2011.8.5 兰州
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Transcript of 2011.8.5 兰州
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 bgBa 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 bgBa 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
ab
c
/k 1/ 2
0Fk
( )s sa k a
1, , / 2
s k s
s k
ka kaka
weak interaction
(universality) strong interaction
2
2
/( *)( )/ ( )s bg
mra k ak m B B
Phase shift( )k stg ka k
(ii) narrow:
abc
/k
k0
1
Fk
1/ *rBB
( 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 a2FU 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)
1Dcgm
1Dmgn
Repulsive upper-branch in 1D---BA solution
1Dcgm
sTG
TG1Dmgn
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
Eiig m
C
1D contact:Barth and Zwerger, arxiv: 1101.5594
2 21 ( ) 0DC m g dR R
From Hellman-Feynman theorem:
1
( )D
E dR Rg
E always increases with -1/g1D !
4( ) ( ) ( )i n k kkC
1D-1/ g
E
1Dg0
1Dg
21Dg
21
0( 1/ )D
E Cg 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
2122
23
24
24.5
23.5
22.521.520.5
FM ground state
2[ ]kR
identical fermions
g 0g
Application in 1D (II)
narrow resonance, abg<0: Yes for EF<<B; No otherwise
2122
23
24
24.5
23.5
22.521.520.5
, FMBFE
Application in 3D
wide resonance: No
21.522.5
23.5
24.5
25
24
232221
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!