.203) is exact. The BCS theory is obtained from
the saddle-point of the effective action.The simplest
case corresponds to taking =
0
= const. Then the
free energy becomes
F =−
1
ˇ
Tr ln[G
−1
(
∗
0
,
0
)] +
§
V
0
|
0
|
2
, (23.204)
where G is the Nambu–Gorkov matrix Green’s func-
tion, V
0
> 0 is the coupling constant, and § is the
volume of the system. Minimization of F gives the
BCS gap equation.
To start the study the effect of phase fluctuations
in superconductors we make the change of variables
(, r)=(, r)e
i(,r)
. (23.205)
Let us write (, r)=
0
+ ı(, r) and for the phase
(, r)=
0
+ ı(, r) and expand the effective ac-
tion up to quadratic fluctuations in ı and ı.The
effective action then becomes
S
eff
≈
ˇ
2
q,n
[¢
(!
n
, q)ı(!
n
, q)ı(−!
n
, −q)
+¢
(!
n
, q)ı(!
n
, q)ı(−!
n
, −q)] .
(23.206)
Note that we do not consider terms involving both
the phase and amplitude fluctuations. These terms
are often very small at low frequencies. Thus, to low-
est order we can write
F = F
cond
+ F
phase
, (23.207)
where
F
cond
=−
1
ˇ
ln
Dı
exp
⎡
⎣
−
ˇ
2
q,n
¢
(!
n
, q)ı(!
n
, q)ı(−!
n
, −q)
⎤
⎦
(23.208)
gives the condensation energy, and
F
phase
=−
1
ˇ
ln
Dı
exp
⎡
⎣
−
ˇ
2
q,n
¢
(!
n
, q)ı(!
n
, q)ı(−!
n
, −q)
⎤
⎦
,
(23.209)
is the Cooper-pair energy gained due to phase coher-
ence. In the low frequency and momentum limit we
have
¢
(!
n
, q)=−(!
n
, q)!
2
n
+
(!
n
, q)q
q
,
(23.210)
where (!
n
, q) is the density–density correlation
function and
=
n
m
ı
− ˇj
(!
n
, q)j
(−!
n
, q) (23.211)
is the phase stiffness. In the limit ! → 0and
q → 0 the expression
gives the superfluid
density divided by the effective mass, n
s
/m.Thus,
one gets approximately from F
phase
(0) ∼ k
B
T
c
and
F
phase
∝ n
s
, which results for F
cond
> F
phase
,that
T
c
∝ n
s
.
Quantum mechanically the role of the phase fluc-
tuations is expressed by phase and particle number
which are conjugate operators:
[
ˆ
,
ˆ
N]=i . (23.212)
Here,
ˆ
N denotes theCooper-pair particle number op-
erator. Thus, an immediate consequence of (23.212)
is the uncertainty relation
N ∼ . (23.213)
A rough estimate gives N ∼ n
s
(0) and thus
∼
1
n
s
(0)
. (23.214)
Therefore, phase fluctuations are very important in
systems having a small value of the zero-temperature
superfluid density. This is precisely the case for the
cuprate superconductors in the underdoped regime
[79]. Indeed, since n
s
(0) ∝ x,wherex is the hole
concentration, we see that ∼ 1/x.Asa resultof
Cooper-pair phase fluctuations and vortex conden-
sation we have that
∝
1
T
c
. (23.215)
Therefore, we obtain from (23.214) that the reduc-
tionofT
c
inunderdopedcupratesdueto Cooper-pair
phase fluctuations is given by
T
c
∼ n
s
(0) . (23.216)
Therefore, for small doping phase fluctuations are
very important.
1506 D. Manske,I. Eremin, and K.H.Bennemann
Meissner Effect
Let us now estimate the role played by the coherent
phase fluctuations for the Meissner effect. Writing
=
√
n
s
e
i
and taking into account the phase fluc-
tuations only, one finds that the free energy is given
by
F[n
s
, ]=
ˇ
0
d
d
2
rn
s
(23.217)
×
$
1
4
∇
+
2e
c
A
2
+ (∂
)
2
&
.
Then using
=
!
d
2
r
d
ı
2
ln Z
ıA
(, r)ıA
(0)
e
i!
+iqr
,
(23.218)
where Z is the partition function (Z =
D
∗
D
× exp(−F[
∗
, ])), we find for the conductivity af-
ter phase averaging
(q, i!
)=
n
s
R
Q
!
−ı
+
q
q
4!
2
+ q
2
.
(23.219)
The analytic continuation to the real frequencies
yields for q → 0
(q =0, !
)=−
n
s
R
Q
ı
i
!
+ ı(!)
, (23.220)
which implies perfect conductivity and for ! → 0
< j
>=−
n
s
R
Q
ı
−
q
q
q
2
A
. (23.221)
This demonstrates the occurrence of a Meissner ef-
fect below T
c
in the superconducting phase in pres-
ence of the phase fluctuations and shows that phase
fluctuations are indeed important.
Effect of the Phase Fluctuations on the Elementary
Excitations: Dyson’s Equation
In order to calculate the density of states for a d-
wave superconductor in the presence of Cooper-pair
phase fluctuations, one has to derive the correspond-
ing Dyson equation for the elementary excitations. If
one considers the Fourier transformation ofthe spec-
tral density of the pairing interaction P
s
,itturnsout
that the effective pairing potential V
eff
is attractive
on nearest neighbor sitesin real space [166].Thuswe
start with
H =−t
i,j
c
†
i
c
j
+h.c. (23.222)
−
i
n
i
+ V
eff
i,j
c
†
i↑
c
†
j↓
c
j↓
c
i↑
,
where V
eff
< 0. In order to evaluate the free energy
and correlation functions of the system one has to
evaluate the partition function (¥ , ¥
∗
= Grassmann
variables):
Z =
D(¥ , ¥
∗
)e
−S[¥
∗
,¥ ]
,
where the action is given by
S[¥
∗
, ¥ ]=
ˇ
0
d
×
⎡
⎣
i
¥
∗
i
(∂
− )¥
i
+ t
ij
¥
∗
i
¥
j
+V
eff
ij
¥
∗
i↑
¥
∗
j↓
¥
j↓
¥
i↑
⎤
⎦
.
We now come to the derivation of the propaga-
tor for Cooper-pair phase fluctuations and the cor-
responding self-energy equations. For this purpose
one has to integrate out the electronic degrees of
freedom. This leads to
S
eff
[']=S
BCS
eff
(
0
)+
1
ˇ
qn
'(q, i
n
)
×
D
'
(q, i
n
)
−1
'(−q, −i
n
) ,
(23.223)
or diagrammatically
23 Electronic Theory for Superconductivity 1507
S
eff
[']−S
BCS
eff
(
0
) = (23.224)
+
,
where the vertices, which couple the quasiparticles
(holes) and the phases, are given by
ˆ
1
0
(k, k
)=
i
2
(ik − ik
) ˆ
3
−(
k
−
k
) ˆ
0
=
(23.225)
ˆ
2
0
(k, k
, k
)=
1
8
k
−
k−k
−
k
+k
+
k
ˆ
3
= . (23.226)
and ˆ
i
are the corresponding Pauli matrices. The BCS
part of the effective action is given by
S
BCS
eff
(
0
)=−Trln
−
ˆ
G
−1
0
+ ˇN
|
0
|
2
|V
eff
|
. (23.227)
In (23.223) D
'
(q, i
n
)=' (q, i
n
) '(−q, −i
n
)
is the propagator of the corresponding phase of a
Cooper-pair given by [93]
D
'
(q, i
n
)=8
−
0
(q, i
n
)(i
n
)
2
(23.228)
−(i
n
)
˛
¢
˛
(q, i
n
)+¢
˛
(q, i
n
)
2sin
q
˛
2
+
˛ˇ
˛ˇ
(q, i
n
)(2 sin
q
˛
2
)(2 sin
q
ˇ
2
)
−1
,
with the phase stiffness
˛ˇ
(q, i
n
)=T
˛ˇ
+ ¢
˛ˇ
. ¢
and T are the current–current correlation function
and mass operator, respectively [73]. It is
¢
˛ˇ
(q, i
n
)=
1
ˇN
kn
v
˛
k−
q
2
v
ˇ
k−
q
2
Tr (23.229)
×
ˆ
G
0
(k, i!
n
)
ˆ
G
0
(k − q, i!
n
− i
n
)
,
T
˛ˇ
=
1
N
k
t
˛ˇ
k
(n
k↑
+ n
k↓
) . (23.230)
In analogy to Kadanoff and Baym we are now able
to calculate the dressed Green’s functions and their
correspondingself-energies [177,178].Ifwe consider
Cooper-pair phase fluctuations and treat them up to
second order, it is [93]
ˆ
G(k, i!
n
)=
ˆ
G
0
(k, i!
n
) (23.231)
−
ˆ
G
0
(k, i!
n
)
ˆ
£
''
(k, i!
n
)
ˆ
G(k, i!
n
) ,
or diagrammatically
=
+
+
(23.232)
where the normal and anomalous part of the self-
energy due to Cooper-pair phase fluctuations are
given by [179]
£
G
(k, i!
n
)=
1
4ˇN
q,n
[(i
n
)
2
−2i
n
(23.233)
× (
k
−
k−q
)+(
k
−
k−q
)
2
]
× G
0
(k − q, i!
n
− i
n
) ¢
''
(q, i
n
) ,
£
F
(k, i!
n
)=
1
4ˇN
q,n
[(i
n
)
2
+(
k
−
k−q
)
2
]
× F
0
(k − q, i!
n
− i
n
) ¢
''
(q, i
n
) .
(23.234)
The vertices are given above.Furthermore, in the di-
agrammatic representation
ˆ
G
0
is the dressed Green’s
1508 D. Manske,I. Eremin, and K.H.Bennemann
function (given in Appendix A) within the FLEX ap-
proximation that includes, in particular, the scatter-
ing by spin fluctuations.
Finally, these self-energy corrections must be
added to their counterparts £
G
and £
F
resulting from
the exchange of spin fluctuations alone in (23.158).
Then, for
ˆ
G one can perform an analytical continua-
tion to the real !-axis and solve the corresponding
equations.Of course,the current–current correlation
function ¢ has to be calculated self-consistently us-
ing (23.229) describing the propagation of Cooper-
pair phase fluctuations.
Interplay of Phase and Amplitude Fluctuations
So far we assume that the phase and amplitude fluc-
tuations are decoupled. The next step is to study the
interplay between them.At low frequencies and mo-
menta this is equivalent to consider an incompress-
iblechargedfluidwith∇·v
s
=0,wherev
s
isthesuper-
fluid velocity. Therefore, in the hydrodynamic limit
the phase-decoupled free energy leads to a constant
charge density,i.e.the amplitude fluctuations are es-
sentially frozenin the lowenergy limit.To see this in a
simpleway,note that the hydrodynamics of a charged
fluid is described by the non-linear Schr¨odinger-like
equation of the Cooper-pair wave function:
∂
2
∂t
+ ∇·(
2
v
s
) = 0 (23.235)
and
∂
∂t
=−
m
2
v
2
s
+2eV −
2
2m
1
∇
2
.
(23.236)
Here, V is the electric potential and generally v
s
=
[∇ −(2e/c)A]/m,withA being the vector poten-
tial minimally coupled to the phase gradient. Equa-
tion (23.235) is just the continuity equation. In the
above hydrodynamic equations the phase and ampli-
tude of the Cooper pair are coupled. When the am-
plitude is frozen,the continuityequation gives us the
incompressibility condition ∇·v
s
= 0, which implies
other gauge-equivalent solution ∇ =2eA/(c). In
this case (23.235) yields
∂
∂t
=2eV, (23.237)
which is just Josephson’s voltage-phase relation.
What happens to the interplay if the doping con-
centration is changed? At optimal doping the phase
and amplitude fluctuations are equally important.
Note that the critical temperature T
c
is then at its
highest value. However,with further increasing dop-
ing the critical temperature decreases, now not due
to phase fluctuationswhich become negligible in the
overdoping regime, but due to increasing itinerancy
with → 0, i.e. the pairing field disappears.
Note that, in general, like in magnetism (ferro-
magnetism) phase and amplitude fluctuations of the
order parameter are coupled and one would get for
= ||e
i
that ||→0 affects ∇∇ and vica
versa.
Using Ginzburg–Landau theory it is easy to see
that amplitude fluctuations decrease T
c
.Consider
the inverse susceptibility
−1
(T), with T > T
c
,in
a Ginzburg–Landau (GL) description.We have
−1
(T)=a
0
(T − T
0
)+u| |
2
, (23.238)
where T
0
is the mean-field critical temperature and
u is the GL interaction. In the overdoped regime
the CuO
2
planes are more strongly coupled and the
Cooper-pairs tunnel coherently between the planes.
Thus, we calculate | |
2
in three dimensions. Con-
sidering an isotropic system for simplicity we have
the self-consistent equation
−1
(T)=a
0
(T − T
0
)+u
d
3
p
(2)
3
1
p
2
+
−1
(T)
= a
0
(T − T
0
)+
u
2
2
−
u
4
−1
(T) ,
(23.239)
where is a cutoff.We see that the critical tempera-
ture is
T
c
= T
0
−
u
2
2
, (23.240)
which is indeed lower than T
0
.
This T
c
reductionand the one due to phase fluctu-
ations are, however, fundamentally different regard-
ing the way by which the superconducting state is de-
stroyed.In the underdopedcupratesthephase fluctu-
ations are expected to relate to the formation of vor-
tices and the superconducting phase is destroyed due
23 Electronic Theory for Superconductivity 1509
to the proliferation of thermally and quantum ex-
cited vortices. Thus, the non-superconducting phase
can be thought of as a phase where the vortices are
condensed. Note that the vortex condensation in un-
derdoped cuprates may in part be responsible for the
suppression of spectral weight and non-Fermi-liquid
behavior.
In a simple Eliashberg-like treatment of T
c
in
which fluctuations of the superconducting conden-
sate are neglected (see Fig. 23.48), the temperature
T
∗
c
∝ (T → 0) is equal to the mean-field tempera-
ture T
0
.Therefore,fluctuations are reducing T
c
< T
∗
c
.
The Role of Transverse Phase Fluctuations in the BKT
Theory and in Determining T
c
How can one connect the Berezinskii–Kosterlitz–
Thouless (BKT) theory to the Eliashberg-like the-
ory in the presence of vortices? For this let us write
the following simple effective Hamiltonian density
containing both the phase and Cooper-pair particle
number:
H =
1
2g
ˆ
N
2
+
n
s
(T)
2
4m
(∇
ˆ
)
2
. (23.241)
Note that the above Hamiltonian describing a con-
tinuum limit is also an array of Josephson weak-link
junctions.If weidentify
ˆ
N as the“momentum”opera-
tor conjugated to the phase operator
ˆ
(see (23.212)),
we see that in (23.241) the first term on the RHS cor-
responds to the kinetic energy, while the second one
is the potential energy. The partition function of the
system can be written as the functional integral
Z =
[
A
,r
d(, r)dN(, r)] exp
−
1
S(, N)
(23.242)
and the action is given by
S(, N)=
ˇ
0
d
d
2
r(−iN ∂
+ H) . (23.243)
After performing the Gaussian integral over N(, r)
we obtain the effective action containing only the
phase (, r):
S
phase
=
ˇ
0
d
d
2
rL() , (23.244)
where the Lagrangian is given by
L =
g
2
(∂
)
2
+
n
s
(T)
2
4m
(∇)
2
. (23.245)
Let us now turn to a determination of T
c
with the
help of the BKT theory. We make the approximation
that is independent of the Matsubara time. Then
we can write the phase action as
S
phase
=
˜
K
2
d
2
r(∇)
2
, (23.246)
where the phase stiffness
˜
K is related to the super-
fluid density through
˜
K = n
s
(T)
2
/(4mT). The su-
perfluid density can be computed from a Hamilto-
nian like the Hubbard model employing a many-
body treatment as the FLEX approximation and the
Kubo formula. In the case of the cuprates it is use-
ful to introduce an additional parameter d givingthe
average distance between the CuO
2
planes such that
the new phase stiffness is given by
K(T)=ˇ
2
n
s
(T)
m
d
4
, (23.247)
In our calculations we set d to half the height of the
unit cell of YBa
2
Cu
3
O
6+x
. The renormalized phase
stiffness can be expressed as
1
K
R
=
4m
2
T
2
dn
s
(T)
=
4m
2
2
d
2
rv
s
(r) ·v
s
(0). (23.248)
In order to compute the correlation function in-
volving the superfluid velocity, we have to split the
gradient of the phase between the longitudinal (reg-
ular) and transverse (singular) contributions. The
transverse fluctuations account for the vortices. We
then have
v
s,
(r)=∂
'(r)+2"
∂
G(r −r
)n(r
) , (23.249)
where G(r − r
) satisfies the equation −∇
2
G(r − r
)=
ı
2
(r − r
). The vortex density canbewrittenas
1510 D. Manske,I. Eremin, and K.H.Bennemann
n(r)=
i
q
i
ı
2
(r − r
i
), where q
i
= ±1. As is well
known,the statistical mechanics of two-dimensional
vortices can be mapped on a Coulomb gas [180].Us-
ing the Coulomb gas scenario the superfluid density
can be related to the vortex density correlation func-
tion in momentum space. One finds [181]
n
s
n
0
s
=1−
2
T
2
lim
k→0
n(k)n(−k), (23.250)
where n
0
s
is the bare superfluid density and is a
cutoff.
Starting from the smallest vortex–antivortex pairs
of size r
0
, the pairs are integrated out and their ef-
fect is incorporated by an approximate renormaliza-
tion of K and the fugacity y = e
−ˇE
c
. Then, from
(23.248) and (23.249) we obtain the BKT recursion
relations [180–182]:
dy
dl
=
(
2−K
)
y ,
dK
dl
=−4
3
y
2
K
2
. (23.251)
For the vortex core energy we use the approximate
result of Blatter et al., i.e.
E
core
= k
B
TK ln , (23.252)
where is the Ginzburg parameter. l =ln(r/r
0
)is
a logarithmic length scale which relates K to the
strength of the vortex–antivortex interaction. For
T > T
c
, K tends to zero for l →∞, so that the
interaction at large distances is screened and the
largest vortex–antivortex pair unbind. This destroys
the Meissner effect and leads to dissipation. On the
other hand, bound Cooper-pairs reduce K and thus
n
s
, but do not destroy superconductivity. After solv-
ing (23.251) and (23.252) it turns out that the renor-
malization of K is very small. Thus, within a good
approximation, one can obtain T
c
(x)fromthesim-
ple criterion
K(T
c
)=
2
, or
n
s
(T
c
, x)
m
=
2
4k
B
T
c
2
d
, (23.253)
which we have used in the main text.
Let us turn now to the !-dependence of the
Cooper-pair phase fluctuations. A dynamical gener-
alization of the BKT theory was developed by Am-
begaokar et al. [95,181]. It turns out that the critical
size for a vortex–antivortex pair is given by
r
!
=
7
D
2!
,
where D
denotes the corresponding diffusion con-
stant. In the absence of pinning, the theory by
Bardeen and Stephen yields [96]
D
0
=
2c
2
2
ab
n
T
2
0
˜
d
,
where c is the speed of light,
ab
∼ r
0
/2corresponds
to the coherence length,
n
denotes the normal-state
resistivity,
0
= hc/2e is the elementary supercon-
ducting flux quantum, and
˜
d corresponds to an ef-
fective layer thickness. For the pinning we assume an
Arrhenius law. Then, one finds
n
s
(!)
m
=
1
e
2
! Im
S
(!) , (23.254)
where
S
(!) is the difference between the total dy-
namical conductivity in the superconducting state
and its value in the normal state. It can be obtained
from the current–current correlation function and
the Kubo formula using the generalized Eliashberg
equations described in Appendix A.
Acknowledgements
WethankL.Tewordt,C.Timm,J.Schmalian,F.
Nogueira, D. Morr, J. Fink and M. Peter for many use-
ful discussions.
23 Electronic Theory for Superconductivity 1511
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