Why strongly interacting fermion gases are interesting to a many-body theorist? Aurel Bulgac University of Washington, Seattle

Transcription

1 Why strongly interacting fermion gases are interesting to a many-body theorist? Aurel Bulgac University of Washington, Seattle

2 People I have been lucky to work with on these problems: Clockwise (starting top left corner) : G.. Bertsch (Seattle), J.E. J Drut (Seattle), P. Magierski (Warsaw, Seattle), Y. Yu (Seattle, Lund, Wuhan), M.M. orbes (Seattle), A. Schwenk (Seattle, Vancouver), A. onseca (Lisbon), P. Bedaque (Seattle, Berkeley, College Park),

3 What will be covered in this talk: Lots of others people results (experiment and theory) throughout the entire presentation What is the unitary regime? The two-body problem, how one can manipulate the two-body interaction? What many/some theorists know and suspect that is going on? What experimentalists have managed to put in evidence so far and how that agrees with theory?

4 rom a talk of J.E. Thomas (Duke)

5 rom a talk of J.E. Thomas (Duke)

6 rom a talk of J.E. Thomas (Duke)

7 rom a talk of J.E. Thomas (Duke)

8 eshbach resonance B S S V S S a V V P r V P r V V V p H n z n e z e Z n e hf hf d Z i i hf i r ) (, ) ( ) ( ) ( γ γ µ = = = = Tiesinga, Verhaar, Stoof Phys. Rev. A47 Regal and Jin Phys. Rev. Lett. 90, (003), 4114 (1993) Atomic seperation Energy E res E th Channel coupling

9 Bartenstein et al. Phys. Rev. Lett. 94,, (005)

10 Halo dimer (open channel) Pr ( > r0 ) a >> Pr ( < r) r Most of the time two atoms are at distances greatly exceeding the range of the interaction! Köhler et al. Phys. Rev. Lett. 91, (003), inspired by Braaten et al. cond-mat/

11 Z measured probability to find the two atoms in a singlet state (closed channel) Dots - experiment of Partridge et al. cond-mat/

12 What is the unitary regime? 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. The system is very dilute, but strongly interacting! n r r 0 n a 3 1 n-1/3 1/3 λ / a n - number density r 0 - range of interaction a - scattering length

13 Bertsch Many-Body X challenge, Seattle, 1999 What are the ground state properties of the many-body system composed of spin ½ fermions interacting via a zero-range, range, infinite scattering-length contact interaction. In 1999 it was not yet clear, either theoretically or experimentally, whether such fermion matter is stable or not. - systems of bosons are unstable (Efimov effect) - systems of three or more fermion species are unstable (Efimov effect) - a number of theorists believed that the two species fermions systems s are unstable as well Baker (winner of the MBX challenge) concluded that the system is stable. See also Heiselberg (entry to the same competition) Chang et al (003) ixed-node Green unction Monte Carlo and Astrakharchik et al. (004) N-DMC provided best the theoretical estimates for the ground state energy of such systems. Thomas Duke group (00) demonstrated experimentally that such systems are (meta)stable.

14 BCS BEC crossover Leggett (1980), Nozieres and Schmitt-Rink (1985), Randeria et al. (1993), If a<0 at T=0 a ermi system is a BCS superfluid e 7 /3 k m π exp ka << ε, iff k a << 1 and ξ = 1 k ε >> 1 k If a = and nr 03 1 a ermi system is strongly coupled and its properties are universal. Carlson et al. PRL 91, (003) E N normal 3 Esuperfluid ε, 0.44 ε and ξ = O = 5 N 5 ( λ ), O( ε ) If a>0 (a r 0 ) and na 3 1 the system is a dilute BEC of tightly bound dimers n ε = << = = > ma 3 f and na 1, where and b nb abb 0.6a 0

15 rom a talk of J.E. Thomas (Duke)

16 Ground state properties of unitary gases

17 Consider Bertsch s MBX challenge (1999): ind the ground state of infinite homogeneous neutron matter interacting with an infinite scattering length. r 0 << λ << 0 a Carlson, Morales, Pandharipande and Ravenhall, PRC 68, 0580 (003), with Green unction Monte Carlo (GMC) EN N 3 ε 5 = α N, α N = 0.54 normal state Carlson, Chang, Pandharipande and Schmidt, PRL 91, (003), with GMC E S N 3 ε 5 = α S, α S = 0.44 superfluid state This state is half the way from BCS BEC crossover, the pairing correlations are in the strong coupling limit and HB invalid again.

18 Theory (QMC) E 3 = εξ, ξ=0.4() N 5 = εη, η=0.504(4) Experiment ξ = 0.74(4), Duke (00) 0.51(4), Duke (005) 0.3(1), Innsbruck (004) 0.36( 15 ), Paris (004) 0.46(5), Rice (005) 0.45(5), JILA (006) 0.41(15), Paris (007) Note, no reliable experimental determination of the pairing gap yet!

19 BEC side BCS side Solid line with open circles Chang et al. physics/ Dashed line with squares - Astrakharchik et al. cond-mat/

20 E G = 3 5 k m Green unction Monte Carlo with ixed Nodes S.-Y. Chang, J. Carlson, V. Pandharipande and K. Schmidt physics/

21 = = a k m k e a k m k e BCS Gorkov exp 8 exp 7/3 π π ixed node GMC results, S.-Y. Chang et al. (003)

22 Determining the critical temperature for the superfluid to normal phase transition in theory and experiment

23 Grand Canonical Path-Integral Monte Carlo calculations on 4D-lattice 3 3 H = T + V = d x ψ ( x) ψ ( x) ψ ( x) ψ ( x) g d x n ( x) n ( x) + m m 3 N = d x n ( x) + n ( x) Trotter expansion (trotterization( of the propagator) Nτ 1 Z( β) = Tr exp β ( H µ N) = Tr { exp τ ( H µ N) }, β = = Nττ T Recast the propagator at each time slice and use T ( H N) ( T N) ( ) 3 exp τ µ exp τ µ / exp( τv)exp τ T µ N / + O( τ ) Discrete Hubbard-Stratonovich transformation exp( τv) = { 1 + σ ± ( x) A n ( x) + n ( x) }, A= exp( τg) 1 x σ ( x ) =± 1 m 1 mk = + 4π a g π cut off ± σ-fields fluctuate both in space and imaginary time Running coupling constant g defined by lattice A. Bulgac,, J.E. Drut and P.Magierski

24 Superfluid to Normal ermi Liquid Transition Bogoliubov-Anderson phonons and quasiparticle contribution (red line ) Bogoliubov-Anderson phonons contribution only (magenta line) People never consider this??? Quasi-particles contribution only (green line) 4 3 3π T Ephonons( T) = εn, ξ / s 5 16ξs ε π T E ( T) = ε N exp T quasi-particles 4 5 ε 7/3 π = ε exp e ka 4 Lattice size: from 6 3 x 11 at low T to 6 3 x 30 at high T Number of samples: Several 10 5 s s for T Also calculations for 4 3 lattices Limited results for 8 3 lattices

25 Experimentally one needs a thermometer! Solution: Measure the entropy S as a function of the of the energy E and use T = de/ds Chen et al, PRL 95, (005) Exp: Luo et al, PRL (007)

26 How to use Quantum Monte Carlo results to describe what is going on in an atomic trap at finite temperature and confront theory and experiment? Use Local Density Approximation (LDA) in conjunction with the Universality of ermi Gases /3 ( 3π ) n ( r) T T 5 m n ( r) n ( r) 5/3 3 3 ( T) d r = ξ Tσ n r V /3 + /3 trap r λ n r ( ) ( ( ) ) ( ) energy density entropy density

27 Experiment red symbols Luo et al, PRL (007) Blue curves pure theory

28 Collective modes

29 Sound in infinite fermionic matter ω = v s k Local shape of ermi surface Sound velocity Collisional Regime - high T! Compressional mode Spherical v s v 3 irst sound Superfluid collisionless- low T! Compressional mode Normal ermi fluid collisionless - low T! (In)compressional mode Spherical Elongated along propagation direction v v s s s > 1 v 3 = sv Bogoliubov- Anderson sound Landau s zero sound Need repulsion!!!

30 Grimm, cond-mat/

31 3 k ς 5ι 1 ε( n) = ξ + O 5 m ka ka ka ξ 0.44, ς 1, ι 1 U = δω ω ( + + λ ) mω x y z 0 ς = ξ k 1 K (0) a ( ) ( ) 3 Adiabatic regime Spherical ermi surface Bogoliubov-Anderson modes in a trap Perturbation theory result using GMC equation of state in a trap Only compressional modes are sensitive to the equation of state and experience a shift!

32 Innsbruck s s results - blue symbols Duke s s results - red symbols irst order perturbation theory prediction (blue solid line) Unperturbed frequency in unitary limit (blue dashed line) Identical to the case of non-interacting fermions If the matter at the eshbach resonance would have a bosonic character then the collective modes will have significantly higher frequencies!

33 High precision results Radial breathing mode: theory vs experiment meanfield EoS light curve, MC EoS thick curve Grimm, cond-mat/

34 If we set our goal to prove that these systems become superfluid, there is no other way but to show it! Is there a way to put directly in evidence the superflow? Vortices!

35 The depletion along the vortex core is reminiscent of the corresponding density depletion in the case of a vortex in a Bose superfluid, when the density vanishes exactly along the axis for 100% BEC. rom Ketterle s group ermions with 1/k a = 0.3, 0.1, 0, -0.1, -0.5 Bosons with na 3 = 10-3 and 10-5 Extremely fast quantum vortical motion! Number density and pairing field profiles Local vortical speed as fraction of ermi speed

36 Zweirlein et al. cond-mat/

37 Zweirlein et al. cond-mat/

38 Phases of a two species dilute ermi system BCS-BEC BEC crossover High T, normal atomic (plus a few molecules) phase T Strong interaction weak interactions weak interaction BCS Superfluid a<0 no -body bound state Molecular BEC and Atomic+Molecular Superfluids a>0 shallow -body bound state halo dimers 1/a

39 Until now we kept the numbers of spin-up and spin-down equal. What happens when there not enough partners for everyone to pair with? What theory tells us?

40 Green spin up Yellow spin down If µ µ < the same solution as for µ = µ LO solution (1964) Pairing gap becomes a spatially varying function Translational invariance broken Muether and Sedrakian (00) Translational invariant solution Rotational invariance broken

41 LO and Deformed ermi Surfaces pairing can occur only for relatively small imbalances. What happens if the imbalances become large?

42 Son and Stephanov,, cond-mat/ Pao,, Wu, and Yip, PR B 73, (006) Parish, Marchetti, Lamacraft,, Simons cond-mat/ Sheeny and Radzihovsky,, PRL 96,, (006)

43 What really is happening! Induced p-wave superfluidity in asymmetric ermi gases Two new superfluid phases where before they were not expected Bulgac,, orbes, Schwenk One Bose superfluid coexisting with one P-wave P ermi superfluid Two coexisting P-wave P ermi superfluids

44 BEC regime all minority (spin-down) fermions form dimers and the dimers organize themselves in a Bose superfluid the leftover/un-paired majority (spin-up) fermions will form a ermi sea the leftover spin-up fermions and the dimers coexist and, similarly to the electrons in a solid, the leftover spin-up fermions will experience an attraction due to exchange of Bogoliubov phonons of the Bose superfluid

45 p-wave gap Bulgac,, Bedaque, onseca, cond-mat/03060 n n b f ε exp 0.44, if ka 1 n f nb p 6π n nf b k p ε exp, if ka, x α n fb f nb mc b = ( ka ) ln ( x ) p max!!! 5.6 nf ε exp, if 0.44ka 1 ka n!!! b

46 BCS regime: The same mechanism works for the minority/spin-down component

47 = 1 π p ε exp ε exp NU p k 4kkaL p k 5z z z z + Lp() z = ln1 z ln 4 15z 30z 1 + z 15z π 0.11( ka ) p ε exp, for k 0.77 k and fixed k max p ε 3 exp π k k k ( ka ) ln k for k k p ε 18π k exp k ( ka )

48 T=0 thermodynamics in asymmetric ermi gases at unitarity in a trap Use Local Density Approximation (LDA) ( r) = V ( r) λ µ { ab, } { ab, } trap

49 Now let us concentrate on a spin unbalanced ermi gas at unitarity ty

50 At unitarity almost everything is a function of the densities alone at T=0! x We use both micro-canonical canonical and grand canonical ensembles nb µ b = 1, y = 1 n µ a a 3 5/3 E( na, nb) = α [ nag( x) ] 5 P h y n n E n n E n n 5 3 5/ ( µ, µ ) = β [ µ ( )] = µ + µ (, ) = (, ) a b a a a b b a b a b g'( x) 1 y =, h( y) = gx ( ) xg'( x) gx ( ) xg'( x) h'( y) 1 x =, g( x) = hy () yh'() y hy () yh'() y The functions g(x) ) and h(y) ) determine fully the thermodynamic properties and only a few details are relevant

51 Both g(x) ) and h(y) ) are convex functions of their argument. Non-trivial regions exist! Bounds given by GMC 1 if y y0 hy () = 1 + y if y y 3/5 1,1 ( ξ) h"( y) c 1 [ ] 3/5 1, c ( ξ) 1 y Y < y < Y y = g(0) = 1, g( x) = ( ξ) 3/5 g"( x) 0 g(1) g(1) g'(0) Y0 and g'(1), 1 1+ Y1 Bounds from the energy required to add a single spin-down particle to a fully polarized ermi sea of spin-up particles

52 Now put the system in a trap µ ( r) = λ V( r), y( r) = ab, ab, µ = λ λ a b µ b() r µ () r a n() r = β µ ()(()) rhyr hyr (()) yrh () '(()) yr a 3/ [ ] [ ] a = β [ µ a 3/ ] n () r ()(()) r h y r h'(()) y r b

53 blue - green - red - P = 0 region 0 < P < 1 region P= 1 region

54 Column densities (experiment) Normal Superfluid Zweirlein et al. cond-mat/060558

55 y 1 R R γ = = y R R 1 0 vac vac 0.70(5) Experimental data from Zwierlein et al. cond-mat/060558

56 Rf -spectroscopy Grimm, cond-mat/ Schunck et al, cond-mat/070066

57 The unitary ermi gases (at resonance and off resonance) have an extremely rich structure and a very rich phase diagram The tunability of the interaction is a non-paralleled feature of these systems Theoretically these systems can be described essentially exactly and they present an extraordinary opportunity to test lots of many-body techniques and develop new ones It can have a major impact on other fields One can simulate both theoretically and experimentally lots of systems encountered in condensed matter physics, nuclear physics and astrophysics