1999年日本物理学会（盛岡）

A Multicanonical
Monte Carlo Study of
the 3D ±J Spin Glass
*,**
Naomichi Hatano
**
James E. Gubernatis
*
Aoyama Gakuin University
**
Los Alamos National Laboratory
The ±J Model
(a ) ( a )
Jij s i s j
a 1,2 i, j 
Jij  1 (quenched)



H
Overlap Order Parameter
N
1
(1)
(
2
)
q  s i s i
N i 1
replica 1
replica 2
Low-Temperature Phase
Mean-field picture
Droplet picture
(Parisi et al.)
(Fisher & Huse)
L 
T
Order-Parameter Distribution
5
T = 0.3
L=6
-1
4
L=8
3
P(q)
10
L=4
10
2
1
-2
0
0.2
0.4
q
0.6
0.8
1
0
At higher temperatures
0.5
P(q)
0.4
2
T = 0.7
1.5
0.3
1
0.2
0.1
0
0
0.5
L=8
L=6
L=4
0.2
0.4
q
0.6
0.8
1
0
Flattening the histogram
Microcanonical: Few low-energy states
Canonical: Few high-energy states
Trapped by local minima
Multicanonical: Samples uniformly
Auto-Correlation Time
Making the bivariate histogram h(E,q) flat
10

104
3
10
102
E
q
1
10
2
10
N=L
3

 (*)
3
10
(*) Monovariate multicanonical
Berg & Janke, PRL80, 4771 (‘98)
5
Monovariate Multicanonical
L=6
Histogram
30000
20000
-1.8
-1.6
-1.4
E / L3
-1.2
-1
Density of States D(E,q)
N 
D(E,q)
Summary
• Bivariate Multicanonical
Monte Carlo Method
 Correlation Time: N
• Low-Temperature
Phase
 The result of P(q; T=0.3)
suggests the droplet picture
Aoyama+ Project
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http://www.phys.aoyama.ac.jp/~aoyama+

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1999年日本物理学会（盛岡）

Title Robustness in Biological Systems : Design