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 兰州 Collaborator: Tin-Lun Ho (Ohio State University)

Strong attraction: highest superfluid Tc ～ T F 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 ( )

Part I Narrow Feshbach Resonance --- alternative way to achieve strong interaction Tin-Lun Ho and XL Cui, arxiv:

Wide vs. Narrow Feshbach resonance Many cold atomic isotopes across both wide and narrow resonance For example, Li-6 s-wave FR: Emergence of bound states: E=0, two atoms in open channel 1 st closed-channel molecule: strong coupling  wide FR 2 nd closed-channel molecule 2: weak coupling  narrow FR

Wide vs. Narrow Feshbach resonance s-wave scattering length: (with E-dependence) wide single a s weak E-dependence narrow strong E-dependence : effective range a s, r*

Phase shift (i) wide: weak interaction (universality) strong interaction

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

Two-body spectrum 1 st molecule in wide FR 2 nd molecule in narrow FR free levels

Two-body spectrum free levels

Interaction effect studied by High-T Virial expansion upper: bound state excluded lower: bound state included fugacity: upper branch & lower branch

Interaction effect studied by High-T Virial expansion Comparison between narrow and wide: WideNarrow anti-symmetric Interaction across wide resonance!

Interaction effect studied by High-T Virial expansion Comparison between narrow and wide: Wide Narrow

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

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) Stoner criterion for onset of FM:

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 now, theoretical studies: Duine and MacDonald, PRL (2005) : 2 nd perturbation Zhai, PRA 80, (R) (2009) Cui and Zhai, PRA 81, (2010): variational approach Pilati et al, PRL (2010 ): QMC Chang et al, PANS 108,51 (2011): QMC Heiselberg, arxiv : Jastrow wf Barth and Zwerger, arxiv: : fermion-boson mapping Zhou, Ceperley and Zhang, arxiv: : lattice ED He and Huang, arxiv: : 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 now, theoretical studies: Duine and MacDonald, PRL (2005) : 2 nd perturbation Zhai, PRA 80, (R) (2009) Cui and Zhai, PRA 81, (2010): variational approach Pilati et al, PRL (2010 ): QMC Chang et al, PANS 108,51 (2011): QMC Heiselberg, arxiv : Jastrow wf Barth and Zwerger, arxiv: : fermion-boson mapping Zhou, Ceperley and Zhang, arxiv: : lattice ED He and Huang, arxiv: : 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, g 1D <0 side ， upper branch ! Another possible approach to FM: 2D system, k F a 2D >>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, (2005): DMC M. T. Batchelor et al, J. Stat. Mech. 10, L10001 (2005): BA E. 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, (2010): BA Definition of 1D upper-branch from BA:

Repulsive upper-branch in 1D---BA solution Guan and Chen, PRL 105, (2010)

Repulsive upper-branch in 1D---BA solution Energy of fully polarized Fermi gas! Guan and Chen, PRL 105, (2010)

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

Repulsive upper-branch in 1D---BA solution 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 1D contact: Barth and Zwerger, arxiv: From Hellman-Feynman theorem: E always increases with -1/g 1D !

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 Wide Narrow a bg <0,g bg <0 1D 3D 2D Y Y or N N Narrow a bg >0,g bg >0 N N N

Application in 1D (I) wide resonance: Yes FM ground state identical fermions

Application in 1D (II) narrow resonance, a bg <0 : Yes for E F <<B; No otherwise

Application in 3D wide resonance: No RHS: lowest bound state turn to scattering state, no s-wave upper-branch any more!

Yes for k F a 2d >>1 (but easily decay to lower branch) No for k F a 2d <<1 Application in 2D

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 3D1D2D √ Existence of FM: × maybe remain to be examined by experiment !!

Thanks for attending!