Andrei Bautu and Elena Bautu- Searching Ground States of Ising Spin Glasses with Particle Swarms

8/3/2019 Andrei Bautu and Elena Bautu- Searching Ground States of Ising Spin Glasses with Particle Swarms

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SEARCHING GROUND STATES OF ISING SPIN GLASSESWITH PARTICLE SWARMS

ANDREI BUTU1, ELENA BUTU2

1 Mircea cel Btrn Naval Academy, Constantza, 900218, Romania, [email protected] Ovidius University, Constantza, 900527, Romania, [email protected]

Received September 12, 2006

This paper presents a method for searching ground states of Ising spin glasses.

The Ising model is one of the most commonly used because of its simplicity and itsaccuracy in representing real problems. We tackle the problem of finding groundstates with particle swarm optimization (PSO), a population-based stochastic optimi-zation technique inspired by social behavior of bird flocking. The paper is organizedas follows: the first section briefly presents Ising spin glasses. Section 2 describes thebasic principles of the particle swarm optimization meta-heuristic. Section 3 presentsexperimental results, followed by some conclusion and future work in Section 4.

Key words:spin glass, ground state, Ising model, particle swarm optimization.

1. INTRODUCTION

One of the dominant themes in the history of physics in the last century has

been the effort to understand condensed states of matter. This began with verysimple systems and has gradually developed to include more and more complexand subtle states and phenomena. Spin glasses are the current frontier in thisdevelopment, the most complex kind of condensed state encountered so far insolid state physics [1]. In order to present our experiments and results, someintroduction to the Ising model of spin glasses and particle swarm optimization(PSO) is due.

1.1. SPIN GLASSES

Each atom that composes matter carries a spin generated by the motion of

the electrons around its nucleus, which in turn generates a microscopic magneticfield around the atom. To simplify matters, assume that the spin can take onlytwo opposite directions: up and down. At high temperature, i.e. above somematerial dependent critical temperature, the motion of spins is so erratic that at

Paper presented at the 7th International Balkan Workshop on Applied Physics, 57 July2006, Constana, Romania.

Rom. Journ. Phys., Vol. 52, Nos. 34, P. 337342, Bucharest, 2007


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338 Andrei Butu, Elena Butu 2

any time, about half of them are pointing up and half are pointing down, and theindividual microscopic magnetic fields generated by the spins cancel each otherout, resulting in zero macroscopic magnetization. In ferromagnets (materialscapable of attracting pieces of iron placed in their vicinity), below the criticaltemperature, each spin has a tendency to align with the spins in its neighborhood,and the individual microscopic magnetic fields sum up to create a macroscopicmagnetic field. By contrast, in spin glasses, only some pairs of neighboring spinsprefer to be aligned, while the others prefer to be anti-aligned, resulting in twotypes of interactions: ferromagnetic and anti-ferromagnetic. Because of this mixof interactions these systems are called disordered [2]. Some examples of spinglasses are disordered magnetic alloys, i.e. metals containing random magnetic

impurities, such as AuFe or CuMn.

1.2. THE ISING MODEL

The Ising model was named after the German physicist Ernst Ising. In1925, he introduced in his PhD dissertation this mathematical model for phasetransitions, i.e. abrupt changes of state. The goal of the Ising model is to explainhow long-range correlations are generated by local interactions.

We will follow the presentation of the model found in [3]. The Ising modelcan be formulated in any dimension in graph theoretic terms. Vertices of thegraph G = (V,E) represent atoms in a crystal and edges represent bonds between

adjacent atoms. In the classic model, the graph is the standard square lattice inone, two, or three dimensions, so that each atom has two, four, or six nearestneighbors, respectively. Each edge has assigned a coupling constant, denotedby { }, ,ijJ J J where ij is an edge. If we denote by S the state of the whole

system, then each vertex i can be in one of two states. The interaction betweenneighboring vertices i andj contributes an amount of ij i jJ S S to the total energy

H(the Hamiltonian) of the system in the state S, so that

( )

= ij i jij E

H S J S S (1)

IfJij is positive, then having neighbors in the same state (Si = Sj) decreases

the total energy. If all the coupling constants are positive, the lowest-energy con-figuration of the system is with all its vertices in the same state. If the couplingconstants are a mix of positive and negative numbers as they are for the classof Ising models known as spin glasses finding the ground state can be quite afrustrating experience. The most common configurations in the literature are2-dimensional Ising spin glasses on a grid with nearest neighbor interactions.Periodic boundary conditions are often used as a way to approximate infinitesize spin glasses on a finite number of spins.


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3 Ground states of Ising spin glasses with particle swarms 339

In 1982, Francisco Barahona showed that finding a ground state for thethree-value coupling constant (Jij {1, 0, 1}) on a cubic grid is equivalent tofinding a maximum set of independent edges in a graph for which each vertex hasdegree 3. He also showed that computing the minimum value of the Hamiltonian

of a spin glass with an external magnetic field, ( ) ,i j iij E i V H S S S S = is equivalent to solving the problem of finding the largest set of disconnectedvertices in a planar, degree-3 graph. Hence, Barahona uncovered that findingground states for three-dimensional spin glasses on the standard square latticeand for planar spin glasses with an external field are NP-complete problems.Based on the work of Barahona, Sorin Istrail showed that the essential ingredient

in the NP-completeness of the Ising model is non planarity [4].

2. PARTICLE SWARM OPTIMIZATION

Particle Swarm Optimization (PSO) is a meta-heuristic inspired by thesocial behavior of bird flocking or fish schooling, introduced in 1995 by JamesKennedy and Russel Eberhart [5]. Nature inspired algorithms based on Darwin'stheory of evolution solve optimization problems by evolving a set of solutionsthat compete for survival. In a similar fashion, PSO tries to solve optimizationproblems by using a set (called swarm) of potential solutions (called particles).However, in a PSO algorithm all particles survive and share gatheredinformation for the welfare of the swarm [6]. Therefore the driving force of PSOis the (collective) swarm intelligence. The function describing the problem wetackle generates a landscape in which particles improve their quality of life byflying toward more promising areas.

The process starts with a swarm of particles randomly scattered around thesearch space that move with each iteration toward more promising locations.During the flight, each particle steers, i.e. updates its speed and position,according to its own past experience and that of its most successful neighbor.The binary decision form of the PSO [7] can be applied directly in our research.In this case, the component i of the position vector of a particle encodes the stateof the spin glass i of the system (0 means down, 1 means up), while the

component i of the velocity vector determines the confidence that the spin glass iof the system should be up.

On each iteration, each particle updates its velocity and position vectors(i.e. its confidence and decision) based on the formulae

( ) ( )1 1 1 1 2 2 1t t p t g t v v c p p c p p = + + (2)

( )( )max maxmax , min ,t tv v v v= (3)


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311, if

1 exp( )

0, otherwisett

vp


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5 Ground states of Ising spin glasses with particle swarms 341

Fig. 2 Energy (per spin) with respect to the size (3 dimensional cases).

5% of the number of particles, and maximum number of iterations is 1000. Foreach system, the particle swarm algorithm ran 30 times, since this is a probabi-listic algorithm. The average results of the PSO algorithm are presented in Fig. 1and Fig. 2 by comparison with the known ground states of the test configurations.

4. CONCLUSIONS

We have used a PSO algorithm to estimate the ground states of various 2Dand 3D Ising spin glasses on a square lattice with periodic boundary conditions.Preliminary results look very promising, especially because this approach can beextended easily to higher dimensional systems. A further investigation ofoptimal parameters for the PSO algorithm is due, in the hope that PSO will playan important role in the solution of spin glasses and related problems inStatistical Physics.

Acknowledgements. This paper was supported by the CNMP CEEX-05-D11-25/2005 Grant.

REFERENCES

1. C. D. Simone, M. Diehl, M. Junger, P. Mutzel, G. Reinelt, G. Rinaldi, Exact ground states of Ising spin glasses: New experimental results with a branch and cut algorithm, Journal ofStatistical Physics, 80(3), pp. 487496, 1995.

2. F. den Hollander, F. Toninelli, Spin glasses: A mystery about to be solved, Eur. Math. Soc.

Newsl., 56, pp. 1317, 2005.3. B. A. Cipra, The Ising model is NP-complete, SIAM News, 33(6), 2000.4. S. Istrail, Universality of intractability of the partition functions of the Ising model across

non-planar lattices, in Proceedings of the 32nd ACM Symposium on the Theory of Computing(STOC00), pp. 8796, ACM Press, 2000

5. R. C. Eberhart, J. Kennedy, Y. Shi, Swarm Intelligence, Morgan Kaufmann, 2001.6. M. Clerc, Particle Swarm Optimization, HERMES Science Publishing Ltd, London, April 2006.7. J. Kennedy, R. C. Eberhart, A discrete binary version of the particle swarm algorithm,

in Proceedings of the World Multiconference on Systemics, Cybernetics and Informatics,vol. 5, pp. 41044109, Piscataway NJ, USA, October 1997, IEEE Press.


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8. C. A. Anderson, K. F. Jones, J. Ryan, A two-dimensional genetic algorithm for the Isingproblem. Complex Systems, vol. 5, pp. 327333, 1991.

9. S. Fischer, I. Wegener, The Ising model on the ring: Mutation versus recombination. inGenetic and Evolutionary Computation GECCO-2004, Part I, volume 3102 of LectureNotes in Computer Science, pp. 11131124, Springer-Verlag, 2004

10. A. Prgel-Bennett, J. L. Shapiro, An Analysis of Genetic Algorithms Using StatisticalMechanics. Phys. Rev. Lett., vol. 72(9), pp. 13051309, 1994.

11. A. Prgel-Bennett, J. L. Shapiro, The Dynamics of a Genetic Algorithm for Simple RandomIsing Systems. Physica D, 1462, pp. 76114, 1997.

12. P. Sutton, D. Hunter, N. Jan, The ground state energy of the J spin glass from the geneticalgorithm. Journal de Physique I France, 4, pp. 12811285, 1994.

13. M. Wronski, A. Hartmann, Cluster-exact approximation of spin glass ground states. Physica A,vol. 224(3), pp. 480488, 1996.

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Andrei Bautu and Elena Bautu- Searching Ground States of Ising Spin Glasses with Particle Swarms