Pauli Crystals

Coauthors:

M. Gajda, M. ZaĿuska-Kotur, J. Mostowski

Pauli Crystals

hidden geometric structures

of the quantum statistics

Tomasz Sowiński

Institute of Physics of the Polish Academy of Sciences ArXiv:1511.01036

instead of the

introduction

(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

)

small digression

• Fundamentally distinguishable particles

• Bosons

1

(r) =

(r) = . . . =

(r)

Ground-state is obtained for

-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

0.5 1.5 2.5 3.5

• two identical bosons

density profiles

-3 -2 -1 0 1 2 3

0.2 0.4 0.6

(x

, x

) = 1

p 2 ['

(x

)'

(x

) + '

(x

)'

(x

)]

˜

x

= ± 1

p 2 x ˜

= ˜ x

• the most probable configuration

boson

enhancement

-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

⇢(x) = Z

dx₂ | (x, x²)|²

single-particle density

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

)]

-3 -2 -1 0 1 2 3

0.2 0.4 0.6

⇢(x) = Z

dx₂ | (x, x²)|²

single-particle density

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)

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

0 1 2

-1 -2

0 1 2 -1

-2

N=3

P(r

, . . . , r

) = | (r

, . . . , r

) |

• the most probable configuration

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)

⋮ ⋮ ⋮ ⋮

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)

⋮ ⋮ ⋮ ⋮

⇢(r) =

Z

. . .

Z

dr

. . . dr

| (r, r

, . . . , r

) |

Averaged positions 10⁶ elements

0 1 2

-1 -2

0 1 2 -1

-2

P (r

,. .., r

)= | (r

,. .., r

)|

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

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

N=6

N=10

(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)

⋮ ⋮ ⋮ ⋮

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=10

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 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

instead of

conclusions

RIDDLE!

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)

⋮ ⋮ ⋮ ⋮

Appendix:

numerical

recipes

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 some 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 some random configuration

2. Shift the configuration randomly

3. Calculate corresponding probabilities

4. Conditionally accept a new configuration

5. Go to 2.

Ensemble of configurations

THE MOST PROBABLE
 CONFIGURATION

Ensemble of configurations

P({R

}) and P({r

}

)

{r

}

= RAND

{R

} = {r

}

+ RAND( {

})

1. Start with some 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

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

Ensemble of configurations

P({R

}) and P({r

}

)

{r

}

= RAND

{R

} = {r

}

+ RAND( {

})

1. Start with some 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)

Thank You!