Strongly interacting scale-free matter in cold atoms Yusuke Nishida March 12, 2009 @ MIT Faculty...

Strongly interactingscale-free matter in cold atoms

Yusuke Nishida

March 12, 2009 @ MIT Faculty Lunch

2/32

Fermions at infinite scattering length

3/32Interacting Fermion systems

Attraction Superconductivity / Superfluidity

Metallic superconductivity (electrons)

Kamerlingh Onnes (1911), Tc ~4.2 K Liquid 3He

Lee, Osheroff, Richardson (1972), Tc ~2 mK High-Tc superconductivity (electrons or holes)

Bednorz and Müller (1986), Tc ~100 K Cold atomic gases (40K, 6Li)

Regal, Greiner, Jin (2003), Tc ~ 50 nK

• Nuclear matter (neutron stars): Tc ~ 1 MeV ?

• Color superconductivity (quarks): Tc ~ 100 MeV ??

• Neutrino superfluidity ???

BCS theory

(1957)

4/32

r

Feshbach resonance

S-wave scattering length :

E

interatomic potential

bound level

E=B

40K

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

5/32

40K

S-wave scattering length : a

(Gauss)

a

Weak attractiona<0

Strongattraction

a>0

bound molecule

zero bindingenergy : |a|

Attraction is arbitrarily tunable by magnetic field

r (r)

r

V0

r0a<0|a|a>0

6/32

strong attraction

BCS-BEC crossover

0weak attraction

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

superfluid phase

B (gauss)

scattering length : a

BEC of molecules

BCS state of atoms

kF = (3 n)1/3

Fermi momentum

7/32

strong attraction

BEC of molecules

BCS-BEC crossover

0weak attraction

BCS state of atoms


superfluid phase

M. Zwierlein et al.Nature (2005)

Vortex latticesthroughout BCS-BEC crossover8

80

m

8/32BCS-BEC crossover

add=0.6 a

Bose gas with weak repulsionakF << 1

Fermi gas with weak attraction|akF| << 1


strong attraction

BEC of molecules

weak attraction

BCS state of atoms

0

Strong interaction|akF| >> 1

9/32Unitary Fermi gas

0

weak BCSweak BEC strong interaction

40K

S-wave scattering length : a

B (Gauss)

|akF|

10/32

Strong coupling limit : |akF|• Maximal s-wave cross section Unitarity limit

• No perturbative expansion Challenge for theorists

Scale invariant interaction• a & zero range r00 Nonrelativistic CFT

Universality• Atomic gas @ Feshbach resonance

• Dilute neutron matter : |aNN| ~ 19 fm >> r0 ~ 1 fm

Unitary Fermi gas

0

weak BCSweak BEC strong interaction

expansion !

11/32

d=4

d=2

New approach from d≠3

Strong coupling

Unitary regime

g

• d4 : Weakly-interacting fermions & bosons with small coupling g2~(4-d)

• d2 : Weakly-interacting

fermions with small coupling g~(d-2)

Systematic expansions for various physical observables in terms of “4-d” or “d-2”

weak BEC weak BCS

0

g

12/32

expansion

13/32Scale invariant interaction

r (r)

V0

r0 r r

r (r)

V0 ~ 1/(m r02)

Atomic gas @ Feshbach resonance : 0 r0 << kF-1 << a

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

r (r)

r

14/32Specialty of d=2 & 4 Z.Nussinov and S.Nussinov, cond-mat/0410597

4

2

3

d 2-body wave function in general dimensions

“a” corresponds to zero interaction

Fermions at unitarity in d2 are free fermions

Wave function (r) becomes smooth at r0 for d=2

( Any attractive potential in d=2 leads to bound states )

15/32Specialty of d=2 & 4 Z.Nussinov and S.Nussinov, cond-mat/0410597

4

2

3

d 2-body wave function in general dimensions

Pair wave function is concentrated near its origin

Fermions at unitarity in d4 are free bosons

Normalization

diverges at r0 for d=4

16/32Ground state energy

• Ground state energy of unitary Fermi gas at T=0

Density “N” is the only scale

: fundamental quantity of unitary Fermi gas

Mean field approx., Engelbrecht et al. (1996): <0.59

Simulations

Experiments 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)

• Carlson et al., Phys.Rev.Lett. (2003): =0.44(1)• Astrakharchik et al., Phys.Rev.Lett. (2004): =0.42(1)• Carlson and Reddy, Phys.Rev.Lett. (2005): =0.42(1)

17/32Ground state energy in d = 2 & 4

• Ground state energy of unitary Fermi gas

4

2

3

d

Unitary Fermi gas in d4 is a free Bose gas

Unitary Fermi gas in d2 is a free Fermi gas

in d=3 !?J.Carlson and S.Reddy (2005)

Cf. MC simulation in 3d

18/32Ground state energy in d = 2 & 4

• Ground state energy of unitary Fermi gas

4

2

3

d

Unitary Fermi gas in d4 is a free Bose gas

Unitary Fermi gas in d2 is a free Fermi gas

d=4 & d=2 are starting pointsfor systematic expansions of

19/32

T-matrix in general dimensions

Field theoretical approach

iT =

(p0,p) 1 n

“a”

Scattering amplitude has zeros at d=2,4,…Non-interacting limits

Spin-1/2 fermionswith contact interaction :

2-body scattering at vacuum (=0)

Y.N. and D.T.Son PRL(’06) & PRA(’07)

20/32

When d=4- (<<1)

Field theoretical approach

4

2

3

d

iT = ig igiD(p0,p)

Small coupling between fermions & bosong = (82

)1/2/m



21/32Field theoretical approach

4

2

3

d

iT =ig

Small coupling between fermion & fermiong = 2 /m

When d=2+ (<<1)



22/32

g fermions with small coupling g~(d-2) << 1

Systematic expansions

4

2

3

d

O(1) O()

+ +P () = + O(2)

fermions & bosons with small coupling g2~(4-d) << 1 g

=4-d & =d-2

23/32Systematic expansions

4

2

3

d

Carlson & Reddy (2005)Cf. MC simulation in 3d

NLO correctionis small ~5 %

g

fermions & bosons with small coupling g2~(4-d) << 1

fermions with small coupling g~(d-2) << 1

g

O(1) O()

+P () = + O(2)

=4-d & =d-2

24/32Systematic expansions

4

2

3

d

g

fermions & bosons with small coupling g2~(4-d) << 1

fermions with small coupling g~(d-2) << 1

Carlson & Reddy (2005)Cf. MC simulation in 3d

g

NLO correctionis small ~5 %

=4-d & =d-2

25/32

d

♦=0.42

4d

2d

Matching of two expansions in • Padé approximants ( + Borel transformation)

Interpolations to 3d

free Fermi gas free Bose gas

= Eunitary / Efree

26/32

d

Tc / F

4d

2d

Critical temperature• Critical temperature from d=4 and 2

Monte Carlo simulations• Bulgac et al. (’05): Tc/F = 0.23(2)

• Lee and Schäfer (’05): Tc/F < 0.14

• Burovski et al. (’06): Tc/F = 0.152(7)

• Akkineni et al. (’06): Tc/F 0.25

• Interpolated results to d=3

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

free Fermi gas free Bose gas

27/32

Few body aspects

28/32Correspondence

• Schrödinger equation in free space with E=0

Scaling solution

• Schrödinger equation in a harmonic potential

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

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

= anomalous dimension of operator in nonrelativistic CFT

29/323 fermions in a harmonic potential

2d

2d

4d

4d

Angular momentum l = 0 Angular momentum l = 1


2d

2d

4d

4d

Angular momentum l = 0 Angular momentum l = 1

31/32Summary

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)• …

32/32

• 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

Very simple and useful starting points tounderstand the unitary Fermi gas in d=3 !

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

34/32NNLO correction for • NNLO correction for

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

Fit two expansions using Padé approximants

d

Interpolations to 3d

• NNLO 4d + NNLO 2d

cf. NLO 4d + NLO 2d

Nishida, Ph.D. thesis (2007)

♦=0.40

35/32

unitarity

BCS BEC

Gapped superfluid

1-plane waveFFLO : O(6)

Polarized normal state

Polarized Fermi gas around d=4• Rich phase structure near unitarity point in the plane of and : binding energy

Stable gapless phases (with/without spatially varying condensate) exist on the BEC side of unitarity point

Gapless superfluid

36/32

• Borel summation with conformal mapping=1.23550.0050 & =0.03600.0050

• Boundary condition (exact value at d=2)=1.23800.0050 & =0.03650.0050

expansion in critical phenomena

O(1) 2 3 4 5 Lattice Exper.

1 1.167 1.244 1.195 1.338 0.892 1.239(3)

1.240(7) 1.22(3) 1.24(2)

0 0 0.0185 0.0372 0.0289 0.0545 0.027(5) 0.016(7) 0.04(2)

Critical exponents of O(n=1) 4 theory (=4-d 1)

expansion isasymptotic seriesbut works well !

How about our case???


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


|a|

39/32Quasiparticle spectrum

- i (p) =

• Fermion dispersion relation : (p)

Energy gap :

Location of min. :

LOself-energydiagrams

0

Expansion over 4-d

Expansion over d-2

or

O() O()

40/32Extrapolation to d=3 from d=4-• Keep LO & NLO results and extrapolate to =1

J.Carlson and S.Reddy,

Phys.Rev.Lett. 95, (2005)

Good agreement with recent Monte Carlo data

NLOcorrectionsare small

5 ~ 35 %

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

Strongly interacting scale-free matter in cold atoms Yusuke Nishida March 12, 2009 @ MIT Faculty...

Documents

Transcript of Strongly interacting scale-free matter in cold atoms Yusuke Nishida March 12, 2009 @ MIT Faculty...