1. Yan He 贺言 Sichuan University 四川大学
Establishing Gauge Invariant Linear Response of Fermionic Superuids with Pair Fluctuations
Yan He 贺言
Sichuan University 四川大学

2. Collaborators Chih-Chun Chien 简志钧 UC Merced Hao Guo 郭昊，Southest U东南大学
This talk is based on the following papers
H. Guo, C.C.Chien, YH, J. Low Temp. Phys. 172, 5 (2013)
H. Guo, C.C.Chien, YH, K. Levin, Int. J. Mod. Phys. B,27, (2013)
YH, H. Guo, arXiv: 1505:04080

3. Outline BCS-BEC crossover theory and pair fluctuation theory
Gauge invariant linear response in BCS mean field theory: Consistent Fluctuation of Order parameter (CFOP)
A diagrammatic derivation of CFOP
Gauge invariant linear response with pair fluctuation
Summary

4. A brief introduction to BCS-BEC crossover and pair fluctuation

5. Tunable attractive interaction via Feshbach resonance
Feshbach resonance involves a bound state near a continuum level. R
Unitary limit
Contact potential.
Interaction is described by scattering length. k -k
Increasing attraction

6. What is BCS-BEC Crossover
Interpolation between fermionic superfluids to bosonic superfluids
Transition from loosely bound Cooper pairs to tightly bound molecules.
The schematic excitation spectrum:

7. Physical picture of BCS-BEC crossover
Pairs are formed before condensation.
Energy gap is different from order parameter.
This combination defines BCS-BEC crossover.

8. BCS-BEC crossover at T=0
Based on BCS-Leggett ground state
Self-consistently solve for chemical potential

9. Generalization of BCS-BEC Crossover to finite temperatures: G0G pairing fluctuation
t-matrix (ladder approx.)‏ for non-condensed pairs
Fermion self-energy

10. Properties of G0G Pair fluctuations
It reduce to BCS mean field at T=0
It generates pseudogap at finite T
The superfluid transition is a continuous transition.
Consistency requires a gauge invariant linear response theory

11. BCS mean field theory Nambu spinor BCS full Green’s function
Self-consistent equation

12. Linear response to external EM field
Perturbed Hamiltonian
Perturbed current
Response function
But it cannot satisfy current conservation

13. Consistent Fluctuation of Order parameter (CFOP)
Treat the fluctuation of gap as an external field as external EM field
External field
External field vertex
Perturbed current
Response function
Kulik,etc, JLTP, 43, 591(1981)

14. CFOP
Imposing the self-consistent condition
Perturbed current

15. Current conservation (WI)
WI for bare vertex in the Numbu space
Then it is easy to verify that the response functuon
satisfying current conservation or Ward Identity (WI)

16. Full vertex and GWI
Response function
GWI

17. Q-limit WI and sum rules
It cannot be derived from GWI because one has to take first.
GWI and Q-limit WI implies the f-sum rule and compressibility sum rule
f-sum rule
Compressibility sum rule
H. Guo, YH, C.C.Chien, K.Levin, Phys. Rev. A 88, (2013)

18. Prove WI in QED
From Peskin’s QFT book

19. Derive GWI by inserting vertices to self-energy diagram
BCS full Green’s function
BCS self-energy
Gap equation

20. Full vertex
Should be determined from gap equation

21. Equations for
Full vertex satisfy GWI
This result is the same as CFOP

22. Attempt to include pair fluctuation
F. Palestini, P. Pieri, G. C. Strinati, Phys. Rev. Lett. 108, (2012)
However this theory does not satisfy WI,
see C.C.Chien, H Guo, K Levin, Phys. Rev. Lett. 109, (2012)

23. Include pair fluctuation
BCS full Green’s function
Gap equation

24. Full vertex

25. The Vertex for the pseudogap self-energy
The Vertex for the pseudogap self-energy should satisfy
For example

26. Equations for

27. GWI with Pseudogap One can verify GWI Current-current correlation
Current conservation
Longitudinal sum rule
f-sum rule

28. Some possible application
Calculate the viscosity in the whole BCS-BEC crossover
Collective mode have important contribution to shear and bulk viscosity.

29. Summary
We have construct a full vertex in the G0G pair fluctuation theory which satisfy GWI.
The response function calculated from this vertex will satisfy current conservation in BCS-BEC crossover.
It is useful for calculate transport properties.