Pauli Crystals

Pauli Crystals

hidden geometric structures

of the quantum statistics

Tomasz Sowiński

Institute of Physics of the Polish Academy of Sciences

Europhys. Lett. 115, 20012 (2016)

FEW-BODY PROBLEMS THEORY GROUP

FEW-BODY PROBLEMS THEORY GROUP

FewBody.ifpan.edu.pl

Sci. Rep. 7, 15004 (2017)

Co-authors:

M. Gajda, J. Mostowski, D. Rakshit, M. Załuska-Kotur

(r

, . . . , r

) = Det

2

6 6

6 4

1

(r

)

(r

) . . .

(r

)

1

(r

)

(r

) . . .

(r

)

.. . .. . .. . .. .

1

(r

)

(r

) . . .

(r

)

3

7 7

7 5

ground-state of spinless fermions

V (r)

{ ⁱ(r)}

 ~

2m r

+ V (r)

(r) = E

(r)

• We consider spinless fermions

confined in some external potential

• In principle the corresponding Schrödinger equation can be solved

• The set of eigenstates is known

• Wave function of the noninteracting many-body ground-state

(r

, . . . , r

) = Perm

2

6 6

6 4

1

(r

)

(r

) . . .

(r

)

1

(r

)

(r

) . . .

(r

)

.. . .. . .. . .. .

1

(r

)

(r

) . . .

(r

)

3

7 7

7 5

(r

, . . . , r

) =

(r

)

(r

) · . . . ·

(r

)

• Fundamentally distinguishable particles

• Bosons

1

(r) =

(r) = . . . =

(r)

Ground-state obtained for

small digression

-3 -2 -1 0 1 2 3 0.2

0.4 0.6

0.5 1.5 2.5 3.5

• two distinguishable particles

˜

x

= 0 x ˜

= ±1

• the most probable configuration

density profiles

⇢(x

, x

) = | (x

, x

) |

⇢(˜ x

, ˜ x

) = max [⇢(x

, x

)]

first observation

(x

, x

) = '

(x

)'

(x

)

first observation

-3 -2 -1 0 1 2 3

0.2 0.4 0.6

0.5 1.5 2.5 3.5

• two identical bosons

density profiles

(x

, x

) = 1

p 2 ['

(x

)'

(x

) + '

(x

)'

(x

)]

•the most probable configuration

-3 -2 -1 0 1 2 3 0.2

0.4 0.6

first observation

0.5 1.5 2.5 3.5

• two identical bosons

density profiles

(x

, x

) = 1

p 2 ['

(x

)'

(x

) + '

(x

)'

(x

)]

˜

x

= ± 1

p 2 x ˜

= ˜ x

• the most probable configuration

boson

enhancement

(x1 ,x

2 )

= p 1 2 ['0

(x1

)'1

(x2 ) '1

(x1

)'0

(x2 )]

-3 -2 -1 0 1 2 3 0.2

0.4 0.6

first observation

0.5 1.5 2.5 3.5

• two identical fermions

density profiles

(x₁, x₂) = 1

p2 ['₀(x₁)'₁(x₂) '₁(x₁)'₀(x₂)]

many-body

ground-state

(x

,x

) =

1 p 2

['

(x

)'

(x

) + '

(x

)'

(x

)]

-3 -2 -1 0 1 2 3 0.2

0.4 0.6

first observation

0.5 1.5 2.5 3.5

• two identical fermions

• the most probable configuration

density profiles

(x₁, x₂) = 1

p2 ['₀(x₁)'₁(x₂) '₁(x₁)'₀(x₂)]

˜

x

= ± 1

p 2 x ˜

= x ˜

many-body

ground-state

(x

,x

) =

1 p 2

['

(x

)'

(x

) + '

(x

)'

(x

)]

Two-particle density profile

⇢(x

, x

) = | (x

, x

) |

⇢(˜ x

, ˜ x

) = max [⇢(x

, x

)]

distinguishable

particles identical

bosons identical

fermions

Two-particle density profile

⇢(x

, x

) = | (x

, x

) |

⇢(˜ x

, ˜ x

) = max [⇢(x

, x

)]

distinguishable

particles identical

bosons identical

fermions

general scheme

(r

, . . . , r

) = Det

2

6 6

6 4

1

(r

)

(r

) . . .

(r

)

1

(r

)

(r

) . . .

(r

)

.. . .. . .. . .. .

1

(r

)

(r

) . . .

(r

)

3

7 7

7 5

P(r

, . . . , r

) = | (r

, . . . , r

) |

• probability density of finding given configuration

• we can find its maximum (the most probable configuration)

• we can simulate an experiment with given number of particles!

With Metropolis algorithm we can generate

an ensemble of configurations

according to given density distribution

N. Metropolis et al, J. Chem. Phys. 21, 1087 (1953)

Towards the most probable config.

P({R

}) > P({r

}

) ) {r

}

= {R

}

P({R

})  P({r

}

) ) {r

}

= {r

}

P({R

}) and P({r

}

)

{r

}

= RAND

{R

} = {r

}

+ RAND( {

})

1. Start with random configuration

2. Shift the configuration randomly

3. Calculate corresponding probabilities

4. Conditionally accept a new configuration

5. Go to 2.

Ensemble of configurations

Towards the most probable config.

P({R

}) > P({r

}

) ) {r

}

= {R

}

P({R

})  P({r

}

) ) {r

}

= {r

}

P({R

}) and P({r

}

)

{r

}

= RAND

{R

} = {r

}

+ RAND( {

})

1. Start with random configuration

2. Shift the configuration randomly

3. Calculate corresponding probabilities

4. Conditionally accept a new configuration

5. Go to 2.

Ensemble of configurations

P({R

}) and P({r

}

)

{r

}

= RAND

{R

} = {r

}

+ RAND( {

})

1. Start with random configuration

2. Shift the configuration randomly

3. Calculate corresponding probabilities

5. Go to 2.

P({R

}) > P({r

}

) ) {r

}

= {R

}

P({R

})  P({r

}

) ) {r

}

= RAND ( {r

}

, {R

})

4. Conditionally accept a new configuration _P({r^P({Ri})

i}_t)

1 P({Rⁱ}) P({rⁱ}_t)

N. Metropolis et al, J. Chem. Phys. 21, 1087 (1953)

two-dimensional

harmonic trap

Two-dimensional harmonic trap

nm

(x, y) = N

H

(x)H

(y)e

,

V (r) = m⌦

2 r

= m⌦

2 (x

+ y

)

• single-particle basis

Energy shell Energy (osc. units)

Degeneracy Excitation structure (n,m)

s 1 1 (0,0)

p 2 2 (1,0) (0,1)

d 3 3 (2,0) (1,1) (0,2)

f 4 4 (3,0) (2,1) (1,2) (0,3)

g 5 5 (4,0) (3,1) (2,2) (1,3) (0,4)

⋮ ⋮ ⋮ ⋮

• unique ground-state

N = 1, 3, 6, 10, 15, . . .

N=3

N=3

Degeneracy Excitation structure (n,m)

s 1 1 (0,0)

p 2 2 (1,0) (0,1)

d 3 3 (2,0) (1,1) (0,2)

f 4 4 (3,0) (2,1) (1,2) (0,3)

g 5 5 (4,0) (3,1) (2,2) (1,3) (0,4)

⋮ ⋮ ⋮ ⋮

Two-dimensional harmonic trap

0 1 2

-1 -2

0 1 2 -1

-2

N=3

(osc. units)

Degeneracy Excitation structure (n,m)

s 1 1 (0,0)

p 2 2 (1,0) (0,1)

d 3 3 (2,0) (1,1) (0,2)

f 4 4 (3,0) (2,1) (1,2) (0,3)

g 5 5 (4,0) (3,1) (2,2) (1,3) (0,4)

⋮ ⋮ ⋮ ⋮

P(r

, . . . , r

) = | (r

, . . . , r

) |

N=6

• the most probable configuration

0 1 2

-1 -2

0 1 2 -1

-2 0

1 2

-1 -2

0 1 2 -1

-2

N=3

(osc. units)

Degeneracy Excitation structure (n,m)

s 1 1 (0,0)

p 2 2 (1,0) (0,1)

d 3 3 (2,0) (1,1) (0,2)

f 4 4 (3,0) (2,1) (1,2) (0,3)

g 5 5 (4,0) (3,1) (2,2) (1,3) (0,4)

⋮ ⋮ ⋮ ⋮

P(r

, . . . , r

) = | (r

, . . . , r

) |

N=6

Averaged positions 10⁶ elements

⇢(r) = Z

. . . Z

dr₂ . . . dr_N| (r, r², . . . , r_N)|²

0 1 2

-1 -2

0 1 2 -1

-2

0 1 2

-1 -2

0 1 2 -1

-2 0

1 2

-1 -2

0 1 2 -1

-2

N=3

⇢(r) = Z

. . . Z

dr₂ . . . dr_N| (r, r², . . . , r_N)|²

Energy shell Energy (osc. units)

Degeneracy Excitation structure (n,m)

s 1 1 (0,0)

p 2 2 (1,0) (0,1)

d 3 3 (2,0) (1,1) (0,2)

f 4 4 (3,0) (2,1) (1,2) (0,3)

g 5 5 (4,0) (3,1) (2,2) (1,3) (0,4)

⋮ ⋮ ⋮ ⋮

P(r

, . . . , r

) = | (r

, . . . , r

) |

N=6

Averaged positions 10⁶ elements

N=3

Energy shell Energy (osc. units)

Degeneracy Excitation structure (n,m)

s 1 1 (0,0)

p 2 2 (1,0) (0,1)

d 3 3 (2,0) (1,1) (0,2)

f 4 4 (3,0) (2,1) (1,2) (0,3)

g 5 5 (4,0) (3,1) (2,2) (1,3) (0,4)

⋮ ⋮ ⋮ ⋮

N=6

0 1 2

-1 -2

0 1 2 -1

-2 0

1 2

-1 -2

0 1 2 -1

-2

0 1 2

-1 -2

0 1 2 -1

-2

N=6

(osc. units)

Degeneracy Excitation structure (n,m)

s 1 1 (0,0)

p 2 2 (1,0) (0,1)

d 3 3 (2,0) (1,1) (0,2)

f 4 4 (3,0) (2,1) (1,2) (0,3)

g 5 5 (4,0) (3,1) (2,2) (1,3) (0,4)

⋮ ⋮ ⋮ ⋮

N=10

Degeneracy Excitation structure (n,m)

s 1 1 (0,0)

p 2 2 (1,0) (0,1)

d 3 3 (2,0) (1,1) (0,2)

f 4 4 (3,0) (2,1) (1,2) (0,3)

g 5 5 (4,0) (3,1) (2,2) (1,3) (0,4)

⋮ ⋮ ⋮ ⋮

0 1 2

-1 -2

0 1 2 -1

-2 0

1 2

-1 -2

0 1 2 -1

-2

0 1 2

-1 -2

0 1 2 -1

-2

0 1 2

-1 -2

0 1 2 -1

-2 0

1 2

-1 -2

0 1 2 -1

-2

0 1 2

-1 -2

0 1 2 -1

-2

Energy shell Energy (osc. units)

Degeneracy Excitation structure (n,m)

s 1 1 (0,0)

p 2 2 (1,0) (0,1)

d 3 3 (2,0) (1,1) (0,2)

f 4 4 (3,0) (2,1) (1,2) (0,3)

g 5 5 (4,0) (3,1) (2,2) (1,3) (0,4)

⋮ ⋮ ⋮ ⋮

N=15

0 1 2

-1 -2

0 1 2 -1

-2

0 1 2

-1 -2

0 1 2 -1

-2

0 1 2

-1 -2

0 1 2 -1

-2 0

1 2

-1 -2

0 1 2 -1

-2

0 1 2

-1 -2

0 1 2 -1

-2

0 1 2

-1 -2

0 1 2 -1

-2 0

1 2

-1 -2

0 1 2 -1

-2

0 1 2

-1 -2

0 1 2 -1

-2

0 1 2

-1 -2

0 1 2 -1

-2 0

1 2

-1 -2

0 1 2 -1

-2

0 1 2

-1 -2

0 1 2 -1

-2 0

1 2

-1 -2

0 1 2 -1

-2

N=3 N=6

N=5

Degeneracy Excitation structure (n,m)

s 1 1 (0,0)

p 2 2 (1,0) (0,1)

d 3 3 (2,0) (1,1) (0,2)

f 4 4 (3,0) (2,1) (1,2) (0,3)

g 5 5 (4,0) (3,1) (2,2) (1,3) (0,4)

⋮ ⋮ ⋮ ⋮

N=5

Degeneracy Excitation structure (n,m)

s 1 1 (0,0)

p 2 2 (1,0) (0,1)

d 3 3 (2,0) (1,1) (0,2)

f 4 4 (3,0) (2,1) (1,2) (0,3)

g 5 5 (4,0) (3,1) (2,2) (1,3) (0,4)

⋮ ⋮ ⋮ ⋮

N=5

Degeneracy Excitation structure (n,m)

s 1 1 (0,0)

p 2 2 (1,0) (0,1)

d 3 3 (2,0) (1,1) (0,2)

f 4 4 (3,0) (2,1) (1,2) (0,3)

g 5 5 (4,0) (3,1) (2,2) (1,3) (0,4)

⋮ ⋮ ⋮ ⋮

N=5

Degeneracy Excitation structure (n,m)

s 1 1 (0,0)

p 2 2 (1,0) (0,1)

d 3 3 (2,0) (1,1) (0,2)

f 4 4 (3,0) (2,1) (1,2) (0,3)

g 5 5 (4,0) (3,1) (2,2) (1,3) (0,4)

⋮ ⋮ ⋮ ⋮

What can we do

if don't know the pattern?

J. Javanainen, S. M. Yoo Phys. Rev. Lett. 76, 161 (1996)

... just calculate hierarchy of many-body

conditional probability densities

Example for N=3

L = 10

L = 10

L = 10

Role of a finite number of pictures

N=3 N=6

Thank You!

Pauli Crystals

hidden geometric structures

of the quantum statistics

Tomasz Sowiński

Institute of Physics of the Polish Academy of Sciences

Europhys. Lett. 115, 20012 (2016)

FEW-BODY PROBLEMS THEORY GROUP

FEW-BODY PROBLEMS THEORY GROUP

FewBody.ifpan.edu.pl

Sci. Rep. 7, 15004 (2017)

Co-authors:

M. Gajda, J. Mostowski, D. Rakshit, M. Załuska-Kotur