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1 The Incredible Many Facets of the Unitary Fermi Gas Aurel Bulgac University of Washington, Seattle, WA Collaborators: Joaquin E. Drut (Seattle, now in Columbus) Michael McNeil Forbes (Seattle, soon at LANL) Piotr Magierski (Warsaw/Seattle) Achim Schwenk (Seattle, now at TRIUMF) Gabriel Wlazlowski (Warsaw) Sukjin Yoon (Seattle) Yongle Yu (Seattle, now in Wuhan)

2 Outline: What is a unitary Fermi gas? Thermodynamic properties Pairing gap and pseudo gap EOS for spin imbalanced systems P wave pairing Small systems in traps and (A)SLDA Unitary Fermi supersolid: the Larkin Ovchinnikov phase Time dependent phenomena What else can we expect?

3 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 n a 3 1 r 0 n 1/3 λ F /2 a n number density r 0 range of interaction a scattering length

4 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. Why? Besides pure theoretical curiosity, this problem is relevant to neutron stars! In 1999 it was not yet clear, either theoretically or experimentally, whether such fermion matter is stable or not! A number of people argued that under such conditions fermionic matter is unstable. - systems of bosons are unstable (Efimov effect) - systems of three or more fermion species are unstable (Efimov effect) Baker (winner of the MBX challenge) concluded that the system is stable. See also Heiselberg (entry to the same competition) Carlson et al (2003) Fixed-Node Green Function Monte Carlo and Astrakharchik et al. (2004) FN-DMC provided the best theoretical estimates for the ground state energy of such systems. Thomas Duke group (2002) demonstrated experimentally that such systems are (meta)stable.

5 Superconductivity and superfluidity in Fermi systems 20 orders of magnitude over a century of (low temperature) physics Dilute atomic Fermi gases T c ev Liquid id 3 He T c 10-7 ev Metals, composite materials T c ev Nuclei, neutron stars T c ev QCD color superconductivity T 7 8 c ev units (1 ev 10 4 K)

6 Thermodynamic properties

7 Finite Temperatures Grand Canonical Path Integral Monte Carlo 2 2 ˆ ˆ ˆ 3 Δ Δ 3 H = T + V = d x ψ ( x) ψ ( ) ψ ( ) ψ ( ) ˆ ( ) ˆ ( ) x + x x g 2 2 d x n x n x m m ˆ 3 N = d x nˆ ( x) + nˆ ( x), ˆ ( ) = ψ ( ) ψ ( ), =, ns x s x s x s Trotter expansion (trotterization of the propagator) ( ) ( ) Z ˆ ˆ ˆ ˆ Nτ 1 ( β) = Tr exp β μ = Tr exp τ μ H N { H N }, β = = T N τ τ ( μ ) 1 ET ( ) = Tr Hˆ exp β Hˆ Nˆ ZT ( ) ( μ ) 1 NT ( ) = Tr Nˆ exp β Hˆ Nˆ ZT ( ) No approximations so far, except for the fact that the interaction is not well defined!

8 Recast the propagator at each time slice and put the system on a 3d spatial lattice, in a cubic box of side L=N s l, with periodic boundary conditions 3 ( Hˆ Nˆ) ( Tˆ Nˆ) Vˆ ( Tˆ Nˆ) O exp τ μ exp τ μ / 2 exp( τ )exp τ μ / 2 + ( τ ) Discrete Hubbard Stratonovich t transformation ti ˆ 1 exp( τv ) = 1 + σ ( ) ˆ ( ) 1 σ ( ) ˆ ± x An x + ± ( ), = exp( τ ) 1 x An x A g x σ ± ( x ) =± 12 σ-fields fluctuate both in space and imaginary time m 1 mkc π = +, k < c 4π a g 2π l Running coupling constant g defined by lattice

9 n(k) 2π/L L box size l lattice spacing k max =π/l k How to choose the lattice spacing and the box size?

10 k y π/l π/l/ 2π/L k x 2 2 π εf, Δ, T 2 2 ml π δε > 2 ml π εf, Δ 2 ml ξ L = Momentum space 2 δ p > π L N l s

11 ZT ( ) = Dσ( x, τ) Tr Uˆ ({ σ}) x, τ Uˆ({ σ}) = T exp{ τ[ hˆ({ σ}) μ]} τ τ One body evolution operator in imaginary time ET ( ) = x, τ Dσ(,)Tr x τ Uˆ ({ σ}) Tr HU ˆˆ σ ({ }) ZT ( ) Tr Uˆ ({ σ}) ˆ σ = + ˆ σ = σ > No sign problem! 2 Tr U({ }) {det[1 U({ })]} exp[ S({ })] 0 Uˆ({ σ }) n (, x y) = n (, x y) = * exp( ik x) ϕ () x ( ), ( ) k 1 + Uˆ ϕ y ϕ x = l k kl < k σ, V ({ }) c All traces can be expressed through these single particle density matrices k l

12 More details of the calculations: Typical lattice sizes used from 8 3 x 300 (high Ts) to 8 3 x 1800 (low Ts) Effective use of FFT(W) makes all imaginary time propagators diagonal (either in real space or momentum space) and there is no need to store large matrices Update field configurations using the Metropolis importance sampling algorithm Change randomly at a fraction of all space and time sites the signs the auxiliary fields σ(x, (x,τ) so as to maintain a running average of the acceptance rate between 0.4 and 0.6 Thermalize for 50, ,000 MC steps or/and use as a start up field configuration a σ(x, (x,τ) field configuration from a different T At low temperatures use Singular Value Decomposition of the evolution operator U({σ}) to stabilize the numerics Use 100,000 2,000,000 σ(x, (x,τ) field configurations for calculations MC correlation time time steps at T T c

13 a = ± Normal Fermi Gas (with vertical offset, solid line) Bogoliubov-Anderson phonons and quasiparticle contribution (dot-dashed dashed lien ) Bogoliubov-Anderson phonons contribution only 4 3 3π T E phonons( T) = ε FN, ξ /2 s 5 16 ξs ε F πΔ T Δ E ( T) = ε N exp T quasi-particles F ε F 7/3 2 π Δ= ε F exp e 2 kfa 4 Quasi-particles contribution only (dashed line) µ - chemical potential (circles) Bulgac, Drut, and Magierski PRL 96, (2006)

14 Measurement of the Entropy and CriticalTemperature of a Strongly InteractingFermi Gas, Luo, Clancy, Joseph, Kinast, and Thomas, PRL 98, (2007) Bulgac, Drut, and Magierski PRL 99, (2007)

15 Long range order and condensate fraction ( ) = d 1 d 2 ψ ( 1+ ) ψ ( 2 + ) ψ ( 2 ) ψ ( 2 ) g r d r d r r r r r r r N N α = lim r g 2 ( r ) n ( r ), n ( r ) = d r r + r r 1 ψ ( 1 ) ψ ( 1) 2 N Bulgac, Drut, and Magierski, arxiv:0803:3238

16 Critical temperature for superfluid to normal transition Amherst ETH: Burovski et al. arxiv:0805:3047 Hard and soft bosons: Pilati et al. PRL 100, (2008) Bulgac, Drut, and Magierski, arxiv:0803:3238

17 Critical temperature for superfluid to normal transition Amherst ETH: Burovski et al. arxiv:0805:3047 Hard and soft bosons: Pilati et al. PRL 100, (2008) Bulgac, Drut, and Magierski, arxiv:0803:3238

18 Ground state energy Bulgac, Drut, Magierski, and Wlazowski, arxiv:0801:1504 Bulgac, Drut, and Magierski, arxiv:0803:3238

19 Pairing Gap and Pseudo gap

20 Response of the two component Fermi gas in the unitary regime β d Tr{exp[- β(h-μn+g ψ(p))] ψ (p)} χ( p) = T dτg( p, τ) dg Tr{exp[- β(h-μn+g ψ (p))]} = g= 0 One body temperature (Matsubara) Green s function ka = 0 F ka = F Bulgac, Drut, and Magierski, arxiv:0801:1504

21 Quasi particle il spectrum and pairing ii gap β 1 exp[ β E( p)] 1 χ( p) = dτg( p, τ) = E ( p ) exp[ β E ( p)] Bulgac, Drut, and Magierski, arxiv:0801:1504

22 Quasi particle spectrum at unitarity Bulgac, Drut, and Magierski, arxiv:0801:1504

23 1 2, 1, 0 ka = F E( p) 2 2 α p 2 = + U μ +Δ 2 Bulgac, Drut, and Magierski, arxiv:0801:1504

24 Dynamical Mean Field Theory applied to a Unitary Fermi gas (exact ti in infinite it number of dimensions) i Superfluid to insulator phase transition in a unitary Fermi gas Barnea, arxiv:0803:2293

25 2 2 k Es + hν = + φ 2m E( N) = E( N 1) + Es Using photoemission spectroscopy to probe a strongly interacting Fermi gas Stewart, Gaebler and Jin, arxiv:0805:0026

26 EOS for spin imbalanced systems Normal phase (unpaired spin up fermions) Fullysymmetric phase LOFF phases Gapless BEC superfluid (BEC of dimers + unpaired spin up fermions) Son and Stephanov, PRA 74, (2006)

27 What we think is happening in spin imbalanced systems? Induced P wave superfluidity Two new superfluid phases where before they were not expected One Bose superfluid coexisting with one P wave Fermi superfluid Two coexisting P wave Fermi superfluids Bulgac, Forbes, Schwenk, PRL 97, (2006)

28 Going beyond the naïve BCS approximation Eliashberg approx. (red) BCS approx. (black) Fullmomentum and frequency dependence of the self consistent equations (blue) Bulgac and Yoon, unpublished (2007)

29 What happens at unitarity? Predicted quantum first phase order transition, subsequently observed in MIT experiment, Shin et al.. arxiv:0709:3027 Red points with error bars subsequent DMC calculations for normal state due to Lobo et al, PRL 97, (2006) 5/3 2 2/3 2 3(6 π ) n b a b = a a b En (, n) ng, n n 5 2m na Bulgac and Forbes, PRA 75, (R) (2007)

30 A refined EOS for spin unbalanced systems Red line: Larkin Ovchinnikov phase Black line: normal part of the energy density Blue points: DMC calculations for normal state, Lobo et al, PRL 97, (2006) Gray crosses: experimental EOS due to Shin, arxiv :0801:1523 5/3 2 2/3 2 3(6 π ) n b a b = a 5 2m na En (, n) ng Bulgac and Forbes, arxiv:0804:3364

31 How this new refined EOS for spin imbalanced bl systems was obtained? Through the use of the (A)SLDA, which is an extension of the Kohn Sham LDA to superfluid systems

32 How to construct and validate an ab initio Energy Density Functional (EDF)? Given a many bodyhamiltonian determine the properties of the infinite homogeneous system as a function of density Extract theedf Add gradient corrections, if needed or known how (?) Determine in an ab initio calculation the properties of a select numberof wisely selected finitesystems Apply the energy density functional to inhomogeneous systems and compare with the ab initio calculation, and if lucky declare Victory!

33 The SLDA energy density functional at unitarity for equal numbers of spin up and spin down fermions Only this combination is cutoff independent τ π 2 2/3 5/3 () 3(3 ) ( ) = r c n r ε() r α Δ () r ν () r + β c nr () = 2 v(), r τ () r = 2 v(), r k c 0< E < E 0< E < E * ν () r = u()v() r r c 0< E< E k k k c k c c k 2 2 2/3 2/3 (3 π ) n ( r ) Δ () r Ur () = β + V () r + small correction 2/3 ext 2 3 γn ( r) Δ () r = g () r ν () r eff c a can take any positive value, but the best results are obtained when a is fixed by the qp spectrum

34 Fermions at unitarity in a harmonic trap Total energies E(N) GFMC Chang and Bertsch, Phys. Rev. A 76, (R) (2007) FN DMC von Stecher, Greene and Blume, PRL 99, (2007) PRA 76, (2007) Bulgac, PRA 76, (R) (2007)

35 Fermions at unitarity in a harmonic trap Pairing gaps EN ( + 1) 2 EN ( ) + EN ( 1) Δ ( N) =, for odd N 2 GFMC Chang and Bertsch, Phys. Rev. A 76, (R) (2007) FN DMC von Stecher, Greene and Blume, PRL 99, (2007) PRA 76, (2007) Bulgac, PRA 76, (R) (2007)

36 Quasiparticle spectrum in homogeneous matter solid/dotted blue line - SLDA, homogeneous GFMC due to Carlson et al red circles - GFMC due to Carlson and Reddy dashed blue line - SLDA, homogeneous MC due to Juillet black dashed-dotted dotted line meanfield at unitarity Two more universal parameter characterizing the unitary Fermi gas and its excitation spectrum: effective mass, meanfield potential Bulgac, PRA 76, (R) (2007)

37 Asymmetric SLDA (ASLDA) ( ) u ( ), ( n r r n r) v ( r ), 2 2 a( ) = n( ) b( ) = n( ) E < 0 E > 0 n 2 2 τ ( r) = u ( r), τ ( r) = v ( r), a n b n E < 0 E > 0 n 1 ν = * ( r) sign( En)u n( r)v n( r), 2 E n n 2 Ε ( r) = [ a( r) a( r) b( r) b( r) ] ( r) ( r) 2m α τ + α τ Δ ν + 33π ( ) + [ + ] n 2 2/3 2 5/3 na r nb r β x r 10m [ ] [ ] ( ) ( ) [ ( )], α ( r) = α xr ( ), α ( r) = α 1/ xr ( ), xr ( ) = n( r)/ n( r), a b b a Ω= P( ) = Ε( ) ( ) ( ) [ μ μ ] 3 3 dr r dr r ana r bnb r Bulgac and Forbes, arxiv:0804:3364

38 Unitary Fermi Supersolid: the Larkin Ovchinnikov phase Bulgac and Forbes, arxiv:0804:3364 3/2 2 2m b P[ μa, μb] μ 2 2 ah μ = 30π μa 5/2

39 i TD ASLDA Bulgac, Roche, Yoon u ( rt, ) ˆ (, ) ˆ (, ) ˆ (, ) ˆ hrt+ V rt Δ rt+δ ( rt, ) u ( rt, ) v(,) rt n ext ext n t v(,) ˆ ˆ ˆ ˆ n rt = Δ ( rt, ) +Δext ( rt, ) hrt (, ) Vext ( rt, ) n Q d rdtq r t n r t i t 3 ( ω) (, σ, ) (, σ, )exp( ω ) = σ N 3 x N t, Nx , Nt number of ψn( r, σ, t) O( Nx 40) 4 5 Space time lattice, use of FFTW for spatial derivative No matrix operations (unlike (Q)RPA) All nuclei (odd, even, spherical, deformed) Any quantum numbers of QRPA modes Fully selfconsistent, no self consistent symmetries imposed

40 Higgs mode of the pairing field in a homogeneous unitary Fermi gas A remarkable example of extreme Large Amplitude Collective Motion Bulgac and Yoon, unpublished (2007)

41 A zoo of Higgs like pairing modes The frequency of all these modes is below the 2 qp gap Maximum and minimum oscillation amplitudes versus frequency

42 In case you ve got lost: What did I talk about so far? What is a unitary Fermi gas? Thermodynamic properties Pairing gap and pseudo gap EOS for spin imbalanced systems P wave pairing Small systems in traps and (A)SLDA Unitary Fermi supersolid: the Larkin Ovchinnikov phase Time dependent phenomena

43 What else can we expect?

Is a system of fermions in the crossover BCS-BEC. BEC regime a new type of superfluid?

Is a system of fermions in the crossover BCS-BEC BEC regime a new type of superfluid? Finite temperature properties of a Fermi gas in the unitary regime Aurel Bulgac,, Joaquin E. Drut, Piotr Magierski