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

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,27,1330010 (2013) YH, H. Guo, arXiv: 1505:04080

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

A brief introduction to BCS-BEC crossover and pair fluctuation

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

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:

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

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

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

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

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

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

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)

CFOP Imposing the self-consistent condition Perturbed current

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)

Full vertex and GWI Response function GWI

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, 043644 (2013)

Prove WI in QED From Peskin’s QFT book

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

Full vertex Should be determined from gap equation

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

Attempt to include pair fluctuation F. Palestini, P. Pieri, G. C. Strinati, Phys. Rev. Lett. 108, 080401 (2012) However this theory does not satisfy WI, see C.C.Chien, H Guo, K Levin, Phys. Rev. Lett. 109, 118901 (2012)

Include pair fluctuation BCS full Green’s function Gap equation

Full vertex

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

Equations for

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

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

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.