Exact Solution of Ising Model on a
Small-World Network
J. Viana Lopes, Yu. G. Pogorelov,
J.M.B. Lopes dos Santos
CFP, Departamento de Física, Faculdade de Ciências, Universidade do Porto
and Raul Toral
IMEDEA, CSIC-UIB
cond-mat/0402138
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Outline
The model
The solution
Thermodynamics
Finite size scaling behavior
Discussion and conclusions
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Small World Network
Watts e Strogatz 1998
N
ln(N)
Shortest distance, averaged over pairs:
O (N) for ordered networks
O (ln N) for networks with pN shortcuts ( p > 0)
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Building the network
Start with ring of N connected points (periodic bc).
For each site, with a probability p, we make a
connection with a randomly chosen site.
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Building the network
Start with ring of N connected points (periodic bc).
For each site, with a probability p, we make a
connection with a randomly chosen site.
4/2
Building the network
Start with ring of N connected points (periodic bc).
For each site, with a probability p, we make a
connection with a randomly chosen site.
4/2
Building the network
Start with ring of N connected points (periodic bc).
For each site, with a probability p, we make a
connection with a randomly chosen site.
4/2
Building the network
Start with ring of N connected points (periodic bc).
For each site, with a probability p, we make a
connection with a randomly chosen site.
4/2
The model
H = −J
N−1
∑ σiσi+1 − I ∑
i=0
(i j)∈S
σi σ j = − ∑ Ji j σi σ j
(i, j)
Short range links: interaction strength J .
Long range shortcuts: interaction strength I .
S is the set of pairs with long shortcuts (#S = pN ).
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Previous results
Simulations (Hong et. al 2002, Herrero 2002)
Tc finite for any p > 0 ;
Mean-field critical behavior;
Analytical results (Barratt and Weigt 2000, Gitterman 2000)
Agreement over mean-field behavior;
Gitterman predicts no order for p < 1/2.
Bethe lattice representation (Dorogvtsev, Goltsev and Mendes,
2002)
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Motivation
If it is mean-field, does it have an analytical
solution?
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Solving. . . (please wait)
Spin → Bond transformation
Z = Tr{σ} exp(β ∑ Ji j σi σ j ) = Tr{σ} ∏ exp(βJi j σi σ j )
(i, j)
(i, j)
Identity (σi = ±1)
bi j
exp(βJi j σi σ j )
= 1 + σi σ j tanh(βJi j ) = ∑ tanh(βJi j )σi σ j
cosh(βJi j )
bi j =0,1
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...
Z=
∏ cosh βJi j
(i, j)
Sum over
!
bi j
(σ
σ
tanh
βJ
)
ij
∑∑∏ i j
b
σ (i, j)
σk , for a given bond configuration:
factor 2, if
∑ j bk j
is even;
0, if ∑ j bk j is odd;
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...
Z=
Z0 Tr0{b}
∏ (tanh(βJi j ))
bi j
(i, j)
Z0 = 2N ∏(i, j) cosh βJi j
Tr0 :
Trace restricted to the configurations of b0 s with ∑ j bk j
even, for all k.
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Example: Ising 1D
Each site i has 2 links: surviving bond configurations must have
bi−1,i = bi,i+1 . Only two possible bond configurations contribute to
Z:
(a) :
bi j = 0
(b) :
bi j = 1
N
∀i, j
∀i, j
N
N
Z = 2 cosh (βJ) 1 + tanh (βJ)
↑
↑
(a) (b)
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This model
Restriction: at most 1 shortcut per site.
b12 =0
b23 =1
b12 =1
b23 =0
2
1
3
b24 =1
Fix the b values of shortcuts and
the b value of one short range link
and all the b0 s are determined.
4
1
b12 =0
b23 =0
b12 =1
b23 =1
2
3
b24 =0
4
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Partition function
Z=
Nb
Z0 cI
∑ ∑
b0 {b1, ...b pN }
L[b] M[b]
tJ tI
With cI = cosh(βI), tJ = tanh(βJ), tI = tanh(βI) and
L[b]
M[b]
Number of short range links with b = 1;
Number of shortcuts with b = 1.
Z = Z0 cNI b ∑ ∑ eln Ω(M,L)+L lntJ +M lntI
b0 M,L
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Number of states
Two models solved
A
B
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Number of states
0
l1
l2
l3
l4=2s
l5 N - 1
Number of shortcuts, Nb = pN
Choose 0 ≤ 2M ≤ 2Nb points (filled) .
L/d = l2 + l4 + · · · + l2M
(li , integer, d = 1/2p)
(N − L)/d = l1 + l3 + · · · + l2M+1
L/d−1
(N−L)/d
CM−1 ×CM
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Number of states
Pick bonds, not sites! Normalizing factor:
2pN
C2M →number of ways to pick 2M sites;
pN
CM →number of ways to pick M bonds;
Ω(M, L) =
CMpN
2p(N−L) 2pL−1
×CM
CM−1
2pN
C2M
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Thermodynamical behaviour
Intensive variables
n = M/pN
l = L/N
ln Ω(M, L) + L lntJ + M lntI = N (s(l, n) + l lntJ + pn lntI )
+ O (ln N)
lim N → +∞ =⇒ steepest descent!
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Maximization
1
0.75
n
0.5
1
1
2
3
4
0.25
0
0
0.2
0.4
2
3
4
l
0.6
0.8
1
ln Z = N ln z0 + pN ln cI + pN f1
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Critical temperature (Tc)
Maximum exits at a temperature given by
(2tI + 1)tJd = 1
2
1.5
p=0.25
p=0.1
p=0.01
p=0.001
Tc 1
P
0.5
0
-10
-8
-6
-4
Log(I)
-2
F
0
2
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Asymptotic behaviour
I/Tc 1
I/Tc 1
Tc =
Tc =
ln
2J
1
p ln 3
2J
ln
J
pI ln(J/(pI))
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Partition Function
ln Z = N ln z0 + pN ln cI + pN f1
 2
 ln 4(1−l) (1−n) T < T
c
(2−2l−n)2
f1 =

0
T > Tc
n ∝ TC − T
l ∝ TC − T
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Specific heat
2.5
∞
2
C
1.5
2.5
CN(T)/C∞(T)
8192
4096
2048
1024
3
2
1.5
1
0.5
0
-5
1
0
-2.5
2.5
1/2
(T/Tc−1)N
5
0.5
1
1.2
1.4
T
1.6
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Finite size scaling (FSS)
Does this solution give the FSS behaviour?
MC Simulation
Sum
Analytic
2
c
1.5
1
0.5
-4
-2
0
ζ
2
4
CN (T )
vs ζ ≡ tN 1/2
C∞ (T )
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Analytical Result
ZN = exp[−βN( f 0 + fa )]ZFSS
N
ZFSS =
2πp
Z +∞
0
dl
Z 2
0
du g(l, u) exp(pN h(l, u))
h(l, u) ≈ −c1 (l − l ∗ )2 − c2 l(u − u∗ )2
√ 1/4 Z ∞ −(x+k2 ζ)2
πN
e
dx
√
.
ZF S S (ζ) ≈
k1
x
0
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Bethe Lattice Approach
Network is locally a Bethe lattice (loops
∼ O (log N))
0
hσ0 σ(L0 ,M)iNs (L0 , M) = e−κ(x,T )(L +M),
κmin (x) → 0 at Tc !
L0
x≡ 0
L +M
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Bethe Lattice Approach
L+M
hσ0 σ(L, M)i ∼ exp(−
)
ξ1D
L + M ∼ ln N
hσ0 σ(L, M)i ∼ N −1/ξ1D
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Conclusions
Mean-field ferromagnetic transition;
Finite size scaling not governed by ξ (violation of
hyperscaling);
Lacking
order parameter, χ, finite field;
any way of having
lcd < de f f < ucd .
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