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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).
↑ ↓ ↓ ↓ ↑ ↓ ↓ ↑
↑ ↑ ↓ ↑ ↑ ↓ ↑ ↑
↓ ↓ ↓ ↑ ↑ ↓ ↓ ↑
↓ ↑ ↑ ↓ ↓ ↓ ↑ ↓
9.
XY model on WS networks
Kim & al., PRE 64:056135 2001.
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.
Dynamic XY model on WS networks
Kim & al., PRE 64:056135 2001.
13.
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?
14.
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
15.
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:
16.
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
17.
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
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