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Exact ground states of Ising spin glasses: New experimental results with a branch-and-cut algorithm

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Abstract

In this paper we study two-dimensional Ising spin glasses on a grid with nearest neighbor and periodic boundary interactions, based on a Gaussian bond distribution, and an exterior magnetic field. We show how using a technique called branch and cut, the exact ground states of grids of sizes up to 100×100 can be determined in a moderate amount of computation time, and we report on extensive computational tests. With our method we produce results based on more than 20,000 experiments on the properties of spin glasses whose errors depend only on the assumptions on the model and not on the computational process. This feature is a clear advantage of the method over other, more popular ways to compute the ground state, like Monte Carlo simulation including simulated annealing, evolutionary, and genetic algorithms, that provide only approximate ground states with a degree of accuracy that cannot be determineda priori. Our ground-state energy estimation at zero field is −1.317.

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References

  1. J. C. Angles d'Auriac and R. Maynard,Solid State Commun. 49:785 (1984).

    Google Scholar 

  2. F. Barahona, Unpublished (1981).

  3. F. Barahona,J. Phys. A.: Math. Gen. 15:3241 (1982).

    Google Scholar 

  4. F. Barahona,Phys. Rev. B 49:12864 (1994).

    Google Scholar 

  5. F. Barahona, M. Grötschel, M. Jünger, and G. Reinelt,Oper. Res. 36:493 (1988).

    Google Scholar 

  6. F. Barahona and A. R. Mahjoub,Math. Progr. 36:157 (1986).

    Google Scholar 

  7. J. Bendisch,Physica A 202:48 (1994).

    Google Scholar 

  8. I. Bieche, R. Maynard, R. Rammal, and J. P. Uhry,J. Phys. A: Math. Gen. 13: 2553 (1980).

    Google Scholar 

  9. K. Binder and A. P. Young,Rev. Mod. Phys. 58:801 (1986).

    Google Scholar 

  10. CPLEX, CPLEX Optimization Inc. (1993).

  11. M. Grötschel, M. Jünger, and G. Reinelt, inProceedings of the Heidelberg Colloquium on Glassy Dynamics, J. L. van Hemmen and I. Morgenstern, eds. (Springer-Verlag, New York, 1987), p. 325.

    Google Scholar 

  12. U. Gropengiesser,J. Stat. Phys., in press (1995).

  13. F. Hadlock,SIAM J. Computing 4:221 (1975).

    Google Scholar 

  14. N. Jan and T. S. Ray,J. Stat. Phys. 75:1197 (1994).

    Google Scholar 

  15. N. Kawashima and M. Suzuki,J. Phys. A 25:1055 (1992).

    Google Scholar 

  16. M. Deza and M. Laurent, Preprint, University of Bonn, 1991.

  17. S. Liang,Phys. Rev. Lett. 69:2145 (1992).

    Google Scholar 

  18. I. Morgenstern and K. Binder,Phys. Rev. B 22:288 (1980).

    Google Scholar 

  19. G. I. Orlova and Y. G. Dorfman,Eng. Cybernet. 10:502 (1972).

    Google Scholar 

  20. E. S. Rodrigues and P. M. C. de Oliveira,J. Stat. Phys. 74:1265 (1994).

    Google Scholar 

  21. L. Saul and M. Kardar, Preprint, MIT (1994).

  22. D. Stauffer,J. Stat. Phys. 74:1293 (1994).

    Google Scholar 

  23. P. Sutton, D. L. Hunter, and N. Jan,J. Phys. I (Paris)4:1281 (1994).

    Google Scholar 

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Communicated by D. Stauffer

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De Simone, C., Diehl, M., Jünger, M. et al. Exact ground states of Ising spin glasses: New experimental results with a branch-and-cut algorithm. J Stat Phys 80, 487–496 (1995). https://doi.org/10.1007/BF02178370

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  • DOI: https://doi.org/10.1007/BF02178370

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