THEORY OF SUPERCONDUCTIVITY based on TMATRIX APPROXIMATION

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THEORY OF SUPERCONDUCTIVITYbased onT-MATRIX
APPROXIMATION

• 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

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AIM THEORY OF SUPERCONDUCTIVITY

• Why we need any alternative 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.
To describe fluctuations between normal and
superconducting states, we need a theory which
cover both phases.
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AIM THEORY OF SUPERCONDUCTIVITY

• Basic approaches

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
Motivated exclusively by normal state properties,
we modify the T-matrix so that it applies to
normal and superconducting states.
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HISTORY T-MATRIX IN SUPERCONDUCTIVITY
Two-particle collision
Schrödinger equation incoming waves
scattered waves
expansion in powers
T-matrix
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HISTORY T-MATRIX IN SUPERCONDUCTIVITY
Two-particle collision


.....

Schrödinger equation incoming waves
scattered waves
expansion in powers
T-matrix
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HISTORY T-MATRIX IN SUPERCONDUCTIVITY
Two-particle collision in T-matrix


.....

Schrödinger equation incoming waves
scattered waves
T-matrix
reconstructed wave function
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HISTORY T-MATRIX IN SUPERCONDUCTIVITY
Two-particle collision in T-matrix
V
non-local collision
reconstructed wave function reduces penetration
into strong repulsive potential
reconstructed wave function
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HISTORY T-MATRIX IN SUPERCONDUCTIVITY
Two-particle collision in T-matrix
V
reconstructed wave function resonantly increases
giving higher probability to of two particles
staying together
non-local collision due to finite duration of
collision
reconstructed wave function
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HISTORY T-MATRIX IN SUPERCONDUCTIVITY
Two-particle collision in T-matrix Pauli
principle


.....

Energy expansion (Bruckner Bethe and
Goldstone ...)
particle-particle correlation blocking of
occupied states Cooper problem
Time expansion (Feynman Galitskii Bethe
and Salpeter Klein and Prage)
particle-particle and hole-hole correlations
interfere BCS wave function
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HISTORY T-MATRIX IN SUPERCONDUCTIVITY
Multiple collisions
two-particle process





two sequential two-particle processes
two-particle processes under effect of a third
particle
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HISTORY T-MATRIX IN SUPERCONDUCTIVITY
Galitskii-Feynman T-matrix approximation ladder
approximation of T-matrix Dyson equation
two-particle process in selfconsistent expansion




two sequential two-particle processes
two-particle processes under effect of a third
particle
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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




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HISTORY T-MATRIX IN SUPERCONDUCTIVITY
Galitskii-Feynman T-matrix approximation
superconductivity ladder approximation of
T-matrix Dyson equation approximate Dyson
equation
has pole for bound state
it is singular for T-matrix obeys Bose
statistics in condensation
approximate T-matrix






no pole, no gap
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HISTORY T-MATRIX IN SUPERCONDUCTIVITY
Kadanoff-Martin approach ladder approximation of
T-matrix Dyson equation




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HISTORY T-MATRIX IN SUPERCONDUCTIVITY
Kadanoff-Martin approach ladder approximation of
T-matrix Dyson equation approximate Dyson
equation






BCS gap
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HISTORY T-MATRIX IN SUPERCONDUCTIVITY
Kadanoff-Martin approach ladder approximation of
T-matrix Dyson equation approximate Dyson
equation
Prange paradox




BCS gap
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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
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PROBLEM REPEATED COLLISIONS
Multiple collisions
two-particle process





two sequential two-particle processes
two-particle processes under effect of a third
particle
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PROBLEM REPEATED COLLISIONS
Multiple collisions
two-particle process





two sequential two-particle processes
two-particle processes under effect of a third
particle
Third particle ought to be different from the
interacting pair
and
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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
and
not satisfied
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PROBLEM REPEATED COLLISIONS
Galitskii-Feynman T-matrix approximation ladder
approximation of T-matrix Dyson equation
Standard argument Each momentum contributes as
1/volume. terms with vanish for infinite
volume. holds in the normal state fails in
superconductors for momentum








Third particle ought to be different from the
interacting pair
and
not satisfied
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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
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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
site index
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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
site index
ATA CPA
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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
pair sum momentum
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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
pair sum momentum
The set of equations is closed. Its complexity
compares to the Galitskii-Feynman approximation.
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SOLUTION EFFECTIVE MEDIUM OF SOVEN
Soven trick of the self-consistent medium
superconductivity
normal state
pair sum momentum
is regular
In the normal state the repeated collisions
vanish as expected. One recovers the
Galitskii-Feynman approximation.
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SOLUTION EFFECTIVE MEDIUM OF SOVEN
Soven trick of the self-consistent medium
superconductivity
superconducting state


is singular
In the superconducting state the gap opens as in
the renormalized BCS theory.
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EXAMPLE CORRECTION TO THE BCS GAP
ladder approximation of the T-matrix BCS
potential scalar equation
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EXAMPLE CORRECTION TO THE BCS GAP
ladder approximation of the T-matrix
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EXAMPLE CORRECTION TO THE BCS GAP
ladder approximation of the T-matrix
integration over momentum is expressed via
density of states
sum over Matsubaras frequencies performed
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EXAMPLE CORRECTION TO THE BCS GAP
ladder approximation of the T-matrix
the right hand side is a volume correction
integration over momentum is expressed via
density of states
the left hand side is the BCS gap equation
sum over Matsubaras frequencies performed
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EXAMPLE CORRECTION TO THE BCS GAP
ladder approximation of the T-matrix close to Tc
the right hand side is a volume correction
the left hand side is the BCS gap equation
below Tc
above Tc
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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.

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THANK YOU FOR YOUR ATTENTION