Gaussian fluctuations in the two-dimensional BCS-BEC crossover - PowerPoint PPT Presentation

Gaussian fluctuations in the two-dimensional BCS-BEC crossover Giacomo Bighin in collaboration with: Luca Salasnich Universit` a degli Studi di Padova and INFN Padova, January 11th, 2016

Outline • Introduction and motivation: BCS-BEC crossover in 2D. • Theoretical description of a 2D Fermi gas: mean-field and Gaussian fluctuations. • The role of fluctuations: the composite boson limit. • Results and comparison with experimental data: � First sound � Second sound � Berezinskii-Kosterlitz-Thouless critical temperature. Main reference : GB and L. Salasnich, arXiv:1507.07542. 2 of 21

The BCS-BEC crossover (1/2) In 2004 the BCS-BEC crossover has been observed with ultracold gases made of fermionic 40 K and 6 Li alkali-metal atoms. The fermion-fermion attractive interaction can be tuned (using a Feshbach resonance), from weakly to strongly interacting. BCS regime : weakly interacting BEC regime : tightly bound Cooper pairs. bosonic molecules. 3 of 21

The BCS-BEC crossover (2/2) An additional laser confinement can used to create a quasi-2D pancake geometry. The 2D scattering length is determined by the geometry 1 : p a 2 D ' ` z exp( � ⇡ / 2 ` z /a 3 D ) � 0 1 Bose Strong 0.9 Interaction In 2014 the 2D BCS-BEC crossover 0.8 has been achieved 1 with a quasi-2D 0.7 0.6 Fermi gas of 6 Li atoms with P 2 P 2 ideal 0.5 widely tunable s-wave interaction. 0.4 0.3 The pressure P vs the gas parameter 0.2 a B n 1 / 2 has been measured. Fermi 0.1 0 0.1 1 10 a 2 √ n 2 1 V. Makhalov, K. Martiyanov, and A. Turlapov, PRL 112 , 045301 (2014). 2 ` z is the thickness of each layer. 4 of 21

The BCS-BEC crossover in 2D (1/2) Many properties of 2D Fermi gases are currently being studied: • Imaging of the atomic cloud 1 . • Phase diagram 1 . • Very recently (June 2015) the direct observation of the BKT transition has been reported 2 . • Dynamical properties: sound velocity. imaging beam trapping beams camera 1 M. G. Ries et al., Phys. Rev. Lett. 114 , 230401 (2015) 2 P. A. Murthy et al., Phys. Rev. Lett. 115 , 010401 (2015). 5 of 21

The BCS-BEC crossover in 2D (1/2) Many properties of 2D Fermi gases are currently being studied: • Imaging of the atomic cloud 1 . • Phase diagram 1 . • Very recently (June 2015) the direct observation of the BKT transition has been reported 2 . • Dynamic properties: sound velocity. 0.3 N q /N 0.50 0.2 0.40 0.30 T/T F 0.20 0.1 0.10 0.00 BEC BCS 0.0 -8 -6 -4 -2 0 2 4 ln(k F a 2D ) 1 M. G. Ries et al., Phys. Rev. Lett. 114 , 230401 (2015) 2 P. A. Murthy et al., Phys. Rev. Lett. 115 , 010401 (2015). 5 of 21

The BCS-BEC crossover in 2D (1/2) Many properties of 2D Fermi gases are currently being studied: • Imaging of the atomic cloud 1 . • Phase diagram 1 . • Very recently (June 2015) the direct observation of the BKT transition has been reported 2 . • Dynamic properties: sound velocity. (a) (b) 50 1 ln(k F a 2D ) ∼ -0.5 Power-law 40 Exponential t = 0.31 First-order correlation function g 1 (r) t = 0.42 T c 30 t = 0.45 t = 0.47 t = 0.57 20 2 (arbitrary units) 0.1 10 0 1 ln(k F a 2D ) ∼ 0.5 Power-law 30 Exponential t = 0.37 t = 0.44 T c χ t = 0.47 20 t = 0.49 t = 0.58 10 0.1 0 10 100 0.4 0.6 r ( µ m) T/T 0 BEC 1 M. G. Ries et al., Phys. Rev. Lett. 114 , 230401 (2015) 2 P. A. Murthy et al., Phys. Rev. Lett. 115 , 010401 (2015). 5 of 21

The BCS-BEC crossover in 2D (2/2) Why is the 2D case interesting from the theory point of view? • The fluctuations are more relevant for lower dimensionalities. The mean field theory can correctly describe (to some extent) the crossover in 3D, we expect it not to work at all in 2D. • Berezinskii-Kosterlitz-Thouless mechanism: � Mermin-Wagner-Hohenberg theorem: no condensation at finite temperature, no o ff -diagonal long-range order. � Algebraic decay of correlation functions h exp(i θ ( r )) exp(i θ (0)) i ⇠ | r | − η � Transition to the normal state at a finite temperature T BKT . • The physics of the BCS-BEC crossover is also relevant in the description of many di ff erent systems (bilayers of dipolar gases, exciton condensates). It may also be relevant for the description of high- T c cuprates as the scaled correlation length ( k F ⇠ 0 ⇠ 5 for YBCO and k F ⇠ 0 ⇠ 10 for LSCO) lies between the BCS ( k F ⇠ 0 ⇠ 10 3 ) and BEC ( k F ⇠ 0 ⌧ 1) regimes. 6 of 21

Formalism for a D -dimensional Fermi superfluid (1/4) We adopt the path integral formalism. The partition function Z of the uniform system with fermionic fields s ( r , ⌧ ) at temperature T , in a D -dimensional volume L D , and with chemical potential µ reads Z ⇢ � 1 � D [ s , ¯ Z = s ] exp ~ S , where ( � ⌘ 1 / ( k B T ) with k B Boltzmann’s constant) Z ~ β Z L D d D r L S = d ⌧ 0 is the Euclidean action functional with Lagrangian density: ~ @ τ � ~ 2  � 2 m r 2 � µ L = ¯ s + g 0 ¯ ↑ ¯ s ↓ ↓ ↑ where g 0 is the attractive strength ( g 0 < 0) of the s-wave coupling. 7 of 21

Formalism for a D -dimensional Fermi superfluid (2/4) In 2D the strength of the attractive s-wave potential is g 0 < 0, which can be implicitely related to the bound state energy: � 1 1 1 X = . ✏ k + 1 2 L 2 g 0 2 ✏ b k with ✏ k = ~ 2 k 2 / (2 m ). In 2D, as opposed to the 3D case, a bound state exists even for arbitrarily weak interactions, making ✏ B a good variable to describe the whole BCS-BEC crossover. The binding energy ✏ b and the fermionic (2D) scattering length a 2 D are related by the equation 2 : 4 ~ 2 ✏ B = e 2 γ ma 2 2 D 2 C. Mora and Y. Castin, Phys. Rev. A 67 , 053615 (2003). 8 of 21

Formalism for a D -dimensional Fermi superfluid (3/4) Through the usual Hubbard-Stratonovich transformation the Lagrangian density L , quartic in the fermionic fields, can be rewritten as a quadratic form by introducing the auxiliary complex scalar field ∆ ( r , ⌧ ) so that: � S e ( s , ¯ s , ∆ , ¯ ⇢ � Z ∆ ) D [ s , ¯ s ] D [ ∆ , ¯ Z = ∆ ] exp , ~ where Z ~ β Z S e ( s , ¯ s , ∆ , ¯ L D d D r L e ( s , ¯ s , ∆ , ¯ ∆ ) = d ⌧ ∆ ) 0 and the (exact) e ff ective Euclidean Lagrangian density L e ( s , ¯ s , ∆ , ¯ ∆ ) reads ~ @ τ � ~ 2 ↓ � | ∆ | 2  � 2 m r 2 � µ L e = ¯ s + ¯ ∆ ↓ ↑ + ∆ ¯ ↑ ¯ . s g 0 9 of 21

Formalism for a D -dimensional Fermi superfluid (4/4) We want to investigate the e ff ect of fluctuations of the pairing field ∆ ( r , t ) around its saddle-point value ∆ 0 which may be taken to be real. For this reason we set ∆ ( r , ⌧ ) = ∆ 0 + ⌘ ( r , ⌧ ) , where ⌘ ( r , ⌧ ) is the complex field which describes pairing fluctuations. In particular, we are interested in the grand potential Ω , given by Ω = � 1 � ln ( Z ) ' � 1 � ln ( Z mf Z g ) = Ω mf + Ω g , where � S e ( s , ¯ ⇢ � Z s , ∆ 0 ) D [ s , ¯ Z mf = s ] exp ~ is the mean-field partition function and � S g ( s , ¯ Z ⇢ s , ⌘ , ¯ ⌘ , ∆ 0 ) � D [ s , ¯ Z g = s ] D [ ⌘ , ¯ ⌘ ] exp ~ is the partition function of Gaussian pairing fluctuations. 10 of 21

Single particle and collective excitations One finds that in the gas of paired fermions there are two kinds of elementary excitations: fermionic single-particle excitations with energy s✓ ~ 2 k 2 ◆ 2 + ∆ 2 E sp ( k ) = 2 m � µ 0 , where ∆ 0 is the pairing gap, and bosonic collective excitations with energy s ~ 2 q 2 � ~ 2 q 2 ✓ ◆ 2 m + 2 mc 2 E col ( q ) = , s 2 m where � is the first correction to the familiar low-momentum phonon dispersion E col ( q ) ' c s ~ q and c s is the sound velocity. 11 of 21

The role of Gaussian fluctuations and collective excitations: composite bosons In the strongly interacting limit an attractive Fermi gas maps to a gas of composite bosons with chemical potential µ B = 2( µ + ✏ b / 2) and mass m B = 2 m . Residual interaction. Is this limit correctly recovered 3 at mean-field? And at a Gaussian level? Gaussian fluctuations are crucial in correctly describing the prop- erties of a 2D Fermi gas in the BEC limit (boson-boson scattering length, equation of state). What can be said about the sound ve- locity and the BKT critical temperature? 1 L. Salasnich and F. Toigo, Phys. Rev. A 91 , 011604(R) (2015) 12 of 21

Regularization Many regularization schemes: • Dimensional regularization The contribution from Analytical results 4 in the BEC limit fluctuations does not in 2D converge: E col ( q ) • Counterterms regularization Analytical results 5 in the BEC limit Ω g = 1 X in 3D 2 • Convergence factor regularization q Numerics for the whole crossover 6 , 7 . 4 L. Salasnich and F. Toigo, Phys. Rev. A 91 , 011604(R) (2015). 5 L. Salasnich and GB, Phys. Rev. A 91 , 033610 (2015). 6 R. B. Diener, R. Sensarma, and M. Randeria, Phys. Rev. A 77 , 023626 (2008) 7 L. He, H. L¨ u, G. Cao, H. Hu and X.-J. Liu, arXiv:1506.07156 13 of 21

Regularization Many regularization schemes: • Dimensional regularization The contribution from Analytical results 4 in the BEC limit fluctuations does not in 2D converge: E col ( q ) • Counterterms regularization Analytical results 5 in the BEC limit Ω g = 1 X in 3D 2 • Convergence factor regularization q Numerics for the whole crossover 6 , 7 . 4 L. Salasnich and F. Toigo, Phys. Rev. A 91 , 011604(R) (2015). 5 L. Salasnich and GB, Phys. Rev. A 91 , 033610 (2015). 6 R. B. Diener, R. Sensarma, and M. Randeria, Phys. Rev. A 77 , 023626 (2008) 7 L. He, H. L¨ u, G. Cao, H. Hu and X.-J. Liu, arXiv:1506.07156 13 of 21