1 / 39 Plan of - - PowerPoint PPT Presentation
201637-8 @ 1 / 39 Plan of this talk 2 / 39 1. Universality in physics
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2
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atomic BEC liquid helium superconductor
neutron
star graphene
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E.g. Water vs. Magnet Water and magnet have the same exponent β≈ 0.325
solid liquid gas
temperature pressure temperature magnetic field
↑↑↑↑↑ ↓↓↓↓↓
M↑ − M↓ ∼ (Tc − T)β ρliq − ρgas ∼ (Tc − T)β
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a/r0
potential depth V
V r0
a < 0
a > 0 V(r)
scattering length
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E.g. 4He atoms vs. proton/neutron van der Waals force : nuclear force : a ≈ 1×10-8 m ≈ 20 r0 a ≈ 5×10-15 m ≈ 4 r0 Ebinding ≈ 1.3×10-3 K Ebinding ≈ 2.6×1010 K
He
Atoms and nucleons have the same form of binding energy
He
p n
Ebinding → − 2 m a2 (a/r0 → ∞)
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Volume 33B, number 8 PHYSICS LETTERS 21 December 1970
ENERGY LEVELS
ARISING FROM RESONANT
TWO-BODY
FORCES IN A THREE-BODY SYSTEM
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.
The range of nucleon-nucleon forces r o is known to be considerably smaller than the scattering lengts a. This fact is a consequence of the resonant character of nucleon-nucleon forces. Apart from this, many other forces in nuclear physics are resonant. The aim of this letter is to expose an interesting effect of resonant forces in a three-body system. Namely, for a '"r o a series of bound levels appears. In a certain case, the number of levels may become infinite. Let us explicitly formulate this result in the simplest case. Consider three spinless neutral particles of equal mass, interacting through a potential gV(r). At certain g = go two particles get bound in their first s-state. For values of g close to go, the two-particle scattering length a is large, and it is this region of g that we shall confine ourself to. The three-body continuum boundary is shown in the figure by cross-hatching. The effect we are drawing attention to is the fol-
~
g<g. g>g,
The level spectrum of three neutral spinless
ticle bound states emerge one after the other. At g = go (infinite scattering length) their number is
into continuum one after the other (see fig. 1). The number of levels is given by the equation N ~ 1 ln(jal/ro) (1)
7 T
All the levels are of the 0 + kind; corresponding wave funcLions are symmetric; the energies
EN .~ 1/r o 2 (we use~=m
= 1); the range of these bound states is much larger than r o. We want to stress that this picture is valid for a ,-, r o. Three-body levels appearing at a ~ r o
The physical cause of the effect is in the emergence of effective attractive long-range forces of radius a in the three-body system. We can demonstrate that they are of the 1/1~
2 kind;
R 2 =r22 +r23 +r21. This form is valid forR 2: r o. With a ~ o0 the number of levels becomes in- finite as in the case of two particles interacting with attractive 1/r 2 potential. Our result may be considered as a generaliza- 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 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 first excited level from this 0+-series will
563
Efimov (1970)
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Efimov (1970)
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Efimov (1970)
R 22.7×R (22.7)2×R
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g1 g2 g1 g2
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(2004 Nobel prize)
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a=∞
Ferlaino & Grimm, Physics (2010)
(22.7)2 (22.7)2
22.7 22.7 22.7 22.7
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6 s0)
2 s0) =
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△ 4He atoms (a ≈ 1×10-8 m ≈ 20 r0) ? He He He He He He
Eb = 125.8 mK
Eb = 2.28 mK
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215 220 225 230
1000 2000 3000 scattering length (ao) B (gauss)
Ultracold atoms are ideal to study universal quantum physics because of the ability to design and control systems at will
C.A. Regal & D.S. Jin PRL90 (2003)
Universal regime
10 ~ 100 a0
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atom loss rate scattering length a (1000 a0)
Innsbruck group Nature (2006)
Trimer is unstable atom loss
signature of trimer formation
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≈ 25 ≈ 22.5 ≈ 21.1
atom loss rate scattering length a/a0
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r
ˆ e
r S− r+ ˆ e + JzSz r Sz r+ ˆ e) + D(Sz r )2 − BSz r
ARTICLES
PUBLISHED ONLINE: 13 JANUARY 2013 | DOI: 10.1038/NPHYS2523
Yusuke Nishida*, Yasuyuki Kato and Cristian D. Batista
Physics is said to be universal when it emerges regardless of the underlying microscopic details. A prominent example is the Efimov effect, which predicts the emergence of an infinite tower of three-body bound states obeying discrete scale invariance when the particles interact resonantly. Because of its universality and peculiarity, the Efimov effect has been the subject of extensive research in chemical, atomic, nuclear and particle physics for decades. Here we employ an anisotropic Heisenberg model to show that collective excitations in quantum magnets (magnons) also exhibit the Efimov effect. We locate anisotropy-induced two-magnon resonances, compute binding energies of three magnons and find that they fit into the universal scaling law. We propose several approaches to experimentally realize the Efimov effect in quantum magnets, where the emergent Efimov states of magnons can be observed with commonly used spectroscopic measurements. Our study thus
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r
ˆ e
r S− r+ ˆ e + JzSz r Sz r+ ˆ e) + D(Sz r )2 − BSz r
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r
ˆ e
r S− r+ ˆ e + JzSz r Sz r+ ˆ e) + D(Sz r )2 − BSz r
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r
ˆ e
r S− r+ ˆ e + JzSz r Sz r+ ˆ e) + D(Sz r )2 − BSz r
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|r1−r2|→∞ Ψ(r1, r2)
ˆ e
1 ˆ e r 2 ˆ e)
ˆ e
2 ˆ e Jz
ˆ e
e 2Dδr1,r2
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3 2π
3J − Jz J
D 6SJ
J
D 6SJ
3J − Jz J
D 6SJ
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n En/J √En−1/En −2.09 × 10−1 — 1 −4.15 × 10−4 22.4 2 −8.08 × 10−7 22.7 n En/J √En−1/En −5.16 × 10−1 — 1 −1.02 × 10−3 22.4 2 −2.00 × 10−6 22.7 n En/J √En−1/En −5.50 × 10−2 — 1 −1.16 × 10−4 21.8 n En/J √En−1/En −4.36 × 10−3 — 1 −8.88 × 10−6 22.2
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n En/J √En−1/En −2.09 × 10−1 — 1 −4.15 × 10−4 22.4 2 −8.08 × 10−7 22.7 n En/J √En−1/En −5.16 × 10−1 — 1 −1.02 × 10−3 22.4 2 −2.00 × 10−6 22.7
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R 22.7 × R (22.7)2 × R
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Y.N. & S.Tan, Few-Body Syst Y.N. & D.Lee Phys Rev A
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Y.N. & S.Tan, Few-Body Syst Y.N. & D.Lee Phys Rev A
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Super Efimov Effect of Resonantly Interacting Fermions in Two Dimensions
Yusuke Nishida,1 Sergej Moroz,2 and Dam Thanh Son3
1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2Department of Physics, University of Washington, Seattle, Washington 98195, USA 3Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA
(Received 18 January 2013; published 4 June 2013)
PRL 110, 235301 (2013) P H Y S I C A L R E V I E W L E T T E R S
week ending 7 JUNE 2013
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10-9 m 1026 m
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