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Pascal Naidon
Microscopic origin and universality classes of the three-body parameter
Shimpei Endo
Masahito Ueda
The University of Tokyo
3 particles (bosons or distinguishable) with
resonant two-body interactions
β’ single-channel two-body interaction
β’ no three-body interaction
Zero-range condition with
Efimov attraction
R
x
π 2=23(π₯2+π¦2+π§ 2)Hyperradius
The Efimov effect (1970)
y
z
1/π
πΈ
The Efimov 3-body parameter
Th
ree-b
od
y
para
mete
r
Parameters describing particles at low energyScattering length a
π
1/πβ
-
dimertrimer
Ξβ1 3-body parameter
(2-body parameter)
Zero-range condition with
Efimov attraction
R
x
π 2=23(π₯2+π¦2+π§ 2)Hyperradius
The Efimov effect (1970)
y
z
1/π
πΈ
The Efimov 3-body parameter
Th
ree-b
od
y
para
mete
r
Parameters describing particles at low energyScattering length a
π
1/πβ
-
dimertrimer
Ξβ1 3-body parameter
(2-body parameter)
22.72
Microscopic determination?
Universality for atoms
r
short-range details
β1
π6 van der Waals
Two-body potential
Effective three-body potential
Hyperradius R
short-range details
β1
π 2 Efimov
trip
let s
catt
. len
gth
4He
7Li
6Li
39K23Na
87Rb
85Rb
133Csno universality of the scattering length
πβ[π
0]
πββ β10π π£ππ
universality of the 3-body parameter
Three-body with van der Waals interactionsPhys. Rev. Lett. 108 263001 (2012)J. Wang, J. DβIncao, B. Esry, C. Greene
πββ β11π π£ππThree-body repulsion
at
Efimov
Lennard-Jones potentials supporting n = 1, 2, 3, ...10 s-wave bound states
Hyperradius R
Interpretation: two-body correlation
ππ =sin(ππ ΒΏβπΏπ)ΒΏ
Asymptotic behaviour
Two-body correlation
ππ
π (π )
Resonance
ππ
π (π )
Interatomic separation r
Strong depletion
βΌ 12ππ=β«
0
β
(π 0 (π )2 βπ 0 (π )2 )πππ0 βπ π£ππ
Interpretation: two-body correlation
squeezed
equilateral
elongated
Excluded configurations
induced deformation Efimov
Kinetic energy
cost due to
deformation
deformation
π0
Exc
lud
ed
configura
tions
Confirmation 1: pair correlation model
FModel = FEfimov x j(r12) j(r23) j(r31) (product of pair
correlations)(hyperangular wave function)
3-body potential
π (π )= ππ 2
+β«πcosπ ππΌ|πΞ¦ππ |
2
Hyperradius R []
Ene
rgy
E [
] Pair model(for Lennard-Jones two-body interactions)
Exact
Efimov attraction
Reproduces the low-energy 2-body physicsβ’ Scattering lengthβ’ Effective rangeβ’ Last bound stateβ’ β¦.
Confirmation 2: separable model
π=π|π β© β¨ πβ¨ΒΏParameterised to reproduce exactly the two-body correlation at zero energy.
1/πβ
-
π (π)=1βπβ«0
β
(π 0 (π ) βπ 0(π ))sinππ ππ
π=4π ( 1π
β2πβ«0
β
|π (π )|2ππ)
β1
Hyperradius R
Inte
gra
ted
pro
babili
ty
Hyperradius R
Energ
y
Confirmation 2: separable model
π=π|π β© β¨ πβ¨ΒΏParameterised to reproduce exactly the two-body correlation at zero energy.
πβ
ππ£ππ πβ=β10.86 (1)π π£ππ
n
Exact
Pair model
Separable model
Confirmation 2: separable model
π=π|π β© β¨ πβ¨ΒΏParameterised to reproduce exactly the two-body correlation at zero energy.
S. Moszkowski, S. Fleck, A. Krikeb, L. TheuΓΏl, J.-M.Richard, and K. Varga, Phys. Rev. A 62 , 032504 (2000).
Other potentials
Potential
Yukawa -5.73 0.414
Exponential -10.7 0.216
Gaussian -4.27 0.486
Morse () -12.3 0.180
Morse () -16.4 0.131
PΓΆschl-Teller () -6.02 0.367
-6.55
-11.0
-4.47
-12.6
-16.3
-6.23
0.204
-0.366
0.472
0.173
0.128
0.350
Separable model
Exact calculations
at most 10% deviation
Summary
two-body correlation
three-body deformation
three-body repulsion
three-body parameter
universal
universal effective range
effective range
Two-body correlation universality classes
Power-law tails Faster than Power-law tails
Interparticle distance ()
Pro
bab
ility
densi
ty
0.0 1.0 2.0 3.0 4.0
Van der Waals
ββ1
π π (π>3)
Step function correlation limit
Universal correlationπ 0 (π )=Ξ (πβ1
πβ2 ) (π /ππ)1/2 π½ 1/(πβ 2)(2 (π /π π)β(πβ 2)/2)
Separable model
Number of two-body bound states
Bin
din
g w
ave n
um
ber
at
unit
ari
ty3-body parameter in units of the two-body effective range
Atomic physics
Nuclear physics
?
(= size of two-body correlation)
π =β0.2190 (1 )( π π
2 )β1
π =β0.261 (1 )( ππ2 )β1
π =β0.364 (1 )( ππ2 )β 1
P. Naidon, S. Endo, M. Ueda, PRL 112, 105301 (2014)
P. Naidon, S. Endo, M. Ueda, PRA 90, 022106 (2014)
Summary
The 3-body parameter is (mostly) determined by the low-energy 2-body correlation.Reason: 2-body correlation induces a deformation of the 3-body system.Consequences: the 3-body parameterβ’ is on the order of the effective range.β’ has different universal values for distinct classes
of interaction