Skip to main content
SpringerLink
Log in

Ground States of Two-Dimensional Ising Spin Glasses: Fast Algorithms, Recent Developments and a Ferromagnet-Spin Glass Mixture

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

Using advanced numerical approaches based on optimization algorithms, much progress has been achieved for the study of the ground-state and low-temperature behavior of two-dimensional Ising spin glasses. Recent results led to a rather good understanding of these systems in the framework of the droplet-scaling theory. In this work, a pedagogical description of such an optimization-based approach is given and a short review of corresponding recent results is presented.

Furthermore, original results are presented for a special type of system which combines a ferromagnetic sub lattice with a spin-glass sub lattice. Results for exact ground-state calculations up to system size N=1448×1448 are given. Past results of similar systems gave evidence that such a system might exhibit a spin-glass phase at finite temperatures. Nevertheless, the present results do not support this notion. But, for a suitable balance between ferromagnetic and ±J spin-glass couplings, extremely large finite-size effects occur. Thus, when considering intermediate system sizes, the system looks as if it orders. Furthermore, although the system exhibits only a discrete set of interactions, a power-law behavior for the stiffness energy can be observed clearly for a large range of system sizes. This is in contrast to past studies of systems with discrete sets of bond values.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amoruso, C., Hartmann, A.K.: Domain-wall energies and magnetization of the two-dimensional random-bond Ising model. Phys. Rev. B 70, 134425 (2004)

    Article  ADS  Google Scholar 

  2. Amoruso, C., Hartmann, A.K., Moore, M.A.: Determining energy barriers by iterated optimization: the two-dimensional Ising spin glass. Phys. Rev. B 73, 184405 (2006)

    Article  ADS  Google Scholar 

  3. Amoruso, C., Marinari, E., Martin, O.C., Pagnani, A.: Scalings of domain wall energies in two dimensional Ising spin glasses. Phys. Rev. Lett. 91, 087201 (2003)

    Article  ADS  Google Scholar 

  4. Amoruso, C., Hartmann, A.K., Hastings, M.B., Moore, M.A.: Conformal Invariance and SLE in two-dimensional Ising spin glasses. Phys. Rev. Lett. 97, 267202 (2006)

    Article  ADS  Google Scholar 

  5. Aromsawa, A., Poulter, J.: Domain wall entropy of the bimodal two-dimensional Ising spin glass. Phys. Rev. B 76(6), 064427 (2007). doi:10.1103/PhysRevB.76.064427

    Article  ADS  Google Scholar 

  6. Atisattapong, W., Poulter, J.: Excitations of the bimodal Ising spin glass on the brickwork lattice. New J. Phys. 11(6), 063039 (2009). http://stacks.iop.org/1367-2630/11/i=6/a=063039

    Article  ADS  Google Scholar 

  7. Barahona, F.: On the computational complexity of Ising spin glass models. J. Phys. A 15(10), 3241–3253 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  8. Barahona, F., Maynard, R., Rammal, R., Uhry, J.: Morphology of ground states of a two dimensional frustration model. J. Phys. A 15, 673 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  9. Barthel, W., Hartmann, A.K.: Clustering analysis of the ground-state structure of the vertex-cover problem. Phys. Rev. E 70, 066120 (2004)

    Article  ADS  Google Scholar 

  10. Bendisch, J.: Groundstate threshold pc in Ising frustration systems on 2d regular lattices. Physica A 202(1–2), 48–67 (1994)

    Article  ADS  Google Scholar 

  11. Bendisch, J.: Groundstate threshold in triangular anisotropic ±J Ising models. Physica A 245(3–4), 560–574 (1997)

    Article  ADS  Google Scholar 

  12. Bernard, D., Le Doussal, P., Middleton, A.A.: Possible description of domain walls in two-dimensional spin glasses by stochastic Loewner evolutions. Phys. Rev. B 76(2), 020403 (2007). doi:10.1103/PhysRevB.76.020403

    Article  ADS  Google Scholar 

  13. Berthier, L., Young, A.P.: Energetics of clusters in the two-dimensional Gaussian Ising spin glass. J. Phys. A, Math. Gen. 36(43), 10835 (2003). http://stacks.iop.org/0305-4470/36/i=43/a=011

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Bieche, I., Maynard, R., Rammal, R., Uhry, J.P.: On the ground states of the frustration model of a spin glass by a matching method of graph theory. J. Phys. A 13, 2553 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  15. Binder, K., Young, A.: Spin-glasses: experimental facts, theoretical concepts and open questions. Rev. Mod. Phys. 58, 801 (1986)

    Article  ADS  Google Scholar 

  16. Boettcher, S.: Low-temperature excitations of dilute lattice spin glasses. Europhys. Lett. 67(3), 453 (2004). http://stacks.iop.org/0295-5075/67/i=3/a=453

    Article  ADS  Google Scholar 

  17. Boettcher, S., Hartmann, A.K.: Reduction of two-dimensional dilute Ising spin glasses. Phys. Rev. B 72(1), 014429 (2005). doi:10.1103/PhysRevB.72.014429

    Article  ADS  Google Scholar 

  18. Bouchaud, J.P., Krzakala, F., Martin, O.C.: Energy exponents and corrections to scaling in Ising spin glasses. Phys. Rev. B 68, 224404 (2003)

    Article  ADS  Google Scholar 

  19. Bovier, A., Fröhlich, J.: A heuristic theory of the spin glass phase. J. Stat. Phys. 44, 347–391 (1986)

    Article  ADS  Google Scholar 

  20. Bray, A.J., Moore, M.A.: Lower critical dimension of Ising spin glasses: a numerical study. J. Phys. C 17, 463–468 (1984)

    Article  ADS  Google Scholar 

  21. Bray, A.J., Moore, M.A.: Chaotic nature of the spin-glass phase. Phys. Rev. Lett. 58(1), 57–60 (1987). doi:10.1103/PhysRevLett.58.57

    Article  ADS  Google Scholar 

  22. Bray, A.J., Moore, M.A.: Scaling theory of the ordered phase of spin glasses. In: van Hemmen, J.L., Morgenstern, I. (eds.) Heidelberg Colloquium on Glassy Dynamics, p. 121. Springer, Berlin (1987)

    Chapter  Google Scholar 

  23. Campbell, I.A., Hartmann, A.K., Katzgraber, H.G.: Energy size effects of two-dimensional Ising spin glasses. Phys. Rev. B 70, 054429 (2004)

    Article  ADS  Google Scholar 

  24. Cardy, J.: Scaling and Renormalization in Statistical Physics. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  25. Cardy, J.: SLE for theoretical physicists. Ann. Phys. 318, 81–118 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Carter, A.C., Bray, A.J., Moore, M.A.: Aspect-ratio scaling and the stiffness exponent for Ising spin glasses. Phys. Rev. Lett. 88, 077201 (2002)

    Article  ADS  Google Scholar 

  27. Cook, W., Rohe, A.: Calculation of minimum-weighted perfect matchings via the blossom4 extension to the Concorde library. http://www2.isye.gatech.edu/wcook/blossom4/

  28. Cook, W.J., Cunningham, W.H., Pulleyblank, W.R., Schriever, A.: Combinatorial Optimization. Wiley, New York (1998)

    MATH  Google Scholar 

  29. Cormen, T.H., Clifford, S., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press, Cambridge (2001)

    MATH  Google Scholar 

  30. Derigs, U., Metz, A.: Solving (large scale) matching problems combinatorially. Math. Program. 50, 113 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  31. Edwards, S.F., Anderson, P.W.: Theory of spin glasses. J. Phys. F, Met. Phys. 5, 965 (1975)

    Article  ADS  Google Scholar 

  32. Fisch, R.: Finite-size scaling of the domain wall entropy distributions for the 2d±J Ising spin glass. J. Stat. Phys. 125, 793 (2006)

    Article  ADS  Google Scholar 

  33. Fisch, R.: Aspect-ratio scaling of domain wall entropy for the 2d±J Ising spin glass. J. Stat. Phys. 130, 561–569 (2008). http://dx.doi.org/10.1007/s10955-007-9436-4

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Fisch, R., Hartmann, A.K.: Ground-state and domain-wall energies in the spin-glass region of the 2d±J random-bond Ising model. Phys. Rev. B 75, 174415 (2007)

    Article  ADS  Google Scholar 

  35. Fischer, K.H., Hertz, J.A.: Spin Glasses. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

  36. Fisher, D.S., Huse, D.A.: Ordered phase of short-range Ising spin-glasses. Phys. Rev. Lett. 56, 1601 (1986)

    Article  ADS  Google Scholar 

  37. Fisher, D.S., Huse, D.A.: Equilibrium behavior of the spin-glass ordered phase. Phys. Rev. B 38, 386 (1988)

    Article  ADS  Google Scholar 

  38. Galluccio, A., Loebl, M., Vondrák, J.: New algorithm for the Ising problem: partition function for finite lattice graphs. Phys. Rev. Lett. 84(26), 5924–5927 (2000). doi:10.1103/PhysRevLett.84.5924

    Article  ADS  Google Scholar 

  39. Goldenfeld, N.: Lectures on Phase Transitions and the Renormalization Group. Addison-Wesley, Reading (1992)

    Google Scholar 

  40. Hartmann, A.K.: Cluster-exact approximation of spin glass ground states. Physica A 224, 480–488 (1999)

    Article  ADS  Google Scholar 

  41. Hartmann, A.K.: Scaling of stiffness energy for three-dimensional ±J Ising spin glasses. Phys. Rev. E 59, 84 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  42. Hartmann, A.K.: No spin-glass transition in the mobile-bond model. Phys. Rev. B 67(21), 214404 (2003). doi:10.1103/PhysRevB.67.214404

    Article  ADS  Google Scholar 

  43. Hartmann, A.K.: Domain walls, droplets and barriers in two-dimensional Ising spin glasses. In: Janke, W. (ed.) Rugged Free Energy Landscapes, Lecture Notes in Physics, pp. 67–106. Springer, Heidelberg (2007)

    Google Scholar 

  44. Hartmann, A.K.: Droplets in the two-dimensional ±J Ising spin glass. Phys. Rev. B 77, 144418 (2008)

    Article  ADS  Google Scholar 

  45. Hartmann, A.K.: Practical Guide to Computer Simulations. World Scientific, Singapore (2009)

    MATH  Google Scholar 

  46. Hartmann, A.K., Campbell, I.A.: Ordered phase in the two-dimensional randomly coupled ferromagnet. Phys. Rev. B 63, 094423 (2001)

    Article  ADS  Google Scholar 

  47. Hartmann, A.K., Moore, M.A.: Corrections to scaling are large for droplets in two-dimensional spin glasses. Phys. Rev. Lett. 90, 12720 (2003)

    Article  Google Scholar 

  48. Hartmann, A.K., Moore, M.A.: Generating droplets in two-dimensional Ising spin glasses by using matching algorithms. Phys. Rev. B 69, 104409 (2004)

    Article  ADS  Google Scholar 

  49. Hartmann, A.K., Rieger, H.: Optimization Algorithms in Physics. Wiley-VCH, Weinheim (2001)

    Book  Google Scholar 

  50. Hartmann, A.K., Young, A.P.: Lower critical dimension of Ising spin glasses. Phys. Rev. B, Condens. Matter Mater. Phys. 64, 180404 (2001)

    Article  ADS  Google Scholar 

  51. Hartmann, A.K., Young, A.P.: Large-scale, low-energy excitations in the two-dimensional Ising spin glass. Phys. Rev. B, Condens. Matter Mater. Phys. 65, 094419 (2002)

    Article  ADS  Google Scholar 

  52. Hartmann, A.K., Bray, A.J., Carter, A.C., Moore, M.A., Young, A.P.: The stiffness exponent of two-dimensional Ising spin glasses for non-periodic boundary conditions using aspect-ratio scaling. Phys. Rev. B 66, 224401 (2002)

    Article  ADS  Google Scholar 

  53. Kawashima, N.: Fractal droplets in two-dimensional spin glass. J. Phys. Soc. Jpn. 69, 987 (2000)

    Article  ADS  Google Scholar 

  54. Kawashima, N., Aoki, T.: Zero-temperature critical phenomena in two-dimensional spin glasses. J. Phys. Soc. Jpn. 69(Suppl. A), 169 (2000)

    Google Scholar 

  55. Kawashima, N., Rieger, H.: Finite-size scaling analysis of exact gss for + or −j spin glass models in two dimensions. Europhys. Lett. 39, 85 (1997)

    Article  ADS  Google Scholar 

  56. Korte, B., Vygen, J.: Combinatorial Optimization—Theory and Algorithms. Springer, Berlin (2000)

    MATH  Google Scholar 

  57. Landau, D.P., Binder, K.: Monte Carlo Simulations in Statistical Physics. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  58. Landry, J.W., Coppersmith, S.N.: Ground states of two-dimensional ±J Edwards-Anderson spin glasses. Phys. Rev. B 65, 134404 (2002)

    Article  ADS  Google Scholar 

  59. McMillan, W.L.: Domain-wall renormalization-group study of the two-dimensional random Ising model. Phys. Rev. B 29(7), 4026–4029 (1984). doi:10.1103/PhysRevB.29.4026

    Article  ADS  Google Scholar 

  60. McMillan, W.L.: Scaling theory of Ising spin glasses. J. Phys. C 17, 3179 (1984)

    Article  ADS  Google Scholar 

  61. Mehlhorn, K., Näher, S.: The LEDA Platform of Combinatorial and Geometric Computing. Cambridge University Press, Cambridge (1999). http://www.algorithmic-solutions.de

    Google Scholar 

  62. Melchert, O., Hartmann, A.K.: Fractal dimension of domain walls in two-dimensional Ising spin glasses. Phys. Rev. B 76, 174411 (2007)

    Article  ADS  Google Scholar 

  63. Melchert, O., Hartmann, A.K.: Scaling behavior of domain walls at the T=0 ferromagnet to spin-glass transition. Phys. Rev. B 79, 184402 (2009)

    Article  ADS  Google Scholar 

  64. Melchert, O., Hartmann, A.K.: A dedicated algorithm for calculating ground states for the triangular random bond Ising model. Comput. Phys. Commun. 182, 1828–1832 (2011)

    Article  ADS  Google Scholar 

  65. Mézard, M., Parisi, G., Virasoro, M.: Spin Glass Theory and Beyond. World Scientific, Singapore (1987)

    MATH  Google Scholar 

  66. Middleton, A.A.: Computational complexity of determining the barriers to interface motion in random systems. Phys. Rev. E 59(3), 2571–2577 (1999). doi:10.1103/PhysRevE.59.2571

    Article  ADS  Google Scholar 

  67. Middleton, A.A.: Numerical investigation of the thermodynamic limit for ground states in models with quenched disorder. Phys. Rev. Lett. 83(8), 1672–1675 (1999). doi:10.1103/PhysRevLett.83.1672

    Article  ADS  Google Scholar 

  68. Middleton, A.A.: Energetics and geometry of excitations in random systems. Phys. Rev. B 63(6), 060202 (2001). doi:10.1103/PhysRevB.63.060202

    Article  MathSciNet  ADS  Google Scholar 

  69. Newman, C.M., Stein, D.L.: Finite-dimensional spin glasses: states excitations and interfaces. Ann. Henri Poincaré 4, S497–S503 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  70. Newman, M.E.J., Barkema, G.T.: Monte Carlo Methods in Statistical Physics. Clarendon Press, Oxford (1999)

    MATH  Google Scholar 

  71. Ohzeki, M., Nishimori, H.: Analytical evidence for the absence of spin glass transition on self-dual lattices. J. Phys. A, Math. Theor. 42(33), 332001 (2009). http://stacks.iop.org/1751-8121/42/i=33/a=332001

    Article  MathSciNet  Google Scholar 

  72. Ozeki, Y.: Ground state properties of the ±J Ising model in two dimensions. J. Phys. Soc. Jpn. 59, 3531 (1990)

    Article  ADS  Google Scholar 

  73. Palassini, M., Young, A.P.: Evidence for a trivial ground-state structure in the two-dimensional Ising spin glass. Phys. Rev. B 60(14), R9919–R9922 (1999). doi:10.1103/PhysRevB.60.R9919

    Article  ADS  Google Scholar 

  74. Palassini, M., Young, A.P.: Nature of the spin glass state. Phys. Rev. Lett. 85, 3017 (2000)

    Article  ADS  Google Scholar 

  75. Palmer, R.G., Adler, J.: Ground states of large samples of two-dimensional Ising spin glasses. Int. J. Mod. Phys. C 10, 667 (1999)

    Article  ADS  Google Scholar 

  76. Pardella, G., Liers, F.: Exact ground states of large two-dimensional planar Ising spin glasses. Phys. Rev. E 78, 056705 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  77. Picco, M., Ritort, F., Sales, M.: Statistics of lowest droplets in two-dimensional Gaussian Ising spin glasses. Phys. Rev. B 67(18), 184421 (2003). doi:10.1103/PhysRevB.67.184421

    Article  ADS  Google Scholar 

  78. Poulter, J., Blackman, J.A.: Exact algorithm for spin-correlation functions of the two-dimensional ±J Ising spin glass in the ground state. Phys. Rev. B 72, 104422 (2005)

    Article  ADS  Google Scholar 

  79. Ramirez-Pastor, A.J., Romá, F., Nieto, F., Vogel, E.E.: Effect of the ground-state structure on order parameters in ±J Ising lattices. Physica A 336(3–4), 454–460 (2004). doi:10.1016/j.physa.2003.12.038

    Article  ADS  Google Scholar 

  80. Rieger, H., Santen, L., Blasum, U., Diehl, M., Jünger, M., Rinaldi, G.: The critical exponents of the two-dimensional Ising spin glass revisited: exact ground-state calculations and Monte Carlo simulations. J. Phys. A 29, 3939 (1996)

    Article  ADS  MATH  Google Scholar 

  81. Risau-Gusman, S., Romá, F.: Fractal dimension of domain walls in the Edwards-Anderson spin glass model. Phys. Rev. B 77(13), 134435 (2008). doi:10.1103/PhysRevB.77.134435

    Article  ADS  Google Scholar 

  82. Romá, F., Risau-Gusman, S., Ramirez-Pastor, A.J., Nieto, F., Vogel, E.E.: Influence of the ground-state topology on domain-wall energy in the Edwards-Anderson ±J spin glass model. Phys. Rev. B 75(2), 020402 (2007). doi:10.1103/PhysRevB.75.020402

    Article  ADS  Google Scholar 

  83. Romá, F., Risau-Gusman, S., Ramirez-Pastor, A.J., Nieto, F., Vogel, E.E.: The ground state energy of the Edwards-Anderson spin glass model with a parallel tempering Monte Carlo algorithm. Physica A 388(14), 2821–2838 (2009). doi:10.1016/j.physa.2009.03.036

    Article  ADS  Google Scholar 

  84. Romá, F., Risau-Gusman, S., Ramirez-Pastor, A.J., Nieto, F., Vogel, E.E.: Ground-state topology of the Edwards-Anderson ±J spin glass model. Phys. Rev. B 82(21), 214401 (2010). doi:10.1103/PhysRevB.82.214401

    Article  ADS  Google Scholar 

  85. Sedgewick, R.: Algorithms in C. Addison-Wesley, Reading (1990)

    MATH  Google Scholar 

  86. Sherrington, D., Kirkpatrick, S.: Solvable model of a spin glass. Phys. Rev. Lett. 35, 1792 (1975)

    Article  ADS  Google Scholar 

  87. Simone, C., Diehl, M., Jünger, M., Mutzel, P., Reinelt, G., Rinaldi, G.: Exact ground states of Ising spin glasses: New experimental results with a branch-and-cut algorithm. J. Stat. Phys. 80, 487–496 (1995). http://dx.doi.org/10.1007/BF02178370

    Article  ADS  MATH  Google Scholar 

  88. Simone, C., Diehl, M., Jünger, M., Mutzel, P., Reinelt, G., Rinaldi, G.: Exact ground states of two-dimensional ±J Ising spin glasses. J. Stat. Phys. 84, 1363–1371 (1996). http://dx.doi.org/10.1007/BF02174135

    Article  ADS  Google Scholar 

  89. Swamy, M.N.S., Thulasiraman, K.: Graphs, Networks and Algorithms. Wiley, New York (1991)

    Google Scholar 

  90. Thomas, C.K., Huse, D.A., Middleton, A.A.: Zero and low temperature behavior of the two-dimensional ±J Ising spin glass. Preprint arXiv:1103.1946 (2011)

  91. Thomas, C.K., Middleton, A.A.: Matching kasteleyn cities for spin glass ground states. Phys. Rev. B 76, 220406R (2007)

    Article  ADS  Google Scholar 

  92. Weigel, M., Johnston, D.: Frustration effects in antiferromagnets on planar random graphs. Phys. Rev. B 76(5), 054408 (2007). doi:10.1103/PhysRevB.76.054408

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Physik, Universität Oldenburg, 26111, Oldenburg, Germany

    Alexander K. Hartmann

Authors
  1. Alexander K. Hartmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander K. Hartmann.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hartmann, A.K. Ground States of Two-Dimensional Ising Spin Glasses: Fast Algorithms, Recent Developments and a Ferromagnet-Spin Glass Mixture. J Stat Phys 144, 519–540 (2011). https://doi.org/10.1007/s10955-011-0272-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-011-0272-1

Keywords

Search

Navigation