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
1, . . . , r
N) = Det
2
6 6
6 4
1
(r
1)
2(r
1) . . .
N(r
1)
1
(r
2)
2(r
2) . . .
N(r
2)
.. . .. . .. . .. .
1
(r
N)
2(r
N) . . .
N(r
N)
3
7 7
7 5
ground-state of spinless fermions
V (r)
{ i(r)}
~
22m r
2+ V (r)
i(r) = E
i i(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
1, . . . , r
N) = Perm
2
6 6
6 4
1
(r
1)
2(r
1) . . .
N(r
1)
1
(r
2)
2(r
2) . . .
N(r
2)
.. . .. . .. . .. .
1
(r
N)
2(r
N) . . .
N(r
N)
3
7 7
7 5
(r
1, . . . , r
N) =
1(r
1)
2(r
2) · . . . ·
N(r
N)
• Fundamentally distinguishable particles
• Bosons
1
(r) =
2(r) = . . . =
N(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
1= 0 x ˜
2= ±1
• the most probable configuration
density profiles
⇢(x
1, x
2) = | (x
1, x
2) |
2⇢(˜ x
1, ˜ x
2) = max [⇢(x
1, x
2)]
first observation
(x
1, x
2) = '
0(x
1)'
1(x
2)
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
1, x
2) = 1
p 2 ['
0(x
1)'
1(x
2) + '
1(x
1)'
0(x
2)]
•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
1, x
2) = 1
p 2 ['
0(x
1)'
1(x
2) + '
1(x
1)'
0(x
2)]
˜
x
1= ± 1
p 2 x ˜
2= ˜ x
1• 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
1,x
2) =
1 p 2
['
0(x
1)'
1(x
2) + '
1(x
1)'
0(x
2)]
-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= ± 1
p 2 x ˜
2= x ˜
1many-body
ground-state
(x
1,x
2) =
1 p 2
['
0(x
1)'
1(x
2) + '
1(x
1)'
0(x
2)]
Two-particle density profile
⇢(x
1, x
2) = | (x
1, x
2) |
2⇢(˜ x
1, ˜ x
2) = max [⇢(x
1, x
2)]
distinguishable
particles identical
bosons identical
fermions
Two-particle density profile
⇢(x
1, x
2) = | (x
1, x
2) |
2⇢(˜ x
1, ˜ x
2) = max [⇢(x
1, x
2)]
distinguishable
particles identical
bosons identical
fermions
general scheme
(r
1, . . . , r
N) = Det
2
6 6
6 4
1
(r
1)
2(r
1) . . .
N(r
1)
1
(r
2)
2(r
2) . . .
N(r
2)
.. . .. . .. . .. .
1
(r
N)
2(r
N) . . .
N(r
N)
3
7 7
7 5
P(r
1, . . . , r
N) = | (r
1, . . . , r
N) |
2• 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
i}) > P({r
i}
t) ) {r
i}
t+1= {R
i}
P({R
i}) P({r
i}
t) ) {r
i}
t+1= {r
i}
tP({R
i}) and P({r
i}
t)
{r
i}
0= RAND
{R
i} = {r
i}
t+ RAND( {
i})
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
i}) > P({r
i}
t) ) {r
i}
t+1= {R
i}
P({R
i}) P({r
i}
t) ) {r
i}
t+1= {r
i}
tP({R
i}) and P({r
i}
t)
{r
i}
0= RAND
{R
i} = {r
i}
t+ RAND( {
i})
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
i}) and P({r
i}
t)
{r
i}
0= RAND
{R
i} = {r
i}
t+ RAND( {
i})
1. Start with random configuration
2. Shift the configuration randomly
3. Calculate corresponding probabilities
5. Go to 2.
P({R
i}) > P({r
i}
t) ) {r
i}
t+1= {R
i}
P({R
i}) P({r
i}
t) ) {r
i}
t+1= RAND ( {r
i}
t, {R
i})
4. Conditionally accept a new configuration P({rP({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
nmH
n(x)H
m(y)e
(x2+y2)/2,
V (r) = m⌦
22 r
2= m⌦
22 (x
2+ y
2)
• 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
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)
⋮ ⋮ ⋮ ⋮
Two-dimensional harmonic trap
0 1 2
-1 -2
0 1 2 -1
-2
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)
⋮ ⋮ ⋮ ⋮
P(r
1, . . . , r
N) = | (r
1, . . . , r
N) |
2N=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
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
1, . . . , r
N) = | (r
1, . . . , r
N) |
2N=6
Averaged positions 106 elements
⇢(r) = Z
. . . Z
dr2 . . . drN| (r, r2, . . . , rN)|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 . . . drN| (r, r2, . . . , rN)|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
1, . . . , r
N) = | (r
1, . . . , r
N) |
2N=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
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
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 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=10 N=15N=5
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=5
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=5
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=5
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)
⋮ ⋮ ⋮ ⋮
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
5L = 10
4L = 10
3Role 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