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Charged analogsof the Efimov effectYusuke Nishida (Tokyo Tech)
International symposium on Clustering
as a Window on the Hierarchical Structureof Quantum Systems (CLUSHIQ2022)

Oct.31 - Nov.3 (2022) @ Sendai

2 / 18

Plan of this talk
1. Introduction• Efimov effect & discrete scale invariance

2. Charged analog in non-relativistic system• Efimovian states in hydrogen molecular ion
Y. Nishida, Phys. Rev. A 105, L010802 (2022)

3. Charged analog in relativistic system• Atomic collapse resonances& vacuum polarization in graphene
Y. Nishida, Phys. Rev. B 90, 165414 (2014)
Y. Nishida, Phys. Rev. B 94, 085430 (2016)

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Efimov effect &discrete scale inv.

4 / 18

Efimov effect✓ 3 bosons

Infinite bound states✓ 3 dimensions

with universal scaling✓ s-wave resonance
V. Efimov, PLB (1970)<latexit sha1_base64="zmv8Vdo5lYz3oxtvm5D76tQAghI=">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</latexit>

En ⇠ (22.7)2nE04 / 18

Efimov effect✓ 3 bosons

Infinite bound states✓ 3 dimensions

with universal scaling✓ s-wave resonance<latexit sha1_base64="zmv8Vdo5lYz3oxtvm5D76tQAghI=">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</latexit>

En ⇠ (22.7)2nE0

V. Efimov, PLB (1970)

...

...
R
22.7 × R(22.7)2 × R

Discrete scale invariance5 / 18

Efimov effect
3-body Schrodinger equation<latexit sha1_base64="V8laZr5UxpflixsxqIqlDh6YwaE=">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</latexit>[T1 + T2 + T3 + V12 + V23 + V31 ](~r1 , ~
r2 , ~r3 )= E (~
r1 , ~
r2 , ~
r3 )Zero-range and infinite scattering length5 / 18

Efimov effect
3-body Schrodinger equation<latexit sha1_base64="V8laZr5UxpflixsxqIqlDh6YwaE=">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</latexit>[T1 + T2 + T3 + V12 + V23 + V31 ](~r1 , ~
r2 , ~r3 )= E (~
r1 , ~
r2 , ~
r3 )Zero-range and infinite scattering length
Hyperradial motion

✓ 2
◆
2
1
@
1 @
s
+
+
2m @R2R @R
2mR2<latexit sha1_base64="TbiwjFR4UR2DqcA90JSfEaUo1jw=">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</latexit>

(R) =



2

2m

(R)

Scale invariant potential with s2 = -1.013 < 0
induced by hyperangular motion

5 / 18

Efimov effect
3-body Schrodinger equation<latexit sha1_base64="V8laZr5UxpflixsxqIqlDh6YwaE=">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</latexit>[T1 + T2 + T3 + V12 + V23 + V31 ](~r1 , ~
r2 , ~r3 )= E (~
r1 , ~
r2 , ~
r3 )Zero-range and infinite scattering length
Hyperradial motion

✓ 2
◆
2
1
@
1 @
s
+
+
2m @R2R @R
2mR2<latexit sha1_base64="TbiwjFR4UR2DqcA90JSfEaUo1jw=">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</latexit>

(R) =



2

2m

(R)

Scale invariant potential with s2 = -1.013 < 0
induced by hyperangular motion
Its solution<latexit sha1_base64="4SvtW+/gAGje5lgydyxahyF3yjU=">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</latexit>(R) / Ki|s| (R)

Efimov effectIts solution<latexit sha1_base64="3PZc3su0uvPBhLAnmEKP9xyY/Y8=">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</latexit>

0<latexit sha1_base64="4SvtW+/gAGje5lgydyxahyF3yjU=">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</latexit>(R) / Ki|s| (R)

/ |R=r0 has to be fixed by short-range B.C.

6 / 186 / 18

Efimov effectIts solution
0<latexit sha1_base64="3PZc3su0uvPBhLAnmEKP9xyY/Y8=">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</latexit><latexit sha1_base64="4SvtW+/gAGje5lgydyxahyF3yjU=">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</latexit>(R) / Ki|s| (R) ! sin [|s| ln (r0 ) + ]<latexit sha1_base64="cFGKFLmKQnoG2F8IXiZSzAjFrak=">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</latexit>/ |R=r0 has to be fixed by short-range B.C.

If  = ⇤ is a solution for r0 ⌧ 1 ,<latexit sha1_base64="RenxzwT9BRVvzu8grbYlBi6MOkU=">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</latexit><latexit sha1_base64="RRm/QOTxItMj11QAZntNAQ3R4Ik=">AAAC0XicbZHBbhMxEIadBUpaoE2BGxfUXBBEUbaK6KVFUTnAsQjSVqqjaOwdtqvYXsv2Jg1WpIorL8GFQ3kj3gZ3k6rdpCPZ/jX/Nx6NhmmRWdfp/KtFDx4+WntcX9948vTZ5lZj+/mxzQvDsc9zkZtTBhZFprDvMifwVBsEyQSesNHHa/9kjMZmufrmphoHElKVfc84uJAaNl5SJj0dgdZwMH+Gb2fDRrPT7pTxelXEC9Ekizgabtf+0CTnhUTluABrvcGxw4vubIMWFjXwEaToQVoJ7rwVXjuVrMVkywb6HJMKdxakAom2lYwzbUs58OWsgVM44bmUoBJPTag8iwfe09DMlYS/zvlmPJstsalBVCv0pzJ7H89EgSt4mbyhDd7yNFRoMC4DUfkntIVgjZFTBUxA1XznaQJpitW56L6nAlQqcLnJhzDyjXEnXWgfDhiTT6pGkk+UL++5GRYbL69xVRzvtuP37e6XbrN3uFhxnbwiO+QNicke6ZHP5Ij0CSc/yG9yRf5GX6NpdBn9nKNRbVHzglQi+vUfAcTrpg==</latexit><latexit sha1_base64="cBnTitN1c6gNrvQuJIwVVmQzFDE=">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</latexit> = (e⇡/|s| )n⇤ are also solutions

6 / 18

Efimov effectIts solution
0<latexit sha1_base64="3PZc3su0uvPBhLAnmEKP9xyY/Y8=">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</latexit>(R) / Ki|s| (R) ! sin [|s| ln (r0 ) + ]<latexit sha1_base64="4SvtW+/gAGje5lgydyxahyF3yjU=">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</latexit><latexit sha1_base64="cFGKFLmKQnoG2F8IXiZSzAjFrak=">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</latexit>/ |R=r0 has to be fixed by short-range B.C.

If  = ⇤ is a solution for r0 ⌧ 1 ,<latexit sha1_base64="RenxzwT9BRVvzu8grbYlBi6MOkU=">AAAC0HicbZHPattAEMbXapKm+Ve7OfYS6ksgxkjBpJcSQnJoj2mpk0DWmNnVRBHeXS27KztmMaHXvkRPhfaR+jbdKA6t7AxI+vi+32gYhmmRWxfHfxrRi5XVtZfrrzY2t7Z3Xjdbby5sURqOfV6IwlwxsChyhX2XO4FX2iBIJvCSjc4e8ssxGpsX6qubahxIyFR+k3NwwRo2dymTno5Aa9gzw5gKkcyGzXbcjavaWxbJXLTJvM6HrcZPmha8lKgcF2CtNzh2eNebbdDSogY+ggw9SCvB3XbC104l6zDZsYG+xbTGXQepQKLtpONc20oOfLVq4BROeCElqNRTEzqvk4H3NAxzFeEfPN9OZrMFNjOIaon+WLnP8UyUuIRX5hNt8B9PQ4cG43IQtf+EsRCiMXKqgAmohweeppBlWN+LfvBUgMoELg45Dis/Bf/ZpfbhAWOKST1Ii4ny1fsxDIdNFs+4LC4Ou8lRt/e51z45nZ94nbwl78g+Sch7ckI+kXPSJ5xMyQ/yi/yOvkR30X307RGNGvOeXVKr6PtfRlbq8w==</latexit><latexit sha1_base64="RRm/QOTxItMj11QAZntNAQ3R4Ik=">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</latexit><latexit sha1_base64="cBnTitN1c6gNrvQuJIwVVmQzFDE=">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</latexit> = (e⇡/|s| )<latexit sha1_base64="uAuAia+/EYwBwXmtJ07rcLIQl34=">AAADMnicbZHfbtMwFMad8G90wDq45GaiN51Wqqaq4IZNEwjB5ajoNqkuleOcptFsJ7KdlsnK4/AGvAxwhbjlIXCcTJCWI8Xn5Pt+tnV8wowlSg8G3zz/1u07d+/t3G/tPnj4aK+9//hcpbmkMKEpS+VlSBSwRMBEJ5rBZSaB8JDBRXj1pvQvViBVkoqP+jqDGSexSBYJJdpK83aOQ24wg4WePscLSWhghrxwQtf9Gxx9GhZ2PRjbfFQxY5dw5OQCyyRe6sNWZRpV8kNe4pUzw5lKuuPD47d1UczbnUF/4OJguwjqooPqOJvve19wlNKcg9CUEaWMhJWGz6OihXMFGaFXJAZDuOJEL3s2q2se9kLeU5ZeQtTgprYUhIPqRaskU66cGfeYlhOwpinnRNj2pN05DWbGYHuZdoQpNdMJimKDjSWA2KLfOfV/fMhy2MKdeENL+MuXj50RqRPCGufYa4m1VkCxICEjTfPITojEMTT7wq/syImIGWxecmJbvjH+kfPM2I9Ima6bRpSuhXFrZdrBBptj3C7Oh/3gRX/0YdQ5fV2PeAc9Rc9QFwXoJTpF79EZmiCKfnie1/J2/a/+d/+n/6tCfa/e8wQ1wv/9B6NSDx0=</latexit>

1
2m

✓

@

n

2

@R

+
2

⇤ are also solutions
1 @
R @R

◆

+

s

2

2mR2

(R) = E (R)• Scale invariance is broken by short-range B.C.
down to discrete scale invariance• Long-range Coulomb potential is usually obstacle

6 / 18

Efimov effectIts solution
0<latexit sha1_base64="3PZc3su0uvPBhLAnmEKP9xyY/Y8=">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</latexit>(R) / Ki|s| (R) ! sin [|s| ln (r0 ) + ]<latexit sha1_base64="4SvtW+/gAGje5lgydyxahyF3yjU=">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</latexit><latexit sha1_base64="cFGKFLmKQnoG2F8IXiZSzAjFrak=">AAADB3icbZHLbtNAFIYn5lbCLYUlm4gIKVGjKEYR3VSoggUINqUibaVMFB3bB3eUuWlmnFBN8wA8CGtYIba8BBIreBSmbiJwwpHs+fX/3/HR+CSaM+v6/Z+16MrVa9dvbN2s37p95+69xvb9I6sKk+IwVVyZkwQsciZx6JjjeKINgkg4HifTFxf58QyNZUq+c2caxwJyyd6zFFywJo3XNBH+MdWWtQ87VBulnWq+mXh2bs8XbToFraF52KlTp6hlchRsyuUqMJN+Z4dmyB2MF5NGq9/rl9XcFPFStMiyDibbtU80U2khULqUg7Xe4Mzhh8GiTguLGtIp5OhBWAHutBtOeyaSbiK6NtCnmFW4UZASBNpuNmPalnLsyz8UOInzVAkBMvPUhM5RPPaehmGuJPyF51vxYrHG5gZRbtAvS/d/fMIL3MBLc0Ub/MvT0KHBOAa88p0wFkI0w5RKSDhUwx1PM8hzrN6L7nnKQeYc14c8C1deBf/YhfbhAWPUvBpkai59+b4Mw2Lj9TVuiqMnvfhpb/B20Np/vlzxFnlIHpE2icku2SevyAEZkpR8IT/IL/I7+hh9jr5G3y7RqLbseUAqFX3/A+WXAR8=</latexit>/ |R=r0 has to be fixed by short-range B.C.

If  = ⇤ is a solution for r0 ⌧ 1 ,<latexit sha1_base64="RenxzwT9BRVvzu8grbYlBi6MOkU=">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</latexit><latexit sha1_base64="RRm/QOTxItMj11QAZntNAQ3R4Ik=">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</latexit><latexit sha1_base64="cBnTitN1c6gNrvQuJIwVVmQzFDE=">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</latexit> = (e⇡/|s| )<latexit sha1_base64="uAuAia+/EYwBwXmtJ07rcLIQl34=">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</latexit>

1
2m

✓

@

n

2

@R

+
2

⇤ are also solutions
1 @
R @R

◆

+

s

2

2mR2

(R) = E (R)• Scale invariance is broken by short-range B.C.
down to discrete scale invariance• Long-range Coulomb potential is usually obstacle
Discrete scale invariance for charged particles

7 / 18

Non-relativistic
charged particles

L. D. Landau & E. M. Lifshitz, “Quantum Mechanics”

8 / 18

Hydrogen molecular ion
+
1

R

+
2

V(R) ~ R-1

8 / 18

Hydrogen molecular ion
- 3
+
1

R

+
2

V(R) ~ R-1 => R-2

8 / 18

Hydrogen molecular ion
- 3
+
1

R

+

V(R) ~ R-1 => R-2

2

Born-Oppenheimer approximation (M >> m)• Schrodinger equation for a light particle
"
#
2
2
2
r3
kq
kq
(~
r3 ) = E R
r3 )
~ 1R
~ 2 (~
~1 ~
~2 ~
2m
|R
r3 |
|R
r3 |
<latexit sha1_base64="Ok3o6gP3JSoFRRmsvw2Kka2sUUs=">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</latexit>• Schrodinger equation for two heavy particles
"
#
r12
r22
kq 2
~ 1, R
~ 2)
+
+ ER
(R
~ 1R
~2
~1 R
~ 2|
2M
2M
|R
~ 1, R~ 2)= E (R<latexit sha1_base64="7s9CxT7C9Xb3FF6kwSR3gAmRjfM=">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</latexit>9 / 18

Hydrogen molecular ion
r - 3
+
1

+

aB << R

V(R) ~ R-1 => R-2

2<latexit sha1_base64="bfVXJe8BGZnBBJwnrLsguR3UijI=">AAAC9XicbZHNbtNAFIUn5q+Un6awZFORTREmiq0IumhRVYRgWSTSVuqk1p3xrWt1ZmxmxgnVyI8CK2DLS7CFPW/DxE0FTriS7aNzvuvr8WWlyI0dDH53gmvXb9y8tXJ79c7de/fXuusPDkxRaY4jXohCHzEwKHKFI5tbgUelRpBM4CE7fzXLDyeoTV6o9/aixLGETOWnOQfrraS7RZl0VII94yA2Xidq5xk91cAjF0tI9k5idRLX9EMF6abaicI4pCItrHlSJ93eoD9oamNZRHPRI/PaT9Y7X2la8EqislyAMU7jxOLHYb1KK4Ml8HPI0IE0s68J/dNcSBYyGRpPn2Ha4o69VCDRhOkkL00jx675H55TOOWFlKBSR7XvPI7GzlE/zDaEm3muF9X1AptpRLVEv2nc//FMVLiEN+YVrfEvT31HCdrmIFrv8WPBRxPkVAET0A6fOppClmH7XHTbUQEqE7g45KU/8lXwj12Vzl+gdTFtB2kxVa65X4Z+sdHiGpfFQdyPnveH74a93b35ilfII/KYbJKIvCC75C3ZJyPCyWfyg/wkv4Jp8Cn4Eny7RIPOvOchaVXw/Q/cp/jY</latexit>

Hydrogen-like atom E n =1
2ma2B n2(n = 1, 2, . . .)

9 / 18

Hydrogen molecular ion
r - 3
+
1

+

aB << R

V(R) ~ R-1 => R-2

2

1<latexit sha1_base64="bfVXJe8BGZnBBJwnrLsguR3UijI=">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</latexit>

Hydrogen-like atom E n =2ma2B n2(n = 1, 2, . . .)under electric field produced by far separated charge<latexit sha1_base64="DEbaySAOp8M0h4Zth6vHr9Y9TW0=">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</latexit>

V (~
r) =

kq 2
R

kq 2
'
~
|R
~
r|

kq 2
R

R̂
·
~
r
2

9 / 18

Hydrogen molecular ion
r - 3
+
1

+

aB << R

V(R) ~ R-1 => R-2

2

1<latexit sha1_base64="bfVXJe8BGZnBBJwnrLsguR3UijI=">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</latexit>

Hydrogen-like atom E n =2ma2B n2(n = 1, 2, . . .)under electric field produced by far separated charge<latexit sha1_base64="DEbaySAOp8M0h4Zth6vHr9Y9TW0=">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</latexit>

V (~
r) =

kq 2
R

kq 2
'
~
|R
~
r|

1st-order perturbation<latexit sha1_base64="6bnOj8SKTlTzW6TvJcEdF2zDfWE=">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</latexit>

kq 2
R

R̂
·
~
r
2

E n = hV (~r )in = 0 (n = 1)
3
E n = hV (~r )in = ±
,
0(⇥2)
(n
=
2)
mR2
<latexit sha1_base64="q6Ot1XiPzSfeBbybyNpjsl5MywM=">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</latexit>

Scale invariant attraction for n=2,3,…

10 / 18

Hydrogen molecular ion
r - 3
+
1

aB << R+

V(R) ~ R-1 => R-2

2

Infinite bound states obeying discrete scale invarianceEtoward thresholds of (H)n=2,3,…
continuum
n=3
n=2
n=1

Y. Nishida, Phys. Rev. A 105, L010802 (2022)

10 / 18

Hydrogen molecular ion
r - 3
+
1

aB << R+

V(R) ~ R-1 => R-2

2

Infinite bound states obeying discrete scale invarianceEtoward thresholds of (H)n=2,3,…
continuum
n=3
n=2
n=1

• Efimovian states are resonances
embedded into continuum of (H)n=1• Relevant to H2+ ions or trions(nuclear systems?)• Future work:
Their width and experimental probe
Y. Nishida, Phys. Rev. A 105, L010802 (2022)

11 / 18

Relativistic
charged particles

12 / 18

Atomic collapse

Hydrogen-like atom from Dirac equation
Z↵
Coulomb potential
↵
~ ·p
~+ m
(~
r ) = E (~r)
r
is scale invariant<latexit sha1_base64="uLza83KWDdvaiyLzQBluzqrM4GI=">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</latexit>m<latexit sha1_base64="0uOLzTJD92itoJy3fPI5vOia7qM=">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</latexit>

En 0 j = r

1+

[n0 +

p

(Z↵)2
2
(j+ 1
)
2

(Z↵)2 ]2

12 / 18

Atomic collapse

Hydrogen-like atom from Dirac equation
Z↵
Coulomb potential
↵
~ ·p
~+ m
(~
r ) = E (~r)
r
is scale invariant<latexit sha1_base64="uLza83KWDdvaiyLzQBluzqrM4GI=">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</latexit>m<latexit sha1_base64="0uOLzTJD92itoJy3fPI5vOia7qM=">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</latexit>

En 0 j = r

1+

[n0 +

p

(Z↵)2
2
(j+ 1
)
2

(Z↵)2 ]2<latexit sha1_base64="w+8xikUiy5cwxW/Y47S2SsCaylE=">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</latexit>

becomes complex for Z >

1
j+2

↵

> 137

“Atomic collapse” Y. B. Zeldovich & V. S. Popov (1971)12 / 18

Atomic collapse

Hydrogen-like atom from Dirac equation
Z↵
Coulomb potential
↵
~ ·p
~+ m
(~
r ) = E (~r)
r
is scale invariant<latexit sha1_base64="uLza83KWDdvaiyLzQBluzqrM4GI=">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</latexit>m<latexit sha1_base64="0uOLzTJD92itoJy3fPI5vOia7qM=">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</latexit>

En 0 j = r

1+

[n0 +

p

(Z↵)2
2
(j+ 1
)
2

(Z↵)2 ]2<latexit sha1_base64="w+8xikUiy5cwxW/Y47S2SsCaylE=">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</latexit>

becomes complex for Z >

1
j+2

↵

> 137

“Atomic collapse” Y. B. Zeldovich & V. S. Popov (1971)<latexit sha1_base64="5qR/D2LNxaDjnKR/SVnmAsE+wyI=">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</latexit>(~r) ⇠ e

±i

p

(Z↵)2

2 ln r(j+ 1)
2(r ⌧ aB )signals discrete scale invarianceAtomic collapse

13 / 18

Hydrogen-like atom from Dirac equation
Z↵
Coulomb potential
↵
~ ·p
~+ m
(~
r ) = E (~r)
r
is scale invariant<latexit sha1_base64="uLza83KWDdvaiyLzQBluzqrM4GI=">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</latexit>• Z > 137 is not yet achieved with a single nucleus
but may be realized by colliding two heavy nuclei
W. Greiner, B. Muller & J. Rafelski, “Quantum Electrodynamics of Strong Fields”

13 / 18

Atomic collapse

Hydrogen-like atom from Dirac equation
Z↵
Coulomb potential
↵
~ ·p
~+ m
(~
r ) = E (~r)
r
is scale invariant<latexit sha1_base64="uLza83KWDdvaiyLzQBluzqrM4GI=">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</latexit>• Z > 137 is not yet achieved with a single nucleus
but may be realized by colliding two heavy nuclei
W. Greiner, B. Muller & J. Rafelski, “Quantum Electrodynamics of Strong Fields”<latexit sha1_base64="B2mX+vJbFSlmveSYWV4RFOi7h+o=">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</latexit>• Because vF/c ~ O(0.01), ↵e↵ =ke2⇠ O(1)~“vF”“superheavy nucleus” can be realized
by a charged impurity with Z ~ O(1) on graphene
V. M. Pereira, J. Nilsson & A. H. Castro Neto, PRL (2007)
A. V. Shytov, M. I. Katsnelson & L. S. Levitov, PRL (2007)

Graphene
2D massless Dirac equation with a charged impurity

Z↵e↵
~ ·p
~
(~
r ) = E (~r)
r<latexit sha1_base64="bo1PDE+3CbEJHcq3Tl/qzoXBt3w=">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</latexit>

Scale invariance

14 / 18

14 / 18

Graphene
2D massless Dirac equation with a charged impurity

Z↵e↵
~ ·p
~
(~
r ) = E (~r)
r<latexit sha1_base64="bo1PDE+3CbEJHcq3Tl/qzoXBt3w=">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</latexit>

Scale invariance is broken by short-range B.C.
down to discrete scale invariance for Zaeff > 1
E

Infinite bound states14 / 18

Graphene
2D massless Dirac equation with a charged impurity

Z↵e↵
~ ·p
~
(~
r ) = E (~r)
r<latexit sha1_base64="bo1PDE+3CbEJHcq3Tl/qzoXBt3w=">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</latexit>

Scale invariance is broken by short-range B.C.
down to discrete scale invariance for Zaeff > 1

continuum

E

Infinite bound states resonances“Atomic collapse resonances”=> DoS peaks probed by STM

15 / 18

Graphene
Scale invariance is broken by short-range B.C.
down to discrete scale invariance for Zaeff > 1

M. F. Crommie et al.
Science 340, 734 (2013)

Infinite bound states resonances“Atomic collapse resonances”=> DoS peaks probed by STM

continuumE

Cond-mat realization of “superheavy nucleus”

16 / 18

Vacuum polarization
e+ e-

Ze

e- e+

Y. Nishida, Phys. Rev. B 90, 165414 (2014); Phys. Rev. B 94, 085430 (2016)16 / 18

Vacuum polarization
e+ e-

Ze

e- e+
<latexit sha1_base64="sOrE6ASp088z1CmaZgntSg1RQAk=">AAAC5HicbZHNbtQwEMe94auUry0cuSD2wFasVkm1gkuLKlAFxyKxbaV6icbONLVqO5Ht7LZy8wZwQly58QRc4UF4G9x0K8guI8Ue/f+/yWg8rJTCujj+3YmuXb9x89bK7dU7d+/df9Bde7hni8pwHPNCFuaAgUUpNI6dcBIPSoOgmMR9dvLmwt+forGi0B/cWYkTBbkWR4KDC1LafUaZ8rpv1reorVTqdzbj+pyWVqQ7fTpF7k29fv5xo067vXgYN/FkOUnmSY/MYzdd63ynWcErhdpxCdZ6g1OHp6N6lVYWS+AnkKMHZRW440G47ZliA6YGNtDHmLW4w5BqUGgH2VSUtkknvhk/cBpnvFAKdOapCZWHycR7Gpq5hvAXmu8ldb3A5gZRL9FvG/V/PJMVLuGNeEUb/MvTUFGCcQJk6z+hLQQrPC7VwCS0zeeeZpDn2J6LbnoqQecSF5u8CiNfGf/IVenDB8YUs7aRFTPtm/PSDItNFte4nOxtDJMXw9H7UW/79XzFK+QxeUr6JCEvyTZ5R3bJmHDymfwgP8mv6Cj6FH2Jvl6iUWde84i0Ivr2B1p78xw=</latexit>

Charge distribution of electrons n(r) =

X

E<0

|

r )|
E (~

2

Y. Nishida, Phys. Rev. B 90, 165414 (2014); Phys. Rev. B 94, 085430 (2016)16 / 18

Vacuum polarization
e+ e-

e- e+

Ze

<latexit sha1_base64="sOrE6ASp088z1CmaZgntSg1RQAk=">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</latexit>

Charge distribution of electrons n(r) =<latexit sha1_base64="+aB3vsxUBSISuyfFxyiGF2nVe+o=">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</latexit>• Scale invariance => n(r) =

C
r2

X

E<0

|

r )|
E (~

2

Power law

Y. Nishida, Phys. Rev. B 90, 165414 (2014); Phys. Rev. B 94, 085430 (2016)16 / 18

Vacuum polarization
e+ e-

e- e+

Ze

<latexit sha1_base64="sOrE6ASp088z1CmaZgntSg1RQAk=">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</latexit>

Charge distribution of electrons n(r) =<latexit sha1_base64="+aB3vsxUBSISuyfFxyiGF2nVe+o=">AAAC1HicbZHBTttAEIY3bqGBUhrKgUMviFyoGkUxitoLVAgOcKQSCUhsGs2uB2Oxu7Z210mjrU9Vr32Jqjd4IN6GxQQVJx3J9q///8aj0bBMJMZ2One14MXLhcVX9aXl1ytvVt821t71TZprjj2eilSfMzAoEoU9m1iB55lGkEzgGbs+fMjPRqhNkqpTO8lwICFWyWXCwXpr2NigTDq1rT/s0UsN3B0WTn/bKYpho9lpd8ranBfhVDTJtE6Ga7W/NEp5LlFZLsAYp3Fk8Xu3WKa5wQz4NcToQBoJ9qrlv2YiWYvJlvH0FUYV7sJLBRJNKxolmSnlwJXrek7hmKdSgooc1b7zIhw4R/0wWxLuwXPNsChm2Fgjqjn6qHT/xzOR4xxemk+0xn889R0ZaJuAqPzHjwUfjZBTBUxANfzoaARxjNW96K6jAlQscHbIF7/yU/DMzjPnH9A6HVeDKB0rV74fQ3/YcPaM86K/0w4/tbtfu839g+mJ6+Q92SLbJCSfyT45JiekRzgpyB9yQ26DfvAj+Bn8ekSD2rRnnVQq+H0P5Ubsxw==</latexit>• Scale invariance => n(r) =

C
r2

X

E<0

|

r )|
E (~

2

Power law<latexit sha1_base64="vYWGpSzU0UcxrdFVl+cFk/S2rV0=">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</latexit>• Discrete scale invariance => n(r) =F (ln r)r2
Power law + log-periodic oscillation

Y. Nishida, Phys. Rev. B 90, 165414 (2014); Phys. Rev. B 94, 085430 (2016)16 / 18

Vacuum polarization
e+ e-

e- e+

Ze

<latexit sha1_base64="sOrE6ASp088z1CmaZgntSg1RQAk=">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</latexit>

Charge distribution of electrons n(r) =<latexit sha1_base64="+aB3vsxUBSISuyfFxyiGF2nVe+o=">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</latexit>• Scale invariance => n(r) =

X

E<0

C

|

r )|
E (~

2

Power lawr2
<latexit sha1_base64="vYWGpSzU0UcxrdFVl+cFk/S2rV0=">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</latexit>• Discrete scale invariance => n(r) =F (ln r)r2
Power law + log-periodic oscillation2<latexit sha1_base64="sRA5ZcJb83w6hopEMYKW5nM7vfE=">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</latexit>

Fj (r/rj⇤ ) =

2⇡

Re
2

Z 1
0

dz

(1

ig + i ) (1(1 + 2i ) (1⇥ e z U ( ig + i , 1 + 2i , z)U (1(j ig+i ) (1+2i ) (1 ig i )r 2i
1 + (jig i ) (1 2i ) (1 ig+i ) zrj⇤

ig i ) 4
2i )
1

ig

i ,1

(j ig+i ) (1+2i ) (1 ig i )r 2i(j ig i ) (1 2i ) (1 ig+i ) zrj⇤

2i , z)3
5

Y. Nishida, Phys. Rev. B 90, 165414 (2014); Phys. Rev. B 94, 085430 (2016)17 / 18

Vacuum polarization
Comparison to lattice data for Zaeff = 4/3(exact diagonalization on honeycomb lattice with 124x124 sites)
V. M. Pereira, J. Nilsson & A. H. Castro Neto, PRL (2007)

r2 nHrL

Shape & period
are predictions

0.5
0.4

Amplitude & phase
are fit parameters

0.3
0.2

<latexit sha1_base64="vYWGpSzU0UcxrdFVl+cFk/S2rV0=">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</latexit>

0.1
0.0
1

2

5

10

20

50

r
100 a

n(r) =

F (ln r)r2

• Envelop fits well our prediction
but fast oscillation exists with its origin unknown

Summary and problems

18 / 18

1. Introduction• Efimov effect & discrete scale invariance
2. Charged analog in non-relativistic system• Efimovian states in hydrogen molecular ion=> Their width and experimental probe?
3. Charged analog in relativistic system• Atomic collapse resonances& vacuum polarization in graphene=> Possibility in heavy-ion collisions?
Suggestions are appreciated!