1. 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

2. AIM – THEORY OF SUPERCONDUCTIVITY The superconductivity was elucidated in Fermi gases 6 Li and 40 K. 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 T c 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.

3. 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.

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

5. HISTORY – T-MATRIX IN SUPERCONDUCTIVITY Two-particle collision expansion in powers Schrödinger equation incoming waves scattered waves T-matrix + + +..... =

6. HISTORY – T-MATRIX IN SUPERCONDUCTIVITY Two-particle collision in T-matrix + + +..... = Schrödinger equation incoming waves scattered waves T-matrix reconstructed wave function

7. HISTORY – T-MATRIX IN SUPERCONDUCTIVITY Two-particle collision in T-matrix reconstructed wave function non-local collision reconstructed wave function reduces penetration into strong repulsive potential V

8. HISTORY – T-MATRIX IN SUPERCONDUCTIVITY Two-particle collision in T-matrix reconstructed wave function non-local collision due to finite duration of collision V reconstructed wave function resonantly increases giving higher probability to of two particles staying together

9. 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

10. HISTORY – T-MATRIX IN SUPERCONDUCTIVITY Multiple collisions + + +++ two sequential two-particle processes two-particle processes under effect of a third particle two-particle process

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

12. 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

13. HISTORY – T-MATRIX IN SUPERCONDUCTIVITY Galitskii-Feynman T-matrix approximation – superconductivity ladder approximation of T-matrix Dyson equation approximate Dyson equation + = = + = + no pole, no gap has pole for bound state it is singular for T-matrix obeys Bose statistics in condensation approximate T-matrix

14. HISTORY – T-MATRIX IN SUPERCONDUCTIVITY Kadanoff-Martin approach ladder approximation of T-matrix Dyson equation + = = +

15. HISTORY – T-MATRIX IN SUPERCONDUCTIVITY Kadanoff-Martin approach ladder approximation of T-matrix Dyson equation approximate Dyson equation + = = + = + BCS gap

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

17. 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

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

19. 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

20. 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

21. 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 Standard argument: Each momentum contributes as 1/volume. terms with vanish for infinite volume. holds in the normal state fails in superconductors for momentum

22. 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

23. 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

24. 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 ATA CPA site index

25. 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

26. 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.

27. 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.

28. = + 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.

29. EXAMPLE – CORRECTION TO THE BCS GAP ladder approximation of the T-matrix BCS potential scalar equation

30. EXAMPLE – CORRECTION TO THE BCS GAP ladder approximation of the T-matrix

31. EXAMPLE – CORRECTION TO THE BCS GAP ladder approximation of the T-matrix integration over momentum is expressed via density of states sum over Matsubara’s frequencies performed

32. EXAMPLE – CORRECTION TO THE BCS GAP ladder approximation of the T-matrix integration over momentum is expressed 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

33. EXAMPLE – CORRECTION TO THE BCS GAP ladder approximation of the T-matrix – close to T c the left hand side is the BCS gap equation the right hand side is a volume correction below T c above T c

34. 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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