Thermodynamics, pairing properties of a unitary Fermi gas

Transcription

1 Thermodynamics, pairing properties of a unitary Fermi gas Piotr Magierski (Warsaw University of Technology/ University of Washington, Seattle) Collaborators: Aurel Bulgac (Seattle) Joaquin E. Drut (LANL) Timo Lähde (Helsinki) Gabriel Wlazłowski (Warsaw)

2 Outline Path Integral Monte Carlo (PIMC) on the lattice. Equation of state at unitarity. Thermodynamics. Contact. Superfluid to normal phase transition. Critical temperature. Pairing properties. Spectral weight function. Pseudogap.

3 L limit for the spatial correlations in the system Coordinate space Volume L 3 lattice spacing x - Spin up fermion kcut x ; x - Spin down fermion Momentum space Periodic boundary conditions imposed x k y x External conditions: T - temperature - chemical potential UV momentum cutoff IR momentum cutoff IR IR F, 2m 2m UV UV x 2 L k x 2π/L

4 2 ˆ ˆ ˆ 3 3 H T V d r ˆs ( r) ˆs( r) g d r nˆ ( r) nˆ ( r) s 2m ˆ 3 ˆ ˆ ˆs ˆs ˆ s N d r n ( r) n ( r) ; n ( r) ( r) ( r) 1 g mk cut m 4 a Running coupling constant g defined by lattice in 2 the case of spherical momentum cutoff.

5 Provide the link with the finite temperature DFT. Requires better precision. New observables to calculate. Transport properties: e.g. viscosity Path Integral Monte Carlo on the lattice for cold atoms Hybrid Monte Carlo More efficient MC sampling Agressive parallelization Makes possible to consider very large lattices

6 a = ± Equation of state from PIMC (Path Integral Monte Carlo) Deviation from Normal Fermi Gas L Normal Fermi Gas (with vertical offset, solid line) A. Bulgac, J.E. Drut, P. Magierski, PRL96,090404(2006)

7 Diagram. MC Burovski et al. PRL96, (2006) QMC Bulgac, Drut, Magierski, PRL99, (2006) Diagram. + analytic Haussmann et al. PRA75, (2007) Experiment S. Nascimbene et al. Nature 463, 1057 (2010) Courtesy of C. Salomon exp( )

8 Pressure vs temperature: experiment and PIMC theory for various lattice sizes P (, T ) / P FG (, T ) = exp( μβ

9 PIMC, L=8 PIMC, L=10 PIMC, L=12 PIMC, L=14 Bold Diag. MC Goulko & Wingate Burovski et al

10 L Phase transition Ideal Fermi gas entropy S E µ E n 3 T = F( n) N 5 F( n) N kf k, ( ) 2 F n V 3 2m F S( T) S(0) ST ( ) T T 0 / e 0 E dt T T F 3 '( y) N dy 5 y

11 Theory: local density approximation (LDA) Uniform system Nonuniform system (gradient corrections neglected) 3 F N ( x ) F N N d r F ( r ) ( x( r )) U ( r ) n( r ) 5 T 2 x( r ) ; F ( r) 3 n( r) ( r) 2m F The overall chemical potential and the temperature T are constant throughout the system. The density profile will depend on the shape of the trap as dictated by: ( F N) ( x( r)) U( r) 0 n( r) n( r) Using as an input the Monte Carlo results for the uniform system and experimental data (trapping potential, number of particles), we determine the density profiles. 2 2 / 3

12 Unitary Fermi gas ( 6 Li atoms) in a harmonic trap Experiment: Luo, Clancy, Joseph, Kinast, Thomas, Phys. Rev. Lett. 98, , (2007) THEORY EXP. THEORY Superfluid 3 n() r a ho Entropy as a function of energy (relative to the ground state) for the unitary Fermi gas in the harmonic trap. ho E0 N F Ratio of the mean square cloud size at B=1200G to its value at unitarity (B=840G) as a function of the energy. Experimental data are denoted by point with error bars. Normal Full ab initio theory (no free parameters): LDA + QMC input Bulgac, Drut, Magierski, Phys. Rev. Lett. 99, (2007) 2 a ho m max F (0) - Fermi energy at the center of the trap The radial (along shortest axis) density profiles of the atomic cloud at various temperatures.

13 Contact at finite T: C T 4 ( ) lim k n( k, T) k C( T 0) / ( Nk F ) Diffusion Monte Carlo results: C( T 0) / ( Nk F ) 3.4 Combescot, Giorgini,Stringari Europhys.Lett.75,695(2006) Lobo et al. PRL 97,100405(2006 Drut, Lähde,Ten, arxiv: , PRL in press

14 Results in the vicinity of the unitary limit: -Critical temperature -Pairing gap at T=0 BCS theory predicts: ( T 0) T C 1.7 At unitarity: ( T 0) T C 3.3 This is NOT a BCS superfluid! Bulgac, Drut, Magierski, PRA78, (2008)

15 Cold atomic gases and high Tc superconductors

16 Spectral weight function from PIMC at unitarity Superfluid phase 2 Pseudogap regime (T=0.22) Normal Fermi gas p pf 2 Fermi level T 0.12 F T 0.17 F T 0.21 F F

17 From Gabriel Wlazłowski talk

18 Single-particle properties E( p) 2 p U * 2m 2 2 Effective mass: m * ( ) m Self energy: U ( ) m Weak temperature dependence! P.Magierski, G. Wlazłowski, A. Bulgac, J.E. Drut, PRL103, (2009)

19 Gap in the single particle fermionic spectrum from MC calcs. Magierski, Wlazłowski, Bulgac, arxiv

20 Theory vs Experiment (photoemission spectr.) EDC( k, E) A( k, ) f ( ) PIMC Non selfconsistent t-matrix approx.

21 Pseudogap in cold atoms - summary : Theory: Selfconsistent t-matrix approach - NO Nonselfconsistent t-matrix approach - YES (large) Dynamic Mean Field Theory - YES PIMC (AFMC) - YES (moderate)