1. Spin models on networks
revisited
Petter Holme
Tokyo Institute of Technology

2. Spin models of statistical physics
1. An underlying graph G.
Traditionally a d-dimensional
lattice.
2. A spin variable θi associated
with every node in the graph.
3. A function H (the “Hamiltonian”)
that maps G and {θi} to a number.
Typically (always?) ∑f(θi–θj) where the
sum is over edges (i,j).
4. The probability of {θi} is exp(–H/kBT).
↑ ↓ ↓ ↓ ↑ ↓ ↓ ↑
↑ ↑ ↓ ↑ ↑ ↓ ↑ ↑
↓ ↓ ↓ ↑ ↑ ↓ ↓ ↑
↓ ↑ ↑ ↓ ↓ ↓ ↑ ↓

3. Why put spin models on networks

4. Why put spin models on networks

5. Why put spin models on networks

7. The XY model
H(G, {θi}) = –J∑edges (i,j) cos(θi–θj), θi are angles

8. https://www.complexity-explorables.org/explorables/if-you-ask-your-xy/

9. The XY model

10. XY model on WS networks
Kim & al., PRE 64:056135 2001.

11. XY model on WS networks
Kim & al., PRE 64:056135 2001.

12. XY model on WS networks
Kim & al., PRE 64:056135 2001.

13. Dynamic XY model on WS networks
Kim & al., PRE 64:056135 2001.

14. The YX model
Just like XY, but keep (randomly sampled) spins fixed
and vary the links of the graph.
Holme, Wu, Minnhagen, Multiscaling in an YX model
of networks, Phys. Rev. E. 80, 036120 (2009).
Magnetic transitions no longer possible, but maybe
some transition in network structure?

15. The YX model
Just like XY, but keep (randomly sampled) spins fixed
and vary the links of the graph.
(a) H = −199.52
π/2
0π
−π/2
(b) H = −195.86 (c) H = −192.05

16. The YX model
0.4
0.6
0.8
1
1.2
1.4
10 100 103
104
TN
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
10 100 103
104
TN
DNε
δδ
10.110.1
DNε
(b)(a)
400
800
N = 1600
200
δ =1.52, ε = –0.74
The diameter of the largest connected component:

17. The YX model
The largest connected component:
0
0.2
0.4
0.6
0.8
1
10−5
10−4
10−3
0.01
T
1−s1
(a)
400
800
1600
N = 3200
(b)
0.05
0.1
0.15
0.2
3
10100
(1−s1)Nβ
TNα
α =1.6, β = 0.22

18. The YX model
The 2nd largest connected component:
10 100
TN
γ10−3
10−5
10−4
0.01 0.1
0
T
s2
0.1
0.2
(b)(a)
s2
0.1
0.2
N = 3200
1600
800
400
0
γ =1.44

19. The YX model
The 2nd largest connected component:
10 100
TN
γ10−3
10−5
10−4
0.01 0.1
0
T
s2
0.1
0.2
(b)(a)
s2
0.1
0.2
N = 3200
1600
800
400
0
γ =1.44

20. The free XY model
Magnetization (avg. degree 8).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
magnetization
temperature
16
32
64
128
256

21. The free XY model
Magnetization (avg. degree 8).
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
0.01 0.1 1
magnetization*Nnu
temperature
16
32
64
128
256
ν = 0.30

22. The free XY model
Magnetization (avg. degree 8).
0.0000001
0.0000010
0.0000100
0.0001000
0.0010000
0.0100000
0.1000000
1.0000000
10.0000000
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256

23. The free XY model
Size of the largest component (avg. degree 8).
0.97
0.975
0.98
0.985
0.99
0.995
1
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
16
32
64
128
256

24. The free XY model
Number of components (avg. degree 8).
1
10
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
16
32
64
128
256

25. The free XY model
Magnetization (avg. degree 4).
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.01 0.1 1
magnetization*Nnu
temperature
8
16
32
64
128
256
ν = 0.30

26. The free XY model
Size of the largest component (avg. degree 4).
0.7
0.75
0.8
0.85
0.9
0.95
1
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256

27. The free XY model
Number of components (avg. degree 4).
1
10
100
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256

28. 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
0.01 0.1 1
magnetization*Nnu
temperature
16
32
64
128
256
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.01 0.1 1
magnetization*Nnu
temperature
8
16
32
64
128
256
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
0.01 0.1
magnetization*Nnu
temperature
8
16
32
64
128
256
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
0.001 0.01 0.1
magnetization*Nnu
temperature
8
16
32
64
128
256
512
k = 8, ν = 0.30 k = 4, ν = 0.30
k = 2, ν = 0.18 k = 1, ν = 0.02
Magnetization crossing plots

29. 0.97
0.975
0.98
0.985
0.99
0.995
1
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
16
32
64
128
256
0.7
0.75
0.8
0.85
0.9
0.95
1
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
512
Size of LCC
k = 8 k = 4
k = 2 k = 1

30. 2
2.5
3
3.5
4
4.5
5
5.5
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
16
32
64
128
256
2
3
4
5
6
7
8
9
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
2
4
6
8
10
12
14
16
18
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
0
2
4
6
8
10
12
14
16
18
20
22
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
512
Diameter
k = 8 k = 4
k = 2 k = 1

31. The freer XY model
Let both the links and spins be free to update; don’t
conserve the number of links.
π/23π/2
π
0
θ
T = 10–3
T = 1 T = 103

32. Thank you
http://petterhol.me