1 / 39 Plan of - PowerPoint PPT Presentation

西田 祐介（東工大） 第４回 統計物理学懇談会 2016年3月7-8日 @ 学習院大学 量子少数系における普遍性と （スーパー）エフィモフ効果 1 / 39

Plan of this talk 2 / 39 1. Universality in physics 2. What is the Efimov effect ? Keywords: universality scale invariance quantum anomaly RG limit cycle 3. Beyond cold atoms: Quantum magnets 4. New progress: Super Efimov effect

3 / 39 Introduction 1. Universality in physics 2. What is the Efimov effect ? 3. Beyond cold atoms: Quantum magnets 4. New progress: Super Efimov effect

(ultimate) Goal of research Understand physics of few and many particles governed by quantum mechanics atomic 
 BEC liquid helium superconductor neutron star graphene 4 / 39

gas ↓↓↓↓↓ A1. Continuous phase transitions ⇔ ξ/ r 0 → ∞ E.g. Water vs. Magnet Water and magnet have the same exponent β≈ 0.325 When physics is universal ? solid liquid temperature pressure temperature magnetic field ↑↑↑↑↑ 5 / 39 ρ liq − ρ gas ∼ ( T c − T ) β M ↑ − M ↓ ∼ ( T c − T ) β

When physics is universal ? A2. Scattering resonances ⇔ a/r 0 →∞ a/r 0 potential depth V V r 0 a < 0 a→∞ a > 0 V(r) scattering length 6 / 39

When physics is universal ? A2. Scattering resonances ⇔ a/r 0 →∞ E.g. 4 He atoms vs. proton/neutron van der Waals force : nuclear force : a ≈ 1×10 -8 m ≈ 20 r 0 a ≈ 5×10 -15 m ≈ 4 r 0 E binding ≈ 1.3×10 -3 K E binding ≈ 2.6×10 10 K Physics only depends on the scattering length “a” He Atoms and nucleons have the same form of binding energy He 7 / 39 p n E binding → − � 2 ( a/r 0 → ∞ ) m a 2

8 / 39 Efimov effect 1. Universality in physics 2. What is the Efimov effect ? 3. Beyond cold atoms: Quantum magnets 4. New progress: Super Efimov effect

Efimov (1970) Efimov effect 9 / 39 Volume 33B, number 8 PHYSICS LETTERS 21 December 1970 ARISING FROM RESONANT ENERGY LEVELS TWO-BODY FORCES IN A THREE-BODY SYSTEM V. EFIMOV A.F.Ioffe Physico-Technical Institute, Leningrad, USSR Received 20 October 1970 Resonant two-body forces are shown to give rise to a series of levels in three-particle systems. The number of such levels may be very large. Possibility of the existence of such levels in systems of three a-particles (12C nucleus) and three nucleons (3ti) is discussed. ticle bound states emerge one after the other. At The range of nucleon-nucleon forces r o is known to be considerably smaller than the g = go (infinite scattering length) their number is scattering lengts a. This fact is a consequence of infinite. As g grows on beyond go, levels leave the resonant character of nucleon-nucleon forces. into continuum one after the other (see fig. 1). Apart from this, many other forces in nuclear The number of levels is given by the equation physics are resonant. The aim of this letter is to N ~ 1 ln(jal/ro) (1) expose an interesting effect of resonant forces in 7 T a three-body system. Namely, for a '"r o a series of bound levels appears. In a certain case, All the levels are of the 0 + kind; corresponding wave funcLions are symmetric; the energies the number of levels may become infinite. EN .~ 1/r o 2 (we use~=m Let us explicitly formulate this result in the = 1); the range of these simplest case. Consider three spinless neutral bound states is much larger than r o. particles of equal mass, interacting through a We want to stress that this picture is valid for potential gV(r). At certain g = go two particles a ,-, r o. Three-body levels appearing at a ~ r o or with energies E ~ 1/r 2 are not considered. get bound in their first s-state. For values of g close to go, the two-particle scattering length a The physical cause of the effect is in the is large, and it is this region of g that we shall emergence of effective attractive long-range confine ourself to. The three-body continuum forces of radius a in the three-body system. We can demonstrate that they are of the 1/1~ 2 kind; boundary is shown in the figure by cross-hatching. The effect we are drawing attention to is the fol- R 2 =r22 +r23 +r21. This form is valid forR 2: r o. With a ~ o0 the number of levels becomes in- lowing. As g grows, approaching go, three-par- finite as in the case of two particles interacting with attractive 1/r 2 potential. Our result may be considered as a generaliza- -~1 ~ tion of Thomas theorem [1]. According to the latter, when g--~ go' three spinless particles do have a bound state. We assert that in fact there g<g. g>g, are many such states, and for g = go their num- ber is infinite. Note that the effect does not depend on the form of two-body forces - it is only their resonant character that we require. From eq. (1) one finds that the magnitude of the scattering length at which (N+ 1)st level appears is approximately e~ times (~22 times) larger than that for Nth one. Thus, if we assume that the three-body ground state appears at a ~ to, The level spectrum of three neutral spinless Fig. 1. the first excited level from this 0+-series will particles. The scale is not indicative. 563

Efimov effect a→∞ When 2 bosons interact with infinite “a”, 
 3 bosons always form a series of bound states Efimov (1970) 10 / 39

Efimov effect Efimov (1970) R 22.7×R (22.7) 2 ×R . . . Discrete scaling symmetry . . . When 2 bosons interact with infinite “a”, 
 3 bosons always form a series of bound states 11 / 39

Renormalization group limit cycle g 1 g 2 g 1 g 2 Renormalization group flow diagram in coupling space RG fixed point ⇒ Scale invariance E.g. critical phenomena RG limit cycle ⇒ Discrete scale invariance E.g. E????v effect 12 / 39

Renormalization group limit cycle K. Wilson (1971) considered for strong interactions QCD is asymptotic free (2004 Nobel prize) 13 / 39

Renormalization group limit cycle K. Wilson (1971) considered for strong interactions Efimov effect (1970) is its rare manifestation ! 14 / 39

a=∞ Efimov effect at a≠∞ Ferlaino & Grimm, Physics (2010) (22.7) 2 (22.7) 2 Discrete scaling symmetry 22.7 22.7 22.7 22.7 15 / 39

22.7 = exp (π / 1.006...) Why 22.7 ? Just a numerical number given by 22.6943825953666951928602171369... ln ( 22.6943825953666951928602171369...) = 3.12211743110421968073091732438... = π / 1.00623782510278148906406681234... = π / s 0 16 / 39 √ sinh( π 6 s 0 ) 2 π 3 π 2 s 0 ) = cosh( π 4 s 0

E b = 2.28 mK He and observed in 1994 and 2015 2 trimer states were predicted 3 H nucleus (≈ 3 n) and 12 C nucleus (≈ 3α) × Originally, Efimov considered Ultracold atoms ! E b = 125.8 mK He He He He He △ 4 He atoms (a ≈ 1×10 -8 m ≈ 20 r 0 ) ? Where Efimov effect appears ? 17 / 39

Ultracold atoms are ideal to study universal quantum physics because of the ability to design and control systems at will Ultracold atom experiments 18 / 39

10 ~ 100 a 0 ✓ Interaction strength by Feshbach resonances Ultracold atom experiments regime Universal PRL90 (2003) C.A. Regal & D.S. Jin because of the ability to design and control systems at will Ultracold atoms are ideal to study universal quantum physics 19 / 39 3000 2000 scattering length (a o ) 1000 0 -1000 -2000 -3000 215 220 225 230 B (gauss)

Ultracold atom experiments atom loss rate scattering length a (1000 a 0 ) Innsbruck group 
 Nature (2006) Trimer is unstable atom loss First experiment by Innsbruck group for 133 Cs (2006) signature of trimer formation 20 / 39

Ultracold atom experiments Florence group for 39 K (2009) Bar-Ilan University for 7 Li (2009) Rice University for 7 Li (2009) ≈ 25 ≈ 22.5 ≈ 21.1 Discrete scaling & Universality ! atom loss rate scattering length a/a 0 21 / 39

Short summary 22 / 39 Efimov effect: universality, discrete scale invariance, RG limit cycle ? nuclear atomic physics physics prediction realization (1970) (2006) Where else can it be found ?

23 / 39 Beyond cold atoms 1. Universality in physics 2. What is the Efimov effect ? 3. Beyond cold atoms: Quantum magnets 4. New progress: Super Efimov effect