Phase boundary near a magnetic percolation transition

Abstract

Motivated by recent experimental observations [Rowley et al. in Phys Rev 96:020407, 2017] on hexagonal ferrites, we revisit the phase diagrams of diluted magnets close to the lattice percolation threshold. We perform large-scale Monte Carlo simulations of XY and Heisenberg models on both simple cubic lattices and lattices representing the crystal structure of the hexagonal ferrites. Close to the percolation threshold \(p_\mathrm{c}\), we find that the magnetic ordering temperature \(T_\mathrm{c}\) depends on the dilution p via the power law \(T_\mathrm{c} \sim |p-p_\mathrm{c}|^\phi \) with exponent \(\phi =1.09\), in agreement with classical percolation theory. However, this asymptotic critical region is very narrow, \(|p-p_\mathrm{c}| \lesssim 0.04\). Outside of it, the shape of the phase boundary is well described, over a wide range of dilutions, by a nonuniversal power law with an exponent somewhat below unity. Nonetheless, the percolation scenario does not reproduce the experimentally observed relation \(T_\mathrm{c} \sim (x_\mathrm{c} -x)^{2/3}\) in PbFe\(_{12-x}\)Ga\(_x\)O\(_{19}\). We discuss the generality of our findings as well as implications for the physics of diluted hexagonal ferrites.

Graphic abstract

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The datasets generated and analyzed during the current study are available from the corresponding author upon reasonable request.]

Notes

  1. 1.

    The lattice in question is the lattice of exchange interactions between the Fe ions.

  2. 2.

    The crossover exponent has also been computed within an expansion in powers of \(\epsilon =6-d\) yielding \(\phi = 1+ \epsilon /42\) to first order in \(\epsilon \) [23, 24]. The resulting value, \(\phi = 1.071\), is surprisingly close to the best numerical estimate \(\phi =1.12(2)\).

References

  1. 1.

    G. Grinstein, A. Luther, Phys. Rev. B 13, 1329 (1976)

    ADS  Article  Google Scholar 

  2. 2.

    D.S. Fisher, Phys. Rev. Lett. 69, 534 (1992)

    ADS  Article  Google Scholar 

  3. 3.

    D.S. Fisher, Phys. Rev. B 51, 6411 (1995)

    ADS  Article  Google Scholar 

  4. 4.

    R.B. Griffiths, Phys. Rev. Lett. 23, 17 (1969)

    ADS  Article  Google Scholar 

  5. 5.

    M. Thill, D.A. Huse, Phys. A 214, 321 (1995)

    Article  Google Scholar 

  6. 6.

    A.P. Young, H. Rieger, Phys. Rev. B 53, 8486 (1996)

    ADS  Article  Google Scholar 

  7. 7.

    T. Vojta, Phys. Rev. Lett. 90, 107202 (2003)

    ADS  Article  Google Scholar 

  8. 8.

    R. Sknepnek, T. Vojta, Phys. Rev. B 69, 174410 (2004)

    ADS  Article  Google Scholar 

  9. 9.

    G. Schehr, H. Rieger, Phys. Rev. Lett. 96, 227201 (2006)

    ADS  Article  Google Scholar 

  10. 10.

    J.A. Hoyos, T. Vojta, Phys. Rev. Lett. 100, 240601 (2008)

    ADS  Article  Google Scholar 

  11. 11.

    T. Vojta, J. Phys. A 39, R143 (2006)

    ADS  Article  Google Scholar 

  12. 12.

    T. Vojta, J. Low Temp. Phys. 161, 299 (2010)

    ADS  Article  Google Scholar 

  13. 13.

    T. Vojta, Ann. Rev. Condens. Mat. Phys. 10, 233 (2019)

    ADS  Article  Google Scholar 

  14. 14.

    D. Stauffer, A. Aharony, Introduction to Percolation Theory (CRC Press, Boca Raton, 1991)

    Google Scholar 

  15. 15.

    S.E. Rowley, T. Vojta, A.T. Jones, W. Guo, J. Oliveira, F.D. Morrison, N. Lindfield, E. Baggio Saitovitch, B.E. Watts, J.F. Scott, Phys. Rev. B 96, 020407 (2017)

    ADS  Article  Google Scholar 

  16. 16.

    G. Albanese, F. Leccabue, B.E. Watts, S. Díaz-Castañón, J. Mat. Sci. 37, 3759 (2002)

    ADS  Article  Google Scholar 

  17. 17.

    A. Coniglio, Phys. Rev. Lett. 46, 250 (1981)

    ADS  Article  Google Scholar 

  18. 18.

    T. Vojta, J. A. Hoyos, in Recent Progress in Many-Body Theories, ed. by J. Boronat, G. Astrakharchik, F. Mazzanti (World Scientific, Singapore, 2008) p. 235

  19. 19.

    C. Wu, Z. Yu, K. Sun, J. Nie, R. Guo, H. Liu, X. Jiang, Z. Lan, Sci. Rep. 6, 36200 (2016)

    ADS  Article  Google Scholar 

  20. 20.

    E. Shender, B. Shklovskii, Phys. Lett. A 55, 77 (1975)

    ADS  Article  Google Scholar 

  21. 21.

    B. Kozlov, M. Laguës, Phys. A: Statist. Mech. Appl. 389, 5339 (2010)

    ADS  Article  Google Scholar 

  22. 22.

    J. Wang, Z. Zhou, W. Zhang, T.M. Garoni, Y. Deng, Phys. Rev. E 87, 052107 (2013a)

    ADS  Article  Google Scholar 

  23. 23.

    A.B. Harris, T.C. Lubensky, J. Phys. A 17, L609 (1984)

    ADS  Article  Google Scholar 

  24. 24.

    A.B. Harris, A. Aharony, Phys. Rev. B 40, 7230 (1989)

    ADS  Article  Google Scholar 

  25. 25.

    U. Wolff, Phys. Rev. Lett. 62, 361 (1989)

    ADS  Article  Google Scholar 

  26. 26.

    N. Metropolis, S. Ulam, J. Am. Statist. Assoc. 44, 335 (1949)

    Article  Google Scholar 

  27. 27.

    H.G. Ballesteros, L.A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi, J.J. Ruiz-Lorenzo, Phys. Rev. B 58, 2740 (1998)

    ADS  Article  Google Scholar 

  28. 28.

    T. Vojta, R. Sknepnek, Phys. Rev. B 74, 094415 (2006)

    ADS  Article  Google Scholar 

  29. 29.

    Q. Zhu, X. Wan, R. Narayanan, J.A. Hoyos, T. Vojta, Phys. Rev. B 91, 224201 (2015)

    ADS  Article  Google Scholar 

  30. 30.

    K. Binder, Zeitschr. für Phys. B 43, 119 (1981)

    Google Scholar 

  31. 31.

    W. Selke, L.N. Shchur, J. Phys. A 38, L739 (2005)

    ADS  Article  Google Scholar 

  32. 32.

    A.P. Gottlob, M. Hasenbusch, Phys. A Statist. Mech. Appl. 201, 593 (1993)

    ADS  Article  Google Scholar 

  33. 33.

    R.G. Brown, M. Ciftan, Phys. Rev. B 74, 224413 (2006)

    ADS  Article  Google Scholar 

  34. 34.

    S.E. Rowley, Y.-S. Chai, S.-P. Shen, Y. Sun, A.T. Jones, B.E. Watts, J.F. Scott, Sci. Rep. 6, 25724 (2016)

    ADS  Article  Google Scholar 

Download references

Acknowledgements

We acknowledge support from the NSF under Grant nos. DMR-1506152, DMR-1828489, and OAC-1919789. The simulations were performed on the Pegasus and Foundry clusters at Missouri S&T. We also thank Martin Puschmann for helpful discussions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Thomas Vojta.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Khairnar, G., Lerch, C. & Vojta, T. Phase boundary near a magnetic percolation transition. Eur. Phys. J. B 94, 43 (2021). https://doi.org/10.1140/epjb/s10051-021-00056-4

Download citation