Giuseppe Morandi and me. PhD In Bologna years : very fruitful and formative period. Collaboration continued for some years after.

Transcription

1 Giuseppe Morandi and me PhD In Bologna years : very fruitful and formative period. Collaboration continued for some years after. What I have learned from him Follow the beauty (and don t care too much about the experiments) Giuseppe had a great esthetic sense which guided him in deciding what was worth to be investigated and what was not. To some extent beautiful or elegant were synonyms of interesting or important to him. Timeo hominem unius libri Giuseppe was a man of many boos (he was really fond of them) and a deeply learned scholar. At a time when people were still not so obsessed by citations (and thus did not demand them explicitly), he was extremely attentive to other people s wor. Do not oversell your wor Giuseppe preferred understatements to emphatic claims.

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3 Physics & Geometry, Nov. 24, 2017, Bologna Transition temperature of a superfluid Fermi gas throughout the BCS-BEC crossover Leonardo Pisani, Andrea Perali, Pierbiagio Pieri, Giancarlo C. Strinati School of Science and Technology, Physics Division, University of Camerino, Italy

4 The BCS-BEC crossover Gas of fermions interacting via an attractive potential: Wea attraction: Cooper pairs form at low temperature according to BCS theory. Largely-overlapping pairs form and condense at the same temperature ( ). Strong attraction: the pair-size shrins and pair-formation is no longer a cooperative phenomenon. Non-overlapping pairs (composite bosons) undergo Bose-Einstein condensation at low temperature. Pair-formation and condensation become unrelated. How does the system evolves from one regime to the other one?

5 The interaction potential The standard BCS effective potential (constant attraction V within a shell of width ω D about the Fermi energy E F ) is not suitable for studying the BCS-BEC crossover (Fermi surface meaningless in the BEC limit). Two body attractive potential V(- ) is OK. Often approximated by a separable potential V(, '). For example NSR (or Yamaguchi) potential V(, ') = V 0 w()w( ') w() =1/ 1+ ( / 0 ) 2 For 0 >> F ~ n 1/3 (diluteness condition) the attractive potential is effectively short-ranged and can be parametrized completely in terms of the s-wave scattering length a : 1 = m V 0 4πa d m (2π ) 3 < 0 2 [example for a sharp cutoff w() = Θ( 0 ) ] Diluteness condition normally satisfied in experiments with ultracold Fermi atoms for which the dimensionless effective coupling parameter 1/ ( F a) can be tuned from BCS to BEC limit by using appropriate Fano-Feshbach resonances. BCS BEC / ( F a)

6 BCS critical temperature throughout the BCS-BEC crossover BCS equation for (for contact potential): m (1) 4πa = Π pp(0) d tanh(ξ where Π pp (0) / 2 ) m with ξ (2π ) 3 2ξ 2 ε µ. With wea-coupling assumption << µ one gets Π pp (0) = m µ where. Eq. (1) then yields: 2π ln 8eγ 2 µ 2 µ 2mµ π = 8eγ 2 µ π exp π 2 µ a = 8eγ 2 E F π exp π 2 F a where for wea coupling. For stronger coupling becomes of order of (or larger than) T F : necessary to use number equation to adjust chemical potential. BCS number equation at (and above) reduces to free Fermi gas relation (2) Eagles was first to notice (1969) that simultaneous solution of (1) and (2) yields in the BEC limit a pairing temperature T* rather than critical temperature for condensation.

7 Interlude: a threefold role for ladder diagrams Γ (Q) 0 = +Q + + +Q +Q +... Q (Q,Ω ν ) The sum of ladder diagrams Γ 0 (Q) has a threefold role: The BCS equation for corresponds to the equation Γ 1 0 (Q = 0) = 0 (Thouless criterion): the two-particle Green s function diverges at zero center-of-mass momentum and frequency. In the BEC limit, Γ 0 (Q) becomes (except for numerical factors) the free propagator of molecules (composite bosons). For a contact potential, the series of ladder diagrams replaces the bare interaction in diagrammatic theory. This is because the bare potential strength V 0 vanishes as the momentum cutoff 0. Every particle-particle rung in the ladder series is ultraviolet divergent and compensates a vanishing V 0, maing finite for 0. Γ 0

8 Nozières-Schmitt-Rin approach to the BCS-BEC crossover Consider the simplest self-energy which can be constructed with ladder series: Q Σ() = Γ (Q) 0 Use number equation n=tr G, where G = G 0 + G 0 ΣG 0 coupled with the standard BCS equation for critical temperature. In this way approaches BE critical temperature in the BEC limit. True also when the Dyson s equation G 1 = G 0 1 Σ is used to calculate G and then n (T-matrix self-energy approach). / T BEC BCS T-matr NSR ( F a) -1

9 NSR theory in the wea-coupling limit Common belief: The NSR curve for interpolates between the BCS and BE critical temperatures. Not really true: in the BCS limit the NSR approaches the BCS critical temperature divided by a factor e 1/3. Reason? In wea coupling: Γ 0 4πa F / m The number equation in NSR approach then yields: µ 2mµ F π a F Σ() 2πna F / m Σ 0. µ = E F + 2πna / m Insert it in equation: = 8eγ µ πe exp π 2 2 µ a 10 = 8eγ E F πe 2 = 8eγ E F πe 2 = 8eγ E F πe 2 π exp 2 F a F (1+ 2 F a / 3π ) π exp (1 2 F a / 3π ) 2 F a F π exp 1 2 F a e 1/3 / T BEC 1 BCS T 0.1 c T-matr NSR ( F a F ) -1 ( F a) 1

10 Recovering BCS critical temperature in the wea-coupling limit If the fermion propagators within the ladder series are dressed with first-order self-energy Σ 0 = 2πna / m then µ µ ' = µ Σ 0 = E F and the BCS result for is recovered. Diagrammatically, to lowest order in wea coupling, it corresponds to dressing the composite-boson propagator Γ with a composite-boson self-energy insertion Σ Β. Γ 0 Γ = Γ 0 + Γ 0 Γ + Γ0 Γ 0 yielding: Γ 1 (Q) = Γ 1 0 (Q) Σ B (Q) Σ Pop where Pop B = 2 ( Γ 0 )

11 Dressed Thouless criterion Q- Γ 1 (Q = 0) = 0 m + Π pp (0)+ Σ B Pop (0) = 0 4πa F Γ 0 Σ Pop B (0) = 2 G 0 () 2 G 0 ( )G 0 (Q )Γ 0 (Q),Q 4πa m n 0 & m 2 ln 8eγ 2 µ ( 4π 2 µ π T ' ) + * 4πa m µ 3 3π 2 m 2 - π 4π 2 µ 2 µ a = m µ 6π 2 For, ln(µ/τ) originating from G 0 () 2 G 0 ( ) compensates the small parameter a! T Eq. for becomes m 4πa + m µ 2π ln 8eγ 2 µ + m µ 2 π 6π = 0 2 π µ a + ln 8eγ 2 µ + 1 π 3 = 0 Counter-term 1/3 compensates -1/3 from expansion of µ. BCS result for is recovered.

12 The GMB correction in the wea-coupling limit (I) But is BCS expression for really correct in the wea-coupling limit? No! Gorov and Meli-Barhudarov (1961): For a low density attractive Fermi gas, the BCS expression is reduced by a factor (4e) 1/3 ~ 2.2 in the wea-coupling limit. Reason? Same mechanism just discussed: compensation of small parameter a with large factor ln(µ/τ) originating from particle-particle bubble. Dress Γ with following self-energy insertion: G 0 () G 0 ( ) B Σ GMB (0) = which contains two pp bubbles and two Γ 0 ~ a in wea coupling.

13 The GMB correction in the wea-coupling limit (II) Γ Σ GMB (0) = G 0 () G 0 ( )G 0 ( ')G 0 ( ')G 0 ( '')G 0 ( '' ')Γ 0 ( '' )Γ 0 ( '' '),','' ~ 4πa m G 0 ()G 0 ( ) 4πa m ' G 0 ( ')G 0 ( ') '' G 0 ( '')G 0 ( '' ') =1+O( F a) =1+O( F a) and select ( F, 0) and select ' (' F, 0) = 1 2 d( ˆ F ˆ'F )χ ph ( F + ' F, 0) = m F 2π 2 ln(4e)1/3 Equation for : m 4πa + m F 2π ln 8eγ 2 E F 2 π m F 2π 2 ln(4e)1/3 = 0 = 8eγ E F πe 2 π 1 exp 2 F a (4e) 1/3 Note: for standard BCS potential, the GMB reduction is instead exp(ω D /E F ) ~ 1 since ω D << E F. GMB correction irrelevant in this case.

14 GMB (and Popov) correction throughout the BCS-BEC crossover Coupled equations: Γ 1 (Q = 0) = 0 m + Π pp (0)+ Σ B B Pop (0) + Σ GMB (0) = 0 4πa F n = Tr G= 2T n d (2π ) G(,ω n)e iω + n 0 3 Σ() = Q Γ(Q) G 1 () = G 0 1 () Σ() Γ 1 (Q) = Γ 1 0 (Q) Σ B Pop B (0) Σ GMB (0) Γ = Γ Γ Γ + Γ Γ 0 0 Γ Γ Retain full momentum and frequency dependence of Γ and avoid all wea-coupling B B approximations when calculating and : B Σ GMB Σ Pop (0) = G 0 () G 0 ( )G 0 ( ')G 0 ( ')G 0 ( '')G 0 ( '' ')Γ( '' )Γ( '' '),','' B (0) = 2 G 0 () 2 G 0 ( )G 0 (Q )Γ(Q),Q Σ Pop Σ GMB Calculation quite non-trivial!

15 Numerical results: GMB and Popov self-energies at Tc Σ B /(2m F ) POPOV GMB ( F a F ) -1 Inclusion of both self-energies is important from wea coupling to past unitarity. To some extent, they compensate each other.

16 Numerical results for the critical temperature /T F GMB+POPOV QMC Bulgac QMC Burovsi T-matrix BCS-GMB ( F a) -1 A. Bulgac et al., PRA 78, (2008) E. Burovsi et al., PRL 101, (2008) The resulting curve for converges to the GMB result for wea coupling and to the BE critical temperature for strong coupling. In the intermediate coupling region it agrees very well with diag-qmc and lattice QMC.

17 Summary and conclusions GMB found that for a dilute attractive Fermi gas the BCS critical temperature is reduced by a numerical factor. Their wor was formulated in the wea coupling regime of the attraction. Standard diagrammatic approximations for the BCS-BEC crossover (NSR, T- matrix) fail to recover GMB result in wea coupling. We have reformulated GMB s wor in a way which is amenable to extension to the whole BCS-BEC crossover. Consistency in the wea-coupling limit required us to also include Popov twoparticle self-energy. The resulting curve for agrees very well with available QMC data. Than you!