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The ground state energy of the ±J spin glass from the genetic algorithm
P. Sutton, D. Hunter, N. Jan
To cite this version:
P. Sutton, D. Hunter, N. Jan. The ground state energy of the±J spin glass from the genetic algorithm.
Journal de Physique I, EDP Sciences, 1994, 4 (9), pp.1281-1285. �10.1051/jp1:1994112�. �jpa-00246991�
Classification Physics Abstracts
02.50 05.50
Short Communication
The ground state energy of trie +J spin glass &om trie genetic algorithm
P. Sutton, Dl. Hunter and N. Jan
Box 5000, St. Francis Xavier University, Antigonish, Nova Scotia, Canada, B2G 2W5
(Received 27 June 1994, accepted 4 July1994)
Abstract. The ground state energy of the +J 2-dimensional square and 3-dimensional cubic spin glasses are determined by trie Genetic Algorithm. System sizes of linear dimensions 6 to 20
are considered in 2-dimensions and from 4 to 9 in 3-dimensions. We estimate that the ground
state energy of trie infinite system is -1.400 + 0.005 (2d) and -1.765 + 0.01 (3d) in units of J.
The Genetic Algorithm [1-5, 15, 16] is the term used to describe a computing strategy which mimics evolutionary principles in the search for an optimal solution. A population of programs,
or as in our case, configurations of the spm glass, are selected with a certain probability for
contributing off-Springs to the subsequent generation. The selection process is weighted by
the individua1's 'fitness' as compared to the 'fitness' of the other members of the population.
Mutation and 'sex' are included in the reproduction process and the new members' fitnesses
are weighted in the selection for the subsequent generation. We will descnbe below the actual implementation used for the spin glass problem but we refer the reader to references [1-5] for details of the algorithm and its application to a vast array of distinct, novel and challenging problems.
The determination of the ground state energy of the 2-dimensional and to a fesser extent the 3-dimensional +J spin glass has attracted attention over an extended period and has been
a testing ground for several strategies [6-14]. It is generally believed that the ground state energy of the square lattice is -1.403 + 0.005 in units of J [loi and -1.76 or -1.80 [14] for the 3-dimensional simple cubic lattice. We describe below our implementation of the Genetic
Algonthm for this NP problem.
A random arrangement of ferromagnetic and antiferromagnetic bonds are placed on the lattice and a random population (300 to 3000) of spins configurations also created at the start.
The energies of these configurations are measured for the given bond arrangement. The next
generation of the population is constructed in the following manner: potent1al members are selected from the present generation with a probability proportional to their energy dilference
1282 JOURNAL DE PHYSIQUE I N°9
Ground State Energy of the 2-d Spin Glass
-1.3
o -1.32
o
-1.34
-1.36
~
fi -1.38
o
° o
-1.4
~
-l.42
-1.44
o o.05 o-i o,15 o.2 o.25
1/L
Fig. 1. The ground state energy of the 2-dimensional +J spm glass as a function of inverse system size, 1/L. We extrapolate the ground state energy of the co system ta -1.400 + 0.005 in units of J.
with the maximum energy; selected pairs have a probability of 0.5 of interchanging segments
of spin states this mimics the genetic mixing of mater1al; ten percent of the spins are given
random mutations flipped without consideration to increase or decrease in the energy of the system. Next ALL mutations leading to a decrease of the energy are allowed iii, 12]; we visit every site of the lattice in a type-writer fashion and the spin is flipped if this leads to a decrease of the energy of the system otherwise the state of the spin is left unchanged. A mutation is
simply a flip of the spin from its present state to the opposite state. We have now constructed the population of the first generation. This process is continued for a finite set of generations
which we estimate is necessary to find the lowest energy state. We make a generous estimate of this number about ten times the value indicated in figure 2. We store the lowest value of
the energy observed dunng the evolution of the configurations and this is the estimate of the
ground state energy for the configuration of bonds.
We checked that the algorithm is able to successfully accomplish its task: for the smaller system sizes (L
= 5) we enumerated ail the states of the 25 spins (2-dimensions) and checked the ground state energy for a few particular bond arrangements; for larger systems we repeated the search several times with the saine bond configuration but starting with dilferent initial random populations and a different sequence of random numbers. We always arrived at the
same lowest energy in ail our tests. We also increased the number of generations and compared
the final average ground state energy with the previous estimate. There was always a measure
of agreement within the expected statistica1fluctuations; the results were less thon a standard
Generations vs L for 2-d Spin Glass 1000
o
ioo
o 10
o
o
1
6 8 10 12 14 16 18 20 22 24
Fig. 2. The average number of generations required ta find the ground store energy for a fixed population of 300 as a function of system size, L for the 2-dirnensional spm glass.
deviation apart.
Figure 1 shows the data for the average ground 8tate energy for system sizes varying from L
= 4 to L
= 20 in 2-dimensions plotted vs. lIL. We extrapolate the ground state energy of the infinite 8ystem to -1.400 + 0.005 in unit8 of J. Figure 2 8hows the average number of generations required to determine the ground state energy for a population of 300. The
number of generations appear to increase exponentially as the system size for this fixed popu- lation. Another measure of the optimal property of this search method is to compare the total number of possible configurations with a rough estimate of low energy states. The number of
configurations for L = 10 is 2~°° and if we approximate that there are 10~ low energy states then one would expect to generate of the order 10~~ configurations in a random search for one of these low lying states. In fact the genetic algorithm finds the solution for L = 10 with an average of 3500 configurations. Figure 3 shows the ground state energy of the 3-dimensional system as a function of1IL. We estimate that the ground state energy of the infinite system is -1.765 + O.ol in agreement with some of the earlier results.
In summary, we have used the genetic algorithm to determine the ground state energy of the 2-dimensional and 3-dimensional +J spin glasses on the square and cubic lattices. The result agrees with earlier results reported in the literature [10, 14]. An davantage of our approach
is that with little effort the method may be extended to higher dimensional systems. This
luxury18 not afforded by some of the other methods. We have not attempted to calculate the ground 8tate energy of higher dimensional systems because of our rather limited computer
1284 JOURNAL DE PHYSIQUE I N°9
Ground State EneroY of the 3-d Spin Glass
-1.7
-1.72
-1.74
~
c
-1.76 o °
o
o
-1.78
-1.8
0 0.05 ù-1 0.15 0.2 0.25
1/L
Fig. 3. The ground store energy on trie 3-dimensional +J spin glass as a function of inverse sys;tem
size, 1/L. We extrapolate trie ground store energy of the co system to -1.765 + 0.01 in umts of .1.
resources three workstations took about six weeks to complete this project. There are certain
features of our implementation that require further investigation: optimal cntenon for selecting
members for the subsequent generation (our condition depended linearly on the energy of the
configuration) and whether it is more eflicient to optimise random sections of a large system rather that attempting, as we did, to optimise the complete system. We expect that the Genetic Algorithm will play a more important rote m the solution of spin glasses and rel;~ted
problems in Statistical Physics.
This research is supported in part by NSERC.
References
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