Strongly interacting scale-free matter in cold atoms

Yusuke Nishida

March 12, 2009 @ MIT Faculty Lunch

Fermions at infinite scattering length

2/32

Interacting Fermion systems

Attraction Superconductivity / Superfluidity 3/32  Metallic superconductivity (electrons) Kamerlingh Onnes (1911), T c ~4.2 K  Liquid 3 He Lee, Osheroff, Richardson (1972), T c ~2 mK BCS theory (1957)  High-T c superconductivity (electrons or holes) Bednorz and M üller (1986), T c ~100 K 

Cold atomic gases ( 40 K, 6 Li)

Regal, Greiner, Jin (2003), T c ~ 50 nK • Nuclear matter (neutron stars): T c ~ 1 MeV ?

• Color superconductivity (quarks): T c ~ 100 MeV ??

• Neutrino superfluidity ???

Feshbach resonance

S-wave scattering length :   E m m+ D m m D E= Dm B 4/32 40 K

C.A.Regal and D.S.Jin, Phys.Rev.Lett. (2003)

bound level interatomic potential r

Strong attraction a>0 bound molecule a >0 S-wave scattering length : a 40 K |a|  5/32 zero binding energy : |a|  a <0 Weak attraction a<0 a (Gauss) r y (r) r 0 V 0 r Attraction is arbitrarily tunable by magnetic field

BCS-BEC crossover

Eagles (1969), Leggett (1980) Nozières and Schmitt-Rink (1985)

6/32 superfluid phase + strong attraction BEC of molecules 0 scattering length : a - weak attraction BCS state of atoms k F = (3 p n) 1/3 Fermi momentum B (gauss)

BCS-BEC crossover

Eagles (1969), Leggett (1980) Nozières and Schmitt-Rink (1985)

7/32 + strong attraction BEC of molecules superfluid phase 0 - weak attraction BCS state of atoms Vortex lattices throughout BCS BEC crossover

M. Zwierlein et al.

Nature (2005)

BCS-BEC crossover

Eagles (1969), Leggett (1980) Nozières and Schmitt-Rink (1985)

8/32 a dd =0.6a

Strong interaction

|ak F | >> 1 + strong attraction BEC of molecules 0 - weak attraction BCS state of atoms Bose gas with weak ak F << 1 repulsion Fermi gas with weak attraction |ak F | << 1

Unitary Fermi gas

strong interaction weak BEC + 0 |ak F |  S-wave scattering length : a 9/32 weak BCS - 40 K B (Gauss)

Unitary Fermi gas

weak BEC + strong interaction 0 10/32 weak BCS -  Strong coupling limit : |ak F |  • Maximal s-wave cross section Unitarity limit • No perturbative expansion Challenge for theorists  Scale invariant interaction • a  e expansion !

& zero range r 0  0 Nonrelativistic CFT  Universality • Atomic gas @ Feshbach resonance • Dilute neutron matter : |a NN | ~ 19 fm >> r 0 ~1 fm

New approach from d ≠3

g d=4 • d  4 : Weakly-interacting fermions & bosons with small coupling g 2 ~(4-d) 11/32 weak BEC + g

Strong coupling Unitary regime

0 d=2 weak BCS - • d  2 : Weakly-interacting fermions with small coupling g~(d-2) Systematic expansions for various physical observables in terms of

“

4 d” or “d-2”

e

expansion

12/32

Scale invariant interaction

13/32 Atomic gas @ Feshbach resonance : 0  r 0 << k F -1 << a 

spin-1/2 fermions interacting via a zero-range & infinite scattering length contact interaction

 r y (r)  r 0 V 0 r r y (r) V 0 ~ 1/(m r 0 2 ) r  r y (r) r

2 4

Specialty of d=2 & 4

d

Z.Nussinov and S.Nussinov, cond-mat/0410597

14/32 2-body wave function in general dimensions 3 Wave function y (r) becomes smooth at r  0 for d=2 “ a  ” corresponds to zero interaction ( Any attractive potential in d=2 leads to bound states ) Fermions at unitarity in d  2 are free fermions

4

Specialty of d=2 & 4

d

Z.Nussinov and S.Nussinov, cond-mat/0410597

15/32 2-body wave function in general dimensions 3 2 Normalization diverges at r  0 for d=4 Pair wave function is concentrated near its origin Fermions at unitarity in d  4 are free bosons

Ground state energy

• Ground state energy of unitary Fermi gas at T=0 16/32 Density “N” is the only scale x : fundamental quantity of unitary Fermi gas Simulations Experiments Mean field approx., Engelbrecht et al. (1996): • Carlson et al., Phys.Rev.Lett. (2003): • Astrakharchik et al., Phys.Rev.Lett. (2004): • Carlson and Reddy, Phys.Rev.Lett. (2005): x <0.59

x =0.44(1) x =0.42(1) x =0.42(1) Innsbruck(’04): 0.32(13), Duke(’05): 0.51(4), Rice(’05): 0.46(5), JILA(’06): 0.46(12), ENS(’07): 0.41(15)

Ground state energy in d =2 & 4

4 • Ground state energy of unitary Fermi gas d  Unitary Fermi gas in d  4 is a free Bose gas 17/32 3 Cf. MC simulation in 3d in d=3 !?

J.Carlson and S.Reddy (2005)

 Unitary Fermi gas in d  2 is a free Fermi gas 2

Ground state energy in d =2 & 4

4 • Ground state energy of unitary Fermi gas d  Unitary Fermi gas in d  4 is a free Bose gas 18/32 3 d=4 & d=2 are starting points for systematic expansions of x  Unitary Fermi gas in d  2 is a free Fermi gas 2

Field theoretical approach

Spin-1/2 fermions with contact interaction : 2-body scattering at vacuum ( m =0)  (p 0 ,

p

) 

iT

=  T-matrix in general dimensions

Y.N. and D.T.Son PRL(’06) & PRA(’07)

19/32 1 n “a  ” Scattering amplitude has zeros at d=2,4,… Non-interacting limits

2 3 4

Field theoretical approach

T-matrix in general dimensions d

Y.N. and D.T.Son PRL(’06) & PRA(’07)

20/32 When d=4 e ( e <<1)

iT

= ig ig iD( p 0 ,

p

) Small coupling between fermions & boson g = (8 p 2 e ) 1/2 /m

2 3 4

Field theoretical approach

T-matrix in general dimensions d

Y.N. and D.T.Son PRL(’06) & PRA(’07)

21/32 When d=2+ e ( e <<1) ig

iT

= Small coupling between fermion & fermion g = 2 p e /m

2 3 4

Systematic expansions

d fermions & bosons with small coupling g 2 ~(4-d) << 1 P ( m ) = + + O(1) fermions with small coupling g~(d-2) << 1 O( e ) + O( e 2 ) g g 22/32 e =4-d & e =d-2

3 2 4

Systematic expansions

d fermions & bosons with small coupling g 2 ~(4-d) << 1 g 23/32 NLO correction is small ~5 %

Carlson & Reddy (2005)

Cf. MC simulation in 3d fermions with small coupling g~(d-2) << 1 P ( m ) = + O(1) O( e ) g + O( e 2 ) e =4-d & e =d-2

2 3 4

Systematic expansions

d fermions & bosons with small coupling g 2 ~(4-d) << 1 g 24/32 Cf. MC simulation in 3d fermions with small coupling g~(d-2) << 1 NLO correction is small ~5 %

Carlson & Reddy (2005)

g e =4-d & e =d-2

Matching of two expansions in

x • Padé approximants (+ Borel transformation) x = E unitary /E free ♦ =0.42

2d 4d free Fermi gas free Bose gas Interpolations to 3d d 25/32

Critical temperature

Y.N., Phys. Rev. A (2007)

• Critical temperature from d=4 and 2 26/32 T c / e F • Interpolated results to d=3 4d 2d free Fermi gas free Bose gas d Monte Carlo simulations • Bulgac et al. (’05): T c / e F = 0.23(2) • Lee and Schäfer (’05): T c / e F < 0.14

• Burovski et al. (’06): T c / e F • Akkineni et al. (’06): T c / e F = 0.152(7)  0.25

Few body aspects

27/32

Correspondence

S.Tan, cond-mat/0412764 F.Werner & Y.Castin, PRA (2006)

28/32 • Schrödinger equation

in free space with E=0

Scaling solution • Schrödinger equation

in a harmonic potential

g = anomalous dimension of operator in nonrelativistic CFT

Y.N. & D.T.Son, PRD (2007)

3 fermions in a harmonic potential

Angular momentum

l

=0 29/32 Angular momentum

l

=1 2d 4d 4d 2d

3 fermions in a harmonic potential

Angular momentum

l

=0 30/32 Angular momentum

l

=1 2d 4d 4d 2d

Summary

31/32 Fermi gas at infinite scattering length = New strongly interacting matter in cold atoms • Unitary Fermi gas around d=4 becomes weakly-interacting system of fermions & bosons • Weakly-interacting system of fermions around d=2 • Thermodynamics & Quasiparticle spectrum (Y.N. & D.T.Son 2006) • Atom-dimer & dimer-dimer scatterings (G.Rupak 2006) • Phase structure of polarized Fermi gas with (un)equal masses (Y.N. 2007, G.Rupak & T.Schafer & A.Kryjevski 2007) • BCS-BEC crossover (J.W.Chen & E.Nakano 2007) • Momentum distribution & condensate fraction (Y.N. 2007) • Energy of a few atoms in a harmonic potential (Y.N. & D.T.Son 2007) • Low-energy dynamics (A.Kryjevski 2008) • Energy-density functional (G.Rupak & T.Schafer 2009) • …

Summary

32/32 Fermi gas at infinite scattering length = New strongly interacting matter in cold atoms • Unitary Fermi gas around d=4 becomes weakly-interacting system of fermions & bosons • Weakly-interacting system of fermions around d=2 • Thermodynamics & Quasiparticle spectrum (Y.N. & D.T.Son 2006) Very simple and useful starting points to understand the unitary Fermi gas in d=3 !

(Y.N. 2007, G.Rupak & T.Schafer & A.Kryjevski 2007) • BCS-BEC crossover (J.W.Chen & E.Nakano 2007) • Momentum distribution & condensate fraction (Y.N. 2007) • Energy of a few atoms in a harmonic potential (Y.N. & D.T.Son 2007) • Low-energy dynamics (A.Kryjevski 2008) • Energy-density functional (G.Rupak & T.Schafer 2009) • …

NNLO correction for

x • NNLO correction for x 34/32

Nishida, Ph.D. thesis (2007)

Fit two expansions using Pad é approximants Interpolations to 3d • NNLO 4d + NNLO 2d

Arnold, Drut, Son, Phys.Rev.A (2006)

x ♦ =0.40

cf. NLO 4d + NLO 2d d

Polarized Fermi gas around d=4

35/32 • Rich phase structure near unitarity point in the plane of and : binding energy Polarized normal state Gapless superfluid 1-plane wave FFLO : O( e 6 ) Gapped superfluid BCS BEC unitarity Stable gapless phases (with/without spatially varying condensate) exist on the BEC side of unitarity point

e

expansion in critical phenomena

Critical exponents of O(n=1)  4 theory ( e =4-d  1) 36/32 g  O(1) 1 0 +e 0 1 +e 2 1.167

1.244

+e 3 1.195

+e 4 1.338

+e 5 Lattice 0.892

1.239(3) 0.0185 0.0372 0.0289

0.0545

0.027(5) Exper.

1.240(7) 1.22(3) 1.24(2) 0.016(7) 0.04(2) e expansion is asymptotic series but works well ! • Borel summation with conformal mapping g =1.2355

 0.0050 &  =0.0360

 0.0050

• Boundary condition (exact value at d=2) g =1.2380

 0.0050 &  =0.0365

 0.0050

How about our case???

2 fermions in a harmonic potential

37/32

T.Busch et.al., Found. Phys. (1998) T.Stoferle et al., Phys.Rev.Lett. (2006)

2 fermions in a harmonic potential

38/32 |a| 

0

Quasiparticle spectrum

• Fermion dispersion relation : w ( p ) LO self-energy diagrams i S ( p ) = or O( e ) Expansion over 4-d O( e ) Energy gap : Location of min. : Expansion over d-2 39/32

Extrapolation to d=3 from d=4-

e • Keep LO & NLO results and extrapolate to e =1 40/32 NLO corrections are small 5 ~ 35 % Good agreement with recent Monte Carlo data

J.Carlson and S.Reddy, Phys.Rev.Lett. 95, (2005)

cf. extrapolations from d=2+ e NLO are 100 %