Equilibrium and nonequilibrium properties of unitary Fermi gas from Quantum Monte Carlo

Transcription

1 Equilibrium and nonequilibrium properties of unitary ermi gas from Quantum Monte Carlo Piotr Magierski Warsaw University of Technology Collaborators: A. Bulgac - University of Washington J.E. Drut - University of North Carolina K.J. Roche Pacific Northwest National Lab. G. Wlazłowski - Univ. of Washington/Warsaw Univ. of Techn.

2 Outline BCS-BEC crossover. Unitary regime. Theoretical approach: Path Integral Monte Carlo (QMC) Equation of state for the ermi gas in the unitary regime. Pairing gap and pseudogap. Spin susceptibility, conductivity and diffusion. Viscosity.

3 What is a unitary gas? A gas of interacting fermions is in the unitary regime if the average separation between particles is large compared to their size (range of interaction), but small compared to their scattering length. n r 0 3 << 1 n a 3 >> 1 n - particle density a - scattering length r 0 - effective range NONPERTURBATIVE REGIME Universality: E x x E ; x= (0) 0.37(1) EG G System is dilute but strongly interacting! T - Exp. estimate - Energy of noninteracting ermi gas

4 Cold atomic gases and high Tc superconductors 0.5 T c 0.16 rom ischer et al., Rev. Mod. Phys. 79, 353 (2007) & P. Magierski, G. Wlazłowski, A. Bulgac, Phys. Rev. Lett. 107, (2011)

5 L limit for the spatial correlations in the system 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) Path Integral Monte Carlo for fermions on 3D lattice Coordinate space Hamiltonian Volume L lattice spacing x 3 - Spin up fermion: kcut x ; x - Spin down fermion: External conditions: T - temperature - chemical potential Periodic boundary conditions imposed

6 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 Basics of Auxiliary ield Monte Carlo (Path Integral MC) mk cut m 4 a 2 Running coupling constant g defined by lattice 1 kt 0 1 g 2 m 2 x - UNITARY LIMIT Uˆ ({ }) T exp{ d[ hˆ ({ }) ]}; hˆ ({ }) one-body operator U({ }) Uˆ ({ }) ; - single-particle wave function kl k l l S[ ] ˆ D[ ( r, )] e E( T) H E[ U({ })] ZT ( ) EU [ ({ })]- energy associated with a given sigma field 2 Tr Uˆ({ }) {det[1 Uˆ ( )]} exp[ S({ })] 0 - No sign problem for spin symmetric system!

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 Equation of state of the unitary ermi gas - current status Experiment: M.J.H. Ku, A.T. Sommer, L.W. Cheuk, M.W. Zwierlein, Science 335, 563 (2012) QMC (PIMC + Hybrid Monte Carlo): J.E.Drut, T.Lähde, G.Wlazłowski, P.Magierski, Phys. Rev. A 85, (2012)

9 Results in the vicinity of the unitary limit: -Critical temperature -Pairing gap 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)

10 Uniform system Nonuniform system (gradient corrections neglected) Local density approximation (LDA) from QMC 3 N ( x ) N N d r ( r ) ( x( r )) U ( r) n( r) 5 T 2 x( r ) ; ( r) 3 n( r) ( r) 2m 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: ( 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

11 Unitary ermi 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 ermi gas in the harmonic trap. ho E0 N 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. B 1200G 1/ k a 0.75 Normal ull ab initio theory (no free parameters): LDA + QMC input Bulgac, Drut, Magierski, Phys. Rev. Lett. 99, (2007) 2 a ho m max (0) - ermi energy at the center of the trap The radial (along shortest axis) density profiles of the atomic cloud at various temperatures.

12 Constraints : Pairing gap from spectral function::

13 Linear inverse problem G is known from QMC with some error for a number of values of y, usually uniformly distributed within the interval: (0, 1/T) Maximum entropy method (MEM): Bayes theorem: Maximization of conditional probability: Magierski, Wlazłowski, Comp. Phys. Comm. 183 (2012) 2264 Relative entropy term

14 Spectral weight function at unitarity: 1 ( ka) 0 T 0.12 T 0.17 T 0.15 T C T 0.21

15 Spectral weight function at the BEC side: 1 ( ka) 0.2 T 0.13 T 0.19 T C T 0.26

16 Single-particle properties E( p) 2 p U * 2m 2 2 Quasiparticle spectrum extracted from spectral weight function at T 0.1 ixed node MC calcs. at T=0 Effective mass: m * ( ) m Mean-field potential: U ( ) Weak temperature dependence!

17 rom Sa de Melo, Physics Today (2008) Pairing pseudogap: suppression of low-energy spectral weight function due to incoherent pairing in the normal state (T >T c ) Important issue related to pairing pseudogap: - Are there sharp gapless quasiparticles in a normal ermi liquid YES: Landau s ermi liquid theory; NO: breakdown of ermi liquid paradigm

18 Gap in the single particle fermionic spectrum - theory Magierski, Wlazłowski, Bulgac, Phys. Rev. Lett.107,145304(2011) Magierski, Wlazłowski, Bulgac, Drut, Phys. Rev. Lett.103,210403(2009)

19 Energy distribution curves (EDC) from the spectral weight function Unitarity BEC side Unitarity Experiment (blue dots): Gaebler et al. Nature Physics 6, 569(2010) QMC (red line): Magierski, Wlazłowski, Bulgac, Phys. Rev. Lett. 107, (2011)

20 Spin susceptibility and spin drag rate Wlazłowski, Magierski, Bulgac, Drut, Roche, Phys. Rev. Lett. 110, ,(2013) n s ( ) ( q 0, ) / s - spin drag rate 0 s - spin conductivity ˆ z ˆ z ˆ z ˆ z 1 Gs ( q, ) j ( ) j ( ) j (0) j (0) q q q q V cosh ( / 2) Gs( q, ) s( q, ) d sinh / 2

21 Spin diffusion s D s s No minimum is seen in QMC down to 0.1 of ermi energy Estimate from kinetic theory at low T: D p l n n s 1/3 1/3 1 Wlazłowski, Magierski, Bulgac, Drut, Roche, Phys. Rev. Lett. 110, ,(2013)

22 Pseudogap at unitarity theoretical predictions Path Integral Monte Carlo - YES Dynamic Mean ield - YES Selfconsistent T-matrix - NO Nonselfconsistent T-matrix - YES

23 Hydrodynamics at unitarity No intrinsic length scale Uniform expansion keeps the unitary gas in equilibrium Consequence: uniform expansion does not produce entropy = bulk viscosity is zero! Shear viscosity: or any physical fluid: S 4 kb KSS conjecture Kovtun, Son, Starinets, Phys.Rev.Lett. 94, , (2005) from AdS/CT correspondence y x v x A y Maxwell classical estimate: ~ mean free path Perfect fluid - strongly interacting quantum system = S k 4 B No well defined quasiparticles Candidates: unitary ermi gas, quark-gluon plasma

24 Shear viscosity ( ) ( q 0, ) / xyxy G q d r ˆ r ˆ e 3 xyxy (, ) xy (, ) xy (0,0) cosh ( / 2) Gxyxy ( q, ) xyxy ( q, ) d sinh / 2 i ˆ j ( ), ˆ ˆ k r H lkl ( r) 0 iqr Additional symmetries and sum rules: ε energy density

25 Shear viscosity to entropy density ratio T 8 T 3/2 G.Wlazłowski, P.Magierski,J.E.Drut, Phys. Rev. Lett. 109, (2012)

26 Uncertainties related to numerical analytic continuation G.Wlazłowski, P.Magierski,J.E.Drut, Phys. Rev. Lett. 109, (2012)

27 Preliminary results: Wlazłowski et al. C. Chafin, T. Schafer, PRA87,023629(2013) P.Romatschke, R.E. Young, arxiv:

28 Shear viscosity to entropy ratio experiment vs. theory (from A. Adams et al ) Lattice QCD ( SU(3) gluodynamics ): H.B. Meyer, Phys. Rev. D 76, (2007) QMC calculations for UG: G. Wlazłowski, P. Magierski, J.E. Drut, Phys. Rev. Lett. 109, (2012)