Dynamic Density and Spin Responses in the BCS-BEC Crossover: Toward a Theory beyond RPA

Transcription

1 Dynamic Density and Spin Responses in the BCS-BEC Crossover: Toward a Theory beyond RPA Lianyi He ( 何联毅 ) Department of Physics, Tsinghua University 2016 Hangzhou Workshop on Quantum Degenerate Fermi Gases, Zhejiang University arxiv: Many thanks to Joe Carlson, Stefano Gandolfi, and Hui Hu

2 Outline Theory of BCS-BEC crossover: Gapless approximations Dynamic responses and structure factors BCS-Leggett response theory: RPA Gaussian-pair-fluctuation (GPF) response theory Summary

3 Phase diagram of BCS-BEC crossover (3D & s-wave) High T: virial expansion Pseudogap? Low-density expansion for fermions Low-density expansion for composite bosons 1980 From C. A. R. Sa de Melo, Physics Today (2008) BCS smooth crossover v k = N φ 1/ 2 B k 2 ( + N φ ) 1/ 2 1 B k Ψ 0 exp( ) 1/ = φ c 0 N B BEC kc k k k

4 Realizing strongly interacting Fermi gases in cold atoms: Magnetic field tuned (s-wave) Feshbach resonance Alkali atoms ( 6 Li, 40 K) in a magnetic field: Coupled multi-channel scattering Low-energy scattering amplitude of open channel: Scattering energy E << Zeeman splitting Unitary Fermi Gas: A strongly coupled Fermi gas at infinite a Expansion dynamics similar to strongly coupled Quark Gluon Plasma: another nearly perfect liquid! J. E. Thomas et al., Science (2002); Quark Matter 2009 Dilute Neutron Matter: A nearly unitary Fermi gas A. Gezerlis & J. Carlson, PRC 2010

5 Broad Feshbach resonance: large scattering length & negligible effective range Universal many-body physics! Physical properties depend on two parameters: & Low-energy effective theory contact interaction (one-channel model) Renormalization of bare coupling U in terms of physical scattering length a: Computing the scattering amplitude & Matching known result Small k F a & low T: perturbation theory (low-density expansion) Huang, Lee & Yang (1957) Galitskii (1958) Bishop (1973) High T & arbitrary coupling: virial expansion n-body problems are hard! second virial expansion: Beth & Uhlenbeck (1937) Mueller & Ho (2004) third virial expansion: Hu, Liu & Drummond (2009, 2010) Large k F a & low T: Less is known! QMC, T-matrix, epsilon expansion

6 Theory the BCS-BEC crossover: T=0 vs T=Tc At T=0, BCS gap equation + number equation is qualitatively correct to describe the BCS-BEC crossover. BCS-Leggett mean-field theory (1980) Above Tc, we must include the pair fluctuations to get correct superfluid transition temperature! Nozieres Schmitt-Rink (NSR) theory (1985) How to connect the above two different approaches?

7 Experiments & QMC: Quantum fluctuations are important even at T=0, especially for 2D. 3D 2D mean field mean field QMC QMC Exp (Turlapov) From Giorgini, Pitaevskii & Stringari, Rev. Mod. Phys. (2008) QMC: Astrakharchik, Boronat, Casulleras, and Giorgini (2004) See also Carlson et al. (2003), Lobo et al. (2006), Forbes et al. (2011), From Makhalov, Martiyanov & Turlapov, PRL (2014) 2D mean-field theory: Randeria et al. (1989) Experiments: Turlapov et al. (2014) Thomas et al. (2015) QMC: Bertaina & Giorgini (2011) Shi, Chiesa & Zhang (2015) Anderson & Drut (2015)

8 Above Tc: Beyond-mean-field theory: T-matrix approaches Different T-matrix approaches: Different choices of fermion Green s function Below Tc: No clear classification of n-body contributions! Because of condensation: fermion Green s function is a 2x2 matrix G 0 G 0 : Ohashi & Griffin (2003) Pieri, Pisani & Strinati (2004) Hu, Liu & Drummond (2006) Diener, Sensarma & Randeria (2008) gapless approximation, but different treatments of number equation! G 0 G: Chen & Levin et al. (2004) recover mean field at T=0 pseudogap theory: extended mean-field theory at finite T GG: Haussmann, Rantner, Cerrito, & Zwerger (2007) conseving/luttinger-ward approximation: but still need truncation

9 Toward an analytical & quantitative theory at T<Tc Considerations: a quantitative theory should recover known limits. (1) BCS limit: Fermi liquid correction Huang & Yang (1957) Galitskii (1958) (2) BEC limit: weakly interacting Bose condensate Bogoliubov (1947) Lee, Huang & Yang (1957) composite boson scattering length Petrov, Salomon & Shlyapnikov (2004) (3) High-T limit: second virial expansion Beth & Uhlenbeck (1937) Who is the winner under these terms? Surprisingly, it is a G 0 G 0 theory with a careful treatment of the number equation! Gaussian Pair Fluctuation (GPF) Theory Diagrammatic version: Hu, Liu & Drummond, EPL 2006 Functional path integral version: Diener, Sensarma & Randeria, PRA D: mean-field theory fails to describe the strong coupling (BEC) limit GPF is a must! LH, Lü, Cao, Hu & Liu, PRA 2015

10 Functional Path Integral Approach Sa de Melo, Randeria & Engelbrent (1993, 1997) Strotonovich-Hubbard Mean field + fluctuations: mean field Gaussian pair fluctuations Goldstone mode fluctuation at T<Tc Fermi-liquid correction at T=0 and weak coupling

11 Coupling constant renormalization Computing scattering amplitude with contact interaction (LS equation): UV divergence: cutoff regularization (dimensional reg. epsilon expansion) 3D: 2D: Renormalization=Matching known scattering amplitude 3D: 2D: Gas parameters (interaction strength) 2D scattering length 3D: 2D:

12 BCS-Leggett Mean-Field Theory T=0: Gap Equation: Number Equation:

13 BEC limit of mean-field theory at T=0: 3D vs 2D Ginzburg-Landau expansion near the vacuum-bec transition In terms of molecule field: Gross-Pitaevskii free energy Mean-field description of composite boson interaction = Born approximation for four-body scattering 3D: The composite boson coupling is qualitatively correct. with exact four-body result 2D: The composite boson coupling is qualitatively wrong! NOT weakly interacting 2D Bose condensate Logarithmic dependence on energy missing! Note: four-body correlations are included even in T=0 mean field!

14 Gaussian Fluctuations: Collective Modes Goldstone mode in the superfluid state

15 Gaussian Pair Fluctuation (GPF) Theory

16 Gaussian Pair Fluctuation (GPF) Theory: Equation of State thermodynamic consistency correct four-body contribution Note: the order parameter as a function of the chemical potential,, is determined by minimizing the mean-field grand potential! Why: (1) Guarantee Goldstone s theorem: gapless approximation (2) Maintain the Silver Blaze property: vacuum state should have vanishing pressure and density! analogy in QCD: critical baryon & isospin chemical potential

17 Gaussian Pair Fluctuation (GPF) Theory: Results H. Hu et al., EPL (2006) & Nat. Phys (2007) LH et al., PRA (2015) mean-field + 3D 2D Composite boson scattering length: Composite boson scattering length: Exact result: Bertsch parameter: Latest QMC: Stefano et al. (2011) agrees with Petrov s exact result

18 Dynamic density and spin responses Operator definition: Spin-balanced system:

19 In terms of total density and spin density: We define the density and spin responses: In principle, there are off-diagonal responses: For spin-balanced system, they vanish exactly.

20 Experiments Density response: C. Vale group, PRL 2008 (trap) C. Vale group, PRL 2014 (homogeneous) Spin response: C. Vale group, PRL 2012 (trap) RPA cannot fully understand these result quantitatively! Including pair fluctuations would be hopeful to explain experiments.

21 Theories Low temperature: Superfluid Random Phase Approximation (RPA) H. P. Buechler, P. Zoller, and W. Zwerger, PRL, (2004) R. Combescot, S. Giorgini, and S. Stringari, EPL 75, 695 (2006) P. Zou, E. D. Kuhnle, C. J. Vale, and H. Hu, PRA82, (R) (2010) It seems the RPA at T=0 agrees well with experiments at large momentum Finite temperature: T-matrix, virial expansion, etc. H. Guo, C.-C. Chien, and K. Levin, PRL. 105, (2010) F. Palestini, P. Pieri, and G. C. Strinati, PRL. 108, (2012) H. Hu, X.-J. Liu, and P. D. Drummond, Phys. Rev. A81, (2010) H. Hu, X.-J. Liu, Phys. Rev. A85, (2012) Discussion on the gauge invariance: H. Guo, C.-C. Chien, Y. He, and K. Levin, IJMPB27, (2013). B. Anderson, et al., PRB93, (R) (2016).

22 Functional Path Integral Formulation Introduce external sources:

23 Strotonovich-Hubbard transformation:

24 Dynamic response functions from generating functional: Connected correlation functions

25 Momentum space: Expansion of generating functional in powers of j: Grand potential in equilibrium = thermodynamic relation

26 Dynamic response functions given by W (2) [J]: or Spin-balanced system:

27 Dynamic and Static Structure Factors

28 BCS-Leggett Response Theory = RPA Pairing field replaced by its classical field Induced inhomogeneous perturbation determined by

29 In momentum space we need to do the expansion:

30 Second-order expansion gives response functions:

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33 Final Results at T=0: Random Phase Approximation Inverse propagator of collective modes: And define the following functions:

34 Spin response:

35 Some simple numerical results for RPA: (q=4.5k F ) BCS Unitary Density response: signals BCS-BEC BEC Spin response: directly related to neutrino emission rate in neutron matter (Joe Carlson et al., PRC 2013)

36 Why and how to go beyond the RPA? RPA = response of the equilibrium state in BCS- Leggett mean-field theory. Therefore, it is only qualitative. Pair fluctuation effects are not included in RPA Consider the response of equilibrium state described by some beyond-mean-field theory Here let s consider the Gaussian-pair-fluctuation (GPF) theory => GPF response theory

37 GPF Response Theory

38 Nontrivial linear term in GPF generating functional How to eliminating the induced perturbations in GPF? We still use the mean-field gap equation : 1. Natural generalization of the GPF in equilibrium 2. Guarantee the compressibility sum rule

39 Using the generalized gap equation we obtain

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41 Calculation of the GPF generating functional

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43 The GPF generating functional:

44 Expansion of the GPF generating functional

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46 Structure of the expansions LO expansion X NLO expansion LO expansion X NNLO expansion NLO expansion X NLO expansion Then use again derivative expansion!

47 Aslamazov-Larkin Self-energy Maki-Thompson

48 Aslamazov-Larkin Maki-Thompson Self-energy

49 Order-Parameter Induced Contribution

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52 Summary and outlook RPA was reformulated by using path integral Dynamic response theory based on GPF was developed Numerical calculations & comparison with exp Application to other systems: SOC, OFR, Other linear responses: Shear and bulk viscosities, Thank you!