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

• single-channel two-body interaction

• no three-body interaction

Zero-range condition with

Efimov attraction

R

x

𝑅2=23(𝑥2+𝑦2+𝑧 2)Hyperradius

The Efimov effect (1970)

y

z

1/𝑎

𝐸

The Efimov 3-body parameter

Th

ree-b

od

y

para

mete

r

Parameters describing particles at low energyScattering length a

𝑎

1/𝑎−

-

dimertrimer

Λ−1 3-body parameter

(2-body parameter)

Zero-range condition with

Efimov attraction

R

x

𝑅2=23(𝑥2+𝑦2+𝑧 2)Hyperradius

The Efimov effect (1970)

y

z

1/𝑎

𝐸

The Efimov 3-body parameter

Th

ree-b

od

y

para

mete

r

Parameters describing particles at low energyScattering length a

𝑎

1/𝑎−

-

dimertrimer

Λ−1 3-body parameter

(2-body parameter)

22.72

Microscopic determination?

Universality for atoms

r

short-range details

−1

𝑟6 van der Waals

Two-body potential

Effective three-body potential

Hyperradius R

short-range details

−1

𝑅2 Efimov

trip

let s

catt

. len

gth

4He

7Li

6Li

39K23Na

87Rb

85Rb

133Csno universality of the scattering length

𝑎−[𝑎

0]

𝑎−≈ −10𝑟 𝑣𝑑𝑊

universality of the 3-body parameter

Three-body with van der Waals interactionsPhys. Rev. Lett. 108 263001 (2012)J. Wang, J. D’Incao, B. Esry, C. Greene

𝑎−≈ −11𝑟 𝑣𝑑𝑊Three-body repulsion

at

Efimov

Lennard-Jones potentials supporting n = 1, 2, 3, ...10 s-wave bound states

Hyperradius R

Interpretation: two-body correlation

𝜓𝑘  =sin(𝑘𝑟 ¿−𝛿𝑘)¿

Asymptotic behaviour

Two-body correlation

𝜓𝑘

𝑉 (𝑟 )

Resonance

𝜓𝑘

𝑉 (𝑟 )

Interatomic separation r

Strong depletion

∼ 12𝑟𝑒=∫

0

∞

(𝜓 0 (𝑟 )2 −𝜓 0 (𝑟 )2 )𝑑𝑟𝑟0 ≈𝑟 𝑣𝑑𝑊

Interpretation: two-body correlation

squeezed

equilateral

elongated

Excluded configurations

induced deformation Efimov

Kinetic energy

cost due to

deformation

deformation

𝑟0

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

𝑈 (𝑅 )= 𝜆𝑅2

+∫𝑑cos𝜃 𝑑𝛼|𝜕Φ𝜕𝑅|

2

Hyperradius R []

Ene

rgy

E [

] Pair model(for Lennard-Jones two-body interactions)

Exact

Efimov attraction

Reproduces the low-energy 2-body physics• Scattering length• Effective range• Last bound state• ….

Confirmation 2: separable model

𝑉=𝜉|𝜒 ⟩ ⟨ 𝜒∨¿Parameterised to reproduce exactly the two-body correlation at zero energy.

1/𝑎−

-

𝜒 (𝑞)=1−𝑞∫0

∞

(𝜓 0 (𝑟 ) −𝜓 0(𝑟 ))sin𝑞𝑟 𝑑𝑟

𝜉=4𝜋 ( 1𝑎

−2𝜋∫0

∞

|𝜒 (𝑞 )|2𝑑𝑞)

−1

Hyperradius R

Inte

gra

ted

pro

babili

ty

Hyperradius R

Energ

y

Confirmation 2: separable model

𝑉=𝜉|𝜒 ⟩ ⟨ 𝜒∨¿Parameterised to reproduce exactly the two-body correlation at zero energy.

𝑎−

𝑟𝑣𝑑𝑊 𝑎−=−10.86 (1)𝑟 𝑣𝑑𝑊

n

Exact

Pair model

Separable model

Confirmation 2: separable model

𝑉=𝜉|𝜒 ⟩ ⟨ 𝜒∨¿Parameterised to reproduce exactly the two-body correlation at zero energy.

S. Moszkowski, S. Fleck, A. Krikeb, L. Theuÿl, 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

Pöschl-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

∝−1

𝑟 𝑛 (𝑛>3)

Step function correlation limit

Universal correlation𝜓 0 (𝑟 )=Γ (𝑛−1

𝑛−2 ) (𝑟 /𝑟𝑛)1/2 𝐽 1/(𝑛− 2)(2 (𝑟 /𝑟 𝑛)−(𝑛− 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)

𝜅=−0.2190 (1 )( 𝑟 𝑒

2 )−1

𝜅=−0.261 (1 )( 𝑟𝑒2 )−1

𝜅=−0.364 (1 )( 𝑟𝑒2 )− 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 parameter• is on the order of the effective range.• has different universal values for distinct classes

of interaction

Obrigado pela sua attenção!