Connecting two important issues in cold atoms--Origin of strong interaction and

Existence of itinerant Ferromagnetism

崔晓玲

清华大学高等研究院

2011.8.5兰州

Collaborator: Tin-Lun Ho (Ohio State University)

Strong attraction: highest superfluid Tc ～ TF

Origin? Any way to achieve stronger?

Strong repulsion: itinerant Ferromagnetism

Exist or not?

Connecting two issues:

The answers are both strongly indicated by two-body solutions

Strongly interacting Fermi gas (1999-2011)

Part I

Narrow Feshbach Resonance

--- alternative way to achieve strong interaction

Tin-Lun Ho and XL Cui, arxiv: 1105.4627

Wide vs. Narrow Feshbach resonance

[1 ]s bg

Ba a

B B

(G)B834.1543.25

sa

Many cold atomic isotopes across both wide and narrow resonance

For example, Li-6 s-wave FR:

300B G 0.1B G

Emergence of bound states:

E=0, two atoms in open channel

1st closed-channel molecule: strong coupling wide FR

2nd closed-channel molecule 2: weak coupling narrow FR

Wide vs. Narrow Feshbach resonance

( ) [1 ]( )s bg

Ba E a

E B B

s-wave scattering length: (with E-dependence)

wide

single as

,bg FB E E

weak E-dependence

narrow

,bg FB E E

strong E-dependence

-1* [ ]bgr ma B : effective range

as, r*

Phase shift( )k stg ka k

(i) wide:

BB

sa

ab

c

k

a

b

c

/k 1/ 2

0Fk

( )s sa k a

1,

, / 2s k s

s k

ka ka

ka

weak interaction

(universality) strong interaction

2

2

/( *)( )

/ ( )s bg

mra k a

k m B B

Phase shift( )k stg ka k

(ii) narrow:

abc

/k

k0

1

Fk

1/ *r

BB

( 0)sa k

ab

c

bga

1/ * Fk r k

Phase shift in narrow resonance: (i) strong k-dependence (ii) pi-shift within

Two-body spectrum

( 1, 2...)kkR n n

1st molecule in wide FR

2nd molecule in narrow FR

2[ ]kR

free levels

Two-body spectrum

( 1, 2...)kkR n n

FE

0

/k

k0

1

Fk 0

0.5

FE

Fk

2[ ]kR

shift / 2 shift

free levels

Interaction effect studied by High-T Virial expansion

upper: bound state excluded

lower: bound state included

fugacity: 1z e

upper branch & lower branch

Interaction effect studied by High-T Virial expansion

Comparison between narrow and wide:

WideNarrow 2b

anti-symmetric Interaction across wide resonance!

Interaction effect studied by High-T Virial expansion

Comparison between narrow and wide:

WideNarrow int

Interaction effect studied by High-T Virial expansion

Comparison between narrow and wide:

Narrow

New features in Narrow FR:

(i) interaction effect gained far from resonance

(ii) stronger attraction achieved at resonance than in wide FR------ a bran-new class of universality!(iii) strongly asymmetric around resonance

int(0) 0 : 0sa E

Conclusions for Part I

Basic features of narrow resonance

Strong E-dependence of scattering length

Energy scale of resonance width << Fermi energy

Physical consequences

Interaction effect observed far from resonance

New generation of universality at resonance

preliminary results been achieved in Penn State (K. O’Hara group) and Innsbruck (R. Grimm)

easy accessible in experiment: many samples, high-T…

Experiment on Narrow Resonance:

Part II

Existence of Itinerant Ferromagnetism

--- where to look for? 1,2,3D? wide or narrow

resonance?

XL Cui and Tin-Lun Ho, to be published

A long-standing problem : Whether itinerant Ferromagnetism will show up in spin-1/2 fermions due to strong repulsive interaction?

a. 1933, oldest Stoner theory: (Hartree-Fork approx)

/ 3 / 5( )F F

E V E n E n Un n

CU U

CU U

Stoner criterion for onset of FM:F S

( ) 1 k a2F

U E

b. 2009, experiment at MIT: Science 325, 1521 (2009)

(i) based on mean-field calculation in a trap, which predict large domain structure

(ii) not able to observe any domain

Inconsistence:

c. 2009-now, theoretical studies:

Duine and MacDonald, PRL 95 230403. (2005) : 2nd perturbation

Zhai, PRA 80, 051605 (R) (2009)

Cui and Zhai, PRA 81, 041602(2010): variational approachPilati et al, PRL 105 030405 (2010 ): QMC Chang et al, PANS 108,51 (2011): QMCHeiselberg, arxiv 1012.4569: Jastrow wfBarth and Zwerger, arxiv: 1101.5594: fermion-boson mappingZhou, Ceperley and Zhang, arxiv:1103.3534: lattice EDHe and Huang, arxiv:1106.1345: diagrammatic approach……

d. INT and DAPAR meeting, Apr-June 2011, MIT announcement :

“Absence of Itinerant Ferromagnetism in repulsive Fermi gas”

spin susceptibility is measured which never signals the FM transition!!

c. 2009-now, theoretical studies:

Duine and MacDonald, PRL 95 230403. (2005) : 2nd perturbation

Zhai, PRA 80, 051605 (R) (2009)

Cui and Zhai, PRA 81, 041602(2010): variational approachPilati et al, PRL 105 030405 (2010 ): QMC Chang et al, PANS 108,51 (2011): QMCHeiselberg, arxiv 1012.4569: Jastrow wfBarth and Zwerger, arxiv: 1101.5594: fermion-boson mappingZhou, Ceperley and Zhang, arxiv:1103.3534: lattice EDHe and Huang, arxiv:1106.1345: diagrammatic approach……

Supportive!

Now, though the existence of FM in 3D is still under debate, it seems that nature does NOT prefer FM !

Then, is there any place for the cold atom community to find Itinerant FM?

Yes!

One definite approach to FM:

1D system, g1D<0 side ， upper branch !

Another possible approach to FM:

2D system, kFa2D>>1 ， upper branch !

Repulsive upper-branch in 1D---BA solution

the BAEs also have real solutions for c <0, which, however, correspond to some highly excited states of attractive Fermi systems. The FSTG state corresponds to the lowest real solutions of BAEs with c <0.

(i) Bose gas: Crossover from Tonks-Girardeau to super-TG regime

Astrakharchik et al, PRL. 95, 190407 (2005): DMCM. T. Batchelor et al, J. Stat. Mech. 10, L10001 (2005): BAE. Haller et al., Science 325, 1224 (2009): sTG realized in Innsbruck

(ii) Fermi gas: Crossover from Fermionic TG to sTG regime

Guan and Chen, PRL 105, 175301(2010): BA

Definition of 1D upper-branch from BA:

Repulsive upper-branch in 1D---BA solution

Guan and Chen, PRL 105, 175301(2010)

1D

cg

m

1Dmg

n

Repulsive upper-branch in 1D---BA solution

1D

cg

m

sTG

TG1Dmg

n Energy of fully polarized

Fermi gas!

Guan and Chen, PRL 105, 175301(2010)

1D-1/ g (or )B

E

Repulsive upper-branch in 1D---BA solution

lower branch

n n

0

sTG

TG, 0n n n

Repulsive upper-branch in 1D---BA solution

0

sTG

TG, 0n n n

n n

1D-1/ g (or )B

E

By switching B across quasi-1D resonance to g<0 side, equal and uniform spin mixtures relax to FM state due to large spin fluctuations, and form domains.

transition to FM

Without Bethe Ansatz, any other general approach to predict FM in 1D?

Yes!

Understanding FM transition from Tan’s contact

21

( ) ( 1/ )D

Eii

g m

C

1D contact:Barth and Zwerger, arxiv: 1101.5594

2 21 ( ) 0DC m g dR R

From Hellman-Feynman theorem:

1

( )D

EdR R

g

E always increases with -1/g1D !

4( ) ( ) ( )i n k k

k

C

1D-1/ g

E

1Dg

01Dg

21Dg

21

0( 1/ )D

E C

g m

increased E with -1/g

degenerate energy with FM at g=infty (fermionize)

+

FM emerges right at g=infty, and is favored at g<0 (upper branch)

Understanding FM transition from Tan’s contact

Any other physically-transparent way to judge the existence of FM besides 1D?

Yes! from a two-body perspective

qualitative argument

reproduce established results in 1D and 3D

make predictions to many other systems eg: 1D/ 3D narrow resonance, and 2D.

Existence of Itinerant Ferromagnetism from the two-body perspective

WideNarrow

abg<0,gbg<0

1D

3D

2D

Y

Y or N

N

Y or N

Narrow

abg>0,gbg>0

N N

N

Application in 1D (I)

wide resonance: Yes

,2

g

21

22

23

24

24.5

23.5

22.5

21.520.5

FM ground state

2[ ]kR

identical fermions

g 0g

Application in 1D (II)

narrow resonance, abg<0: Yes for EF<<B; No otherwise

21

22

23

24

24.5

23.5

22.5

21.520.5

, FMBFE

Application in 3D

wide resonance: No

21.5

22.5

23.5

24.5

25

24

23

2221

sa

RHS: lowest bound state turn to scattering state, no s-wave upper-branch any more!

Yes for kFa2d>>1 (but easily decay to lower branch)

No for kFa2d<<1

2 1,2dka

2(5/4)

2(9/4)

2(13/4)

2(17/4)

2(3/4)

2(7/4)

2(11/4)

2(15/4)

2(19/4)

2(21/4)

Application in 2D

FM impossible FM possible

Conclusions for Part II

Existence of Itinerant Ferromagnetism from existing studies

3D No (announced recently by MIT experiment)

1D Yes (supported by BA solution)

Understanding the result from other method and further predictions

Tan’s adiabatic theorem (using Contact)

Two-body spectrum

FM depend on the dimension, resonance width, background interaction, and size of Fermi cloud

3D 1D2D

√Existence of FM: × maybe

remain to be examined by experiment !!

Thanks for attending!