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Ising model on a 2D additive small-world network

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Abstract

In this article, we have employed Monte Carlo simulations to study the Ising model on a two-dimensional additive small-world network (A-SWN). The system model consists of a \(L\times L\) square lattice where each site of the lattice is occupied for a spin variable that interacts with the nearest neighbor and has a certain probability p of being additionally connected at random to one of its farther neighbors. The system is in contact with a heat bath at a given temperature T and it is simulated by one-spin flip according to the Metropolis prescription. We have calculated the thermodynamic quantities of the system, such as the magnetization per spin \(m_{L}\), magnetic susceptibility \(\chi _{L}\), and the reduced fourth-order Binder cumulant \(U_{L}\) as a function of T for several values of lattice size L and additive probability p. We also have constructed the phase diagram for the equilibrium states of the model in the plane T versus p showing the existence of a continuous transition line between the ferromagnetic F and paramagnetic P phases. Using the finite-size scaling (FSS) theory, we have obtained the critical exponents for the system, where varying the parameter p, we have observed a change in the critical behavior from the regular square lattice Ising model to A-SWN.

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Data availability statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The original data of this study are available from the corresponding author upon reasonable request.]

References

  1. S. Mingram, Psychol. Today 2, 60 (1967)

    Google Scholar 

  2. D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)

    Article  ADS  Google Scholar 

  3. D.J. Watts, Small Worlds (Princeton University Press, Princeton, 1999)

    Book  Google Scholar 

  4. S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Rev. Mod. Phys. 80, 1275 (2008)

    Article  ADS  Google Scholar 

  5. M.E.J. Newman, D., J. Watts. Phys. Rev. E 60, 7332 (1999)

  6. A. Barrat, M. Weigt, Eur. Phys. J. B 13, 547 (2000)

    Article  ADS  Google Scholar 

  7. M.E.J. Newman, SIAM Rev. 45, 167 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  8. R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  9. C. Moore, M. E., J. Newman. Phys. Rev. E 61, 5678 (2000)

  10. A.D. Sánchez, J.M. López, M.A. Rodríguez, Phys. Rev. Lett. 88, 048701 (2002)

    Article  ADS  Google Scholar 

  11. M. Dupont, N. Laflorencie, Phys. Rev. B 103, 174415 (2021)

    Article  ADS  Google Scholar 

  12. B.J. Zubillaga, A.L.M. Vilela, M. Wang, R. Du, G. Dong, H.E. Stanley, Sci. Rep. 12, 282 (2021)

    Article  ADS  Google Scholar 

  13. E.M.S. Luz, F.W.S. Lima, Int. J. Mod. Phys. C 18, 1251 (2007)

    Article  ADS  Google Scholar 

  14. A. Pȩkalski, Phys. Rev. E 64, 057104 (2001)

    Article  ADS  Google Scholar 

  15. H. Hong, B.J. Kim, M.Y. Choi, Phys. Rev. E 66, 018101 (2002)

  16. F.W.S. Lima, RMES 03, 000553 (2017)

    Google Scholar 

  17. C.P. Herrero, Phys. Rev. E 65, 066110 (2002)

    Article  ADS  Google Scholar 

  18. M. Gitterman, J. Phys. A 33, 8373 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  19. J.V. Lopes, Y.G. Pogorelov, J.M.B.L. dos Santos, R. Toral, Phys. Rev. E 70, 026112 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  20. X. Zhang, M. Novotny, Braz. J. Phys. 36, 3A (2006). (Rev. E, 70, 026112 (2004))

  21. K. Binder, D. W. Heermann. Monte Carlo Simulation in Statistical Physics. An Introduction, 6rd ed. (Springer, Cham, Switzerland, 2019)

  22. K. Binder, D.P. Landau, A Guide to Monte Carlo Simulations in Statistical Physics, 4th edn. (TJ International Ltd, Padstow, 2015)

    MATH  Google Scholar 

  23. L. Böttcher, H.J. Herrmann, Computational Statistical Physics, 1st edn. (Cambridge University Press, NewYork, 2021)

    Book  Google Scholar 

  24. S.-H. Tsai, S.R. Salinas, Braz. J. Phys. 28, 1 (1998)

    Article  Google Scholar 

  25. G. Ódor, Rev. Mod. Phys. 76, 663 (2004)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the Brazilian agencies CNPq, UFMT, and FAPEMAT.

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Authors and Affiliations

  1. Instituto de Física-Universidade Federal de Mato Grosso, Cuiabá, MT, 78060-900, Brazil

    R. A. Dumer & M. Godoy

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  1. R. A. Dumer
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  2. M. Godoy
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Contributions

R.A. Dumer has carried out the numerical calculations and prepared the initial form of the manuscript. All the authors contributed to the analysis and interpretation of the obtained results as well as to the final form of the manuscript.

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Correspondence to M. Godoy.

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Dumer, R.A., Godoy, M. Ising model on a 2D additive small-world network. Eur. Phys. J. B 95, 159 (2022). https://doi.org/10.1140/epjb/s10051-022-00422-w

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