aim – theory of superconductivity rooted in the normal state history – t-matrix approximation in...
TRANSCRIPT
• Aim – theory of superconductivity rooted in the normal state
• History – T-matrix approximation in various versions
• Problem – repeated collisions
• Solution – self-consistent effective medium of Soven
• Example – volume correction to the BCS gap equation
P. Lipavský
THEORY OF SUPERCONDUCTIVITY
based on
T-MATRIX APPROXIMATION
AIM – THEORY OF SUPERCONDUCTIVITY
The superconductivity was elucidated in Fermi gases 6Li and 40K. Increasing the interaction via magnetic field, the BCS condensate of Cooper pairs transforms into Bose-Einstein condensate of strongly bound pairs. In the crossover, there are strong fluctuations between normal and superconducting states.
Many high Tc materials have gap in the energy spectrum even above the critical temperature. Recent theories suggest that this so called pseudogap is due to fluctuations of the supeconducting condensate in the normal state.
Why we need any alternative theory of superconductivity?
To describe fluctuations between normal and superconducting states, we need a theory which cover both phases.
AIM – THEORY OF SUPERCONDUCTIVITY
1. Trial wave functions with variational treatment
– sophisticated methods but suited for ground states
2. Anomalous Green functions
– work horse of superconductivity but defining anomalous functions we determine the condensate, i.e., we do not allow fluctuations
3. T-matrix (Kadanoff-Martin approach)
– recently studied approximation but it does not describe the normal state, because two particles interacting via T-matrix are not treated identically
Basic approaches:
Motivated exclusively by normal state properties, we modify the T-matrix so that it applies to normal and superconducting states.
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Two-particle collision
EVH 00
0 0
1' V
iVHE
expansion in powers
00
0000
000
00
0
1
...0
1
0
1
0
1
0
1
0
1
0
1'
TiHE
ViHE
ViHE
ViHE
ViHE
ViHE
ViHE
+ + + .....
Schrödinger equation incoming waves scattered waves
000 EH
'0
...0
1
0
1
0
1
000
ViHE
ViHE
VViHE
VVTT-matrix
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Two-particle collision
EVH 00
0 0
1' V
iVHE
expansion in powers
00
0000
000
00
0
1
...0
1
0
1
0
1
0
1
0
1
0
1'
TiHE
ViHE
ViHE
ViHE
ViHE
ViHE
ViHE
Schrödinger equation incoming waves scattered waves
000 EH
'0
...0
1
0
1
0
1
000
ViHE
ViHE
VViHE
VVTT-matrix
+ + + .....T =
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Two-particle collision in T-matrix
+ + + .....T =
Schrödinger equation incoming waves scattered waves
00 0
1' T
iHE
'0
1' 0
0
ViHE
'' 000 EVH
'' 00 EVVH
T-matrix
reconstructed wave function
EVH 0 '0
000 EH
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Two-particle collision in T-matrix
'0
1' 0
0
ViHE
reconstructed wave function
non-local collision
'0
0
reconstructed wave function reduces penetration into strong repulsive potential
V
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Two-particle collision in T-matrix
'0
1' 0
0
ViHE
reconstructed wave function
non-local collision due to finite duration of collision
'0
0
V
reconstructed wave function resonantly increases giving higher probability to of two particles staying together
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Two-particle collision in T-matrix – Pauli principle
+ + + .....T =
Energy expansion (Bruckner; Bethe and Goldstone; ...)
210
110
1ff
iHE
particle-particle correlation blocking of occupied states — Cooper problem
Time expansion (Feynman; Galitskii; Bethe and Salpeter; Klein and Prage)
21210
110
1ffff
iHE
particle-particle and hole-hole correlations interfere — BCS wave function
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Multiple collisions
+ +
++ +
two sequentialtwo-particleprocesses
two-particleprocessesunder effect of a third particle
two-particleprocess
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Galitskii-Feynman T-matrix approximation
ladder approximation of T-matrix
Dyson equation
+=
= +
two sequentialtwo-particleprocesses
two-particleprocessesunder effect of a third particle
two-particleprocess inselfconsistentexpansion
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Galitskii-Feynman T-matrix approximation – superconductivity
ladder approximation of T-matrix
Dyson equation
+=
= +
has pole for bound state it is singular for T-matrix obeys Bose statistics in condensation
approximate T-matrix
)(b pkE),,( kkkp T
),,0( kk0T
T
k
p p
k
k k k
p
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Galitskii-Feynman T-matrix approximation – superconductivity
ladder approximation of T-matrix
Dyson equation
approximate Dyson equation
+=
= +
)(b pkE
T
k
p p
k
k k k
p
= +k k kk
),(1),( 2 k
kk
zGz
zG ),(),( kk zGzG
no pole, no gap
),,0( kk0T
),,( kkkp T has pole for bound state it is singular for T-matrix obeys Bose statistics in condensation
approximate T-matrix
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Kadanoff-Martin approach
ladder approximation of T-matrix
Dyson equation
+=
= +
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Kadanoff-Martin approach
ladder approximation of T-matrix
Dyson equation
approximate Dyson equation
+=
= +
T
= +k k kk
)(1),( 2
kk
k
zzzG
BCS gap
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Kadanoff-Martin approach
ladder approximation of T-matrix
Dyson equation
approximate Dyson equation
+=
= +
T
= +k k kk
)(1),( 2
kk
k
zzzG
BCS gap
Prange paradox
HISTORY – T-MATRIX IN SUPERCONDUCTIVITY
Why paradox?
The worse approximation yields
the better result.
Wrong conclusions
The superconductor and normal metal are two distinct states which cannot be covered by a unified theory.
Pragmatic conclusion
The Galitskii-Feynman approximation includes double-counts which are fatal in the superconducting state.
We will remove double-counts.
Prange paradox
PROBLEM – REPEATED COLLISIONS
Multiple collisions
+ +
++ +
two sequentialtwo-particleprocesses
two-particleprocessesunder effect of a third particle
two-particleprocess
PROBLEM – REPEATED COLLISIONS
Multiple collisions
+ +
++ +
two-particleprocess
two sequentialtwo-particleprocesses
two-particleprocessesunder effect of a third particle
k k
k k k
p p
k
p
p ppq
q
q
Third particle ought to be different from the interacting pair:
andkq pq
PROBLEM – REPEATED COLLISIONS
Galitskii-Feynman T-matrix approximation
ladder approximation of T-matrix
Dyson equation
+
Third particle ought to be different from the interacting pair:
andkq pq
=
= +
= + + +
p
k k kk
k k kkkk
p p q
not satisfied
PROBLEM – REPEATED COLLISIONS
Galitskii-Feynman T-matrix approximation
ladder approximation of T-matrix
Dyson equation
+
Third particle ought to be different from the interacting pair:
andkq pq
=
= +
= + + +
p
k k kk
k k kkkk
p p q
not satisfied
Standard argument:Each momentum contributes as 1/volume. terms withvanish for infinite volume.
holds in the normal statefails in superconductorsfor momentum
pq
kp
SOLUTION – EFFECTIVE MEDIUM OF SOVEN
Soven trick of the self-consistent medium:
1. Split scattering potential into independent channels – the self-energy splits into identical channels
2. Describe all channels but one by the self-energy – for the remaining channel use the scattering theory
3. From the scattering theory evaluate the propagator – from the propagator identify the single-channel self-energy
iiii GSGGG
SOLUTION – EFFECTIVE MEDIUM OF SOVEN
Soven trick of the self-consistent medium: alloy scattering
1. Split scattering potential into independent channels – the self-energy splits into identical channels
2. Describe all channels but one by the self-energy – for the remaining channel use the scattering theory
3. From the scattering theory evaluate the propagator – from the propagator identify the single-channel self-energy
i
iVV
iii GGGG
i
i
ii
ii VGVT 1
ii cTS
ii
ii GSS
1
site index
iiii GSGGG
SOLUTION – EFFECTIVE MEDIUM OF SOVEN
Soven trick of the self-consistent medium: alloy scattering
1. Split scattering potential into independent channels – the self-energy splits into identical channels
2. Describe all channels but one by the self-energy – for the remaining channel use the scattering theory
3. From the scattering theory evaluate the propagator – from the propagator identify the single-channel self-energy
i
iVV
iii GGGG
i
i
ii
ii VGVT 1
ii cTS
ii
ii GSS
1
iii
ii
ii
i
GV
Vc
Gc
)(11)1(0
ATA CPAGV
Vc
i
ii
1
site index
),,(),(),(),(
),(),,(
B kppqppk
kkkk
qqp
q
zTzGGVTk
VzT
SOLUTION – EFFECTIVE MEDIUM OF SOVEN
Soven trick of the self-consistent medium: superconductivity
1. Split scattering potential into independent channels – the self-energy splits into identical channels
2. Describe all channels but one by the self-energy – for the remaining channel use the scattering theory
3. From the scattering theory evaluate the propagator – from the propagator identify the single-channel self-energy
pqpkkqq
q pk
aaaaVV ),(1
),(),( kkq
q
pair summomentum
qqq GGGG
qqqq GSGGG
),(),,(),( B kqkkkq
zGzT
TkS
z
GS
S1
),,(),(),(),(
),(),,(
B kppqppk
kkkk
qqp
q
zTzGGVTk
VzT
SOLUTION – EFFECTIVE MEDIUM OF SOVEN
Soven trick of the self-consistent medium: superconductivity
1. Split scattering potential into independent channels – the self-energy splits into identical channels
2. Describe all channels but one by the self-energy – for the remaining channel use the scattering theory
3. From the scattering theory evaluate the propagator – from the propagator identify the single-channel self-energy
pqpkkqq
q pk
aaaaVV ),(1
),(),( kkq
q
pair summomentum
),(),,(),( B kqkkkq
zGzT
TkS
z
GS
S1
The set of equations is closed.
Its complexity compares to the Galitskii-Feynman approximation.
qqq GGGG
qqqq GSGGG
),,(),(),(),(
),(),,(
B kppqppk
kkkk
qqp
q
zTzGGVTk
VzT
SOLUTION – EFFECTIVE MEDIUM OF SOVEN
Soven trick of the self-consistent medium: superconductivity
normal statepqpkkq
qq pk
aaaaVV ),(1
),(),( kkq
q
pair summomentum
),(),,(),( B kqkkkq
zGzT
TkS
z
GS
S1
qT is regular
1
qS
qq S
GGq qqq GGGG
qqqq GSGGG
In the normal state the repeated collisions vanish as expected. One recovers the Galitskii-Feynman approximation.
= +k k kk
),,(),(),(),(
),(),,(
B kppqppk
kkkk
qqp
q
zTzGGVTk
VzT
SOLUTION – EFFECTIVE MEDIUM OF SOVEN
Soven trick of the self-consistent medium: superconductivity
superconducting state
),(),,(),( B kqkkkq
zGzT
TkS
z
),,0( kk0T
is singular
),(),( kk0
GS
),(),(),(),(),( kkkkk000 GGGGG
qqqq GSGGG
In the superconducting state the gap opens as in the renormalized BCS theory.
EXAMPLE – CORRECTION TO THE BCS GAP
ladder approximation of the T-matrix
BCS potential
scalar equation
),,(),(),(),(),(),,( B kppqppkkkkkqq
pqqq
zTzGGV
TkVzT
cut
),(),(22
B
p
p0
pp GGVVTk
)()(),,0( cutcut2
Bkkq
kk
Tk
T)()(),( cutcut kk0 kk VV
2BB
cut
),(),(1
TkGG
Tk
V
p
p0
pp
EXAMPLE – CORRECTION TO THE BCS GAP
ladder approximation of the T-matrix
2BB
cut
),(),(1
TkGG
Tk
V
p
p0
pp
EXAMPLE – CORRECTION TO THE BCS GAP
ladder approximation of the T-matrix
2BB
cut
),(),(1
TkGG
Tk
V
p
p0
pp
2B
B
22
0220 2
tanh1 cut
Tk
Tk
E
E
dEN
V
integration over momentum isexpressed via density of states
sum over Matsubara’s frequencies performed
EXAMPLE – CORRECTION TO THE BCS GAP
ladder approximation of the T-matrix
2BB
cut
),(),(1
TkGG
Tk
V
p
p0
pp
2B
B
22
0220 2
tanh1 cut
Tk
Tk
E
E
dEN
V
integration over momentum isexpressed via density of states
sum over Matsubara’s frequencies performed
the left hand side is the BCS gap equation
the right hand side is a volume correction
EXAMPLE – CORRECTION TO THE BCS GAP
ladder approximation of the T-matrix – close to Tc
2B
20
2
0 1
Tk
T
TN
c
2B
B
22
0220 2
tanh1 cut
Tk
Tk
E
E
dEN
V
the left hand side is the BCS gap equation
the right hand side is a volume correction
20
B202 4
112 N
Tk
T
T
T
T
ccbelow Tc
cT
T12
02
above Tc
cTT
N
Tk
10
2B2
SUMMARY
• The Galitskii-Feynman T-matrix approximation fails in the
superconducting state because of non-physical repeated
collisions.
• The repeated collisions are removed by reinterpretation of the
T-matrix approximation in terms of Soven effective medium.
• Derived corrections vanish in the normal state but make the
theory applicable to the superconductivity. We have unified
theory of normal and superconducting states.
• Derivation is very recent.
I welcome any idea of possible applications or extensions.
THANK YOU
FOR YOUR ATTENTION