Giacomo Bighin Universit` a degli Studi di Padova June 5, 2014
Plan of the talk • Condensate fraction for a polarized Fermi gas. • A gauge approach to superconductivity in high- T c cuprates. 2 of 31
Ultracold Fermi gases (1/3) • Ultracold gases: experimental observation of quantum properties of matter. Vortices in a superfluid, BEC. • Bose-Einstein condensation (1995), degenerate Fermi gas and fermionic condensate (2003). • Very clean experimental environment: control over the temperature, the number of particles, the interaction. 3 of 31
Ultracold Fermi gases (2/3) Why are ultracold Fermi gases interesting? The fermion-fermion interaction can be tuned (using a Feshbach resonance), from weakly to strongly interacting: the BCS-BEC crossover . BCS regime : coherence in BEC regime : coherence in momentum space. coordinate space. 4 of 31
Ultracold Fermi gases (3/3) Polarized Fermi gas Balanced Fermi gas ✓ ◆ ∂τ � ~ 2 r 2 ~ ∂ X ¯ L = ψ σ ( r , τ ) 2 m � µ σ ψ σ ( r , τ ) + σ = " , # g ¯ ψ " ( r , τ ) ¯ + ψ # ( r , τ ) ψ # ( r , τ ) ψ " ( r , τ ) 5 of 31
Condensate fraction (1/2) • Why? The condensate fraction is the fundamental signature of Bose-Einstein condensation: a finite fraction of particles occupying the ground state. • Definitions: � In terms of the Green functions in Nambu-Gor’kov space: N 0 = 1 X X X G 21 ( p , i ω n ) G 12 ( p , i ω m ) β 2 p n m � In terms of the BCS variational parameters, for T = 0: s✓ ◆ 2 ∆ 2 ~ 2 k 2 X X 0 u 2 k v 2 + ∆ 2 N 0 = 2 k = E k = 2 m � µ 0 4 E 2 k k k 6 of 31
The condensed fraction in the balanced case Usual path-integral treatment: ✓ ∂τ � ~ 2 r 2 ◆ ~ ∂ ¯ ψ σ ( r , τ )+ g ¯ ψ ↑ ( r , τ ) ¯ X L = ψ σ ( r , τ ) 2 m � µ ψ ↓ ( r , τ ) ψ ↓ ( r , τ ) ψ ↑ ( r , τ ) σ = ↑ , ↓ Z β Z Z ψ e − S [ ψ , ¯ ψ ] D ψ D ¯ ψ , ¯ ⇥ ⇤ d 3 r L Z = = d τ S ψ 0 V • Hubbard-Stratonovich transformation: ∆ ( r , τ ) ⇠ ¯ ψψ • Mean field approximation: ∆ ( r , τ ) = ∆ 0 + ⇠⇠⇠ δ ( r , τ ) XXX ⇠ X q k + | ∆ 0 | 2 ξ 2 E k = Gap equation/number equation from the thermodynamic � potential: ∂ Ω n = � ∂ Ω � = 0 � ∂ µ ∂ ∆ 0 � T, ζ 7 of 31
The condensate fraction in the balanced case: theory vs. experiments Figure : Condensate fraction N 0 N/ 2 of Fermi pairs in the uniform two-component dilute Fermi gas as a function of y = ( k F a s ) − 1 (solid line), T=0. The same quantity computed in the LDA for a droplet of N = 6 ⇥ 10 6 fermions in harmonic trap, as in the MIT experiment, plotted against the value of y at the center of the trap (joined diamonds). Open circles with error bars: experimentally determined condensed fraction by MIT group. [L. Salasnich, N. Manini, A. Parola, PRA 72 , 023621 (2005)] 8 of 31
The condensate fraction in the balanced case: theory vs. experiments The comparison between mean-field theory and MIT experiments for the condensate fraction of unpolarized two-component Fermi superfluid suggests that: • There are no relevant di ff erences between uniform and trapped theoretical results. • The experimental data are in good agreement with the theory only in the BCS side of the crossover up to unitarity. • The experimental data on the condensate fraction are not reliable in the BEC regime (inelastic losses?). 9 of 31
Condensate fraction for a unbalanced Fermi gas: theory Two number equations: � � n = � ∂ Ω δ n = � ∂ Ω � � � � ∂ µ ∂ζ � � T, ζ T,µ ✓ ◆ µ = µ ↑ + µ ↓ , ζ = µ ↑ � µ ↓ N, P = N + � N − ⇣ ⌘ N + + N − , y = ( k F a s ) − 1 , ∆ 0 $ 2 2 Condensate fraction (T=0): q ∆ 2 X k + | ∆ 0 | 2 ± ζ 0 E ± ξ 2 N 0 = k = 4 E 2 k | k | / 2 [ k − ,k + ] 10 of 31
Condensate fraction for a unbalanced Fermi gas: theory Phase separation: it is essential to model the trapping. In order to compare our theoretical condensate fraction with the MIT experiment done with trapped 6 Li atoms, we use the local density approximation (LDA), given by: µ ! µ � V ( r ) � � x 2 + y 2 � z z 2 � where V ( r ) = m ω 2 + ω 2 is the external trapping 2 ? potential. In this way the gap ∆ ( r ), the total density n ( r ) and the condensate density n 0 ( r ) become local scalar fields. 11 of 31
Condensate fraction for a unbalanced Fermi gas: theory 1000 1000 1000 800 800 800 600 600 600 n 0 n 0 n 0 400 400 400 200 200 200 0 0 0 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 z z z a z a z a z Figure : Condensate density profile n 0 ( z ) (solid line) and total density profile n ( z ) (dashed line) in the axial direction z for three di ff erent scattering lengths. From left to right: y = � 0 . 44, y = 0 . 0, y = 0 . 11, where y = ( k F a s ) − 1 with k F = (3 π 2 n ( 0 )) 1 3 and n ( 0 ) the total density at the center of the trap. Number of atoms N = 2 . 3 ⇥ 10 7 1 and polarization P = ( N ↑ � N ↓ ) /N = 0 . 2. Here a z = √ m ω z is the characteristic length of the axial harmonic confinement. 12 of 31
Condensate fraction for a unbalanced Fermi gas: theory vs. experiments 0.6 exp data for y = 0 exp data for y =- 0.44 0.5 Figure : Condensate fraction φ as a function of the absolute Condensate fraction H f L value of the polarization | P | for 0.4 two values of the dimensionless interaction parameter y = ( k F a s ) − 1 : y = � 0 . 44 (open 0.3 circles) and y = 0 . 0 (filled circles), T=0. Circles with 0.2 error bars are experimental data of 6 Li atoms taken from MIT experiment. Solid lines 0.1 are our theoretical calculations for the trapped system. 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Polarization H» P »L 13 of 31
Condensate fraction in the unbalanced case • The polarized case is di ff erent! A mean-field theory here generally just gives qualitative agreement. • The condensate fraction as a function of the polarization agrees with experimental data only for low polarization values. • We overestimate the phase boundary (determined by the free energy) and, as a consequence, we also overestimate the condensate fraction of a trapped system. 14 of 31
Further work is needed! Clearly a mean-field description can be satisfying in the balanced case, as far as the condensate fraction is concerned, but it is not enough to correctly reproduce experimental results in the unbalanced case. Current work is on: • Including the Gaussian fluctuations in the condensate fraction (with Giovanni Lombardi). • Are the fluctuations enough to reproduce experimental data? • By doing this we can get more insight on the unbalanced Fermi gas: why is it di ff erent? Why do we need to include the fluctuations? 15 of 31
High- T c superconductivity in cuprates Main reference: P. A. Marchetti, F. Ye, Z. B. Su, and L. Yu Phys. Rev. B 84 , 214525 16 of 31
Cuprates: an overview (1/2) • Superconducting cuprates : a class of superconducting materials with very high critical temperatures (up to 135 � K). • Discovered in 1986 by J. G. Bednorz e K. A. M¨ uller; Nobel prize awarded in 1987, the fastest in history. • Very active research field: more than 100,000 research articles in ⇠ 25 years. Figure : Unitary cell for • Up to date, the microscopical La 2 CuO 4 . mechanism behind SC in cuprates is not completely understood. 17 of 31
Cuprates: an overview (2/2) • Di ff erent chemical compositions (YBCO, LSCO, BSSCO) the only common chemical features being the CuO 2 planes. = ) The CuO 2 planes are believed to be the main seat of superconductivity. • Dependence on doping and universality for the phase diagram . • BCS theory can not account for SC in cuprates. 18 of 31
From the CuO 2 planes to the t/J model CuO 2 planes in terms of Zhang-Rice singlets: ZR: Doping-induced hole reside (primarily) on combinations of four oxygen p orbitals centered around a copper site. From ZR singlets to the t/J model: • Strong on-site repulsion ( P G ) • Nearest neighbour hopping ( t ⇡ 0 . 3 eV) • Anti-ferromagnetic Heisenberg term ( J ⇡ 0 . 1 eV) " # X X c † H t/J = P G � t i α c j α + h.c. + J S i · S j P G h i,j i α “Doping a Mott insulator” , P.A. Lee, N. Nagaosa, X.-G. Wen, Rev. Mod. Phys. 78 , 17 19 of 31
Spin-charge separation The creation/annihilation operators for the electron are rewritten in terms of a product of two operators: s i, α ˆ h † c i, α = ˆ ˆ i • ˆ h i is a spinless fermion (holon): the P G constraint is where always satisfied due to Pauli exclusion principle. s i, α is a spin 1 • ˆ 2 boson (spinon). A new local invariance is introduced by this process: ( s i, α e i φ ( x ) ˆ s i, α � ! ˆ U (1) h/s ˆ ! ˆ h i e i φ ( x ) h i � Emergent U (1) gauge field: A µ ⇡ s ⇤ α ∂ µ s α + · · · 20 of 31
Chern-Simons bosonization What is a bosonization scheme? Simplest example: Jordan-Wigner transformation (1D) on lattice: l<j a † c † ! a † j e � i π P l a l j � Continuum version (1D): γ x A µ ( y )d y µ ! Φ ( x ) e i R Ψ ( x ) � • Basic idea for Chern-Simons bosonization in 2D: just like in the 1D case one can bind to the newly-introduced bosonic operator to a “string” which restores the correct statistics back. • In 2D a gauge field and additional term in the action are needed. 21 of 31
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