Skip to main content
SpringerLink
Log in

Dimers Belonging to Three Orientations on Plane Honeycomb Lattices

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

Abstract

It is well known that there are three types of dimers belonging to the three different orientations in a honeycomb lattice, and in each type all dimers are mutually parallel. Based on a previous result, we can compute the partition function of the dimer problem of the plane (free boundary) honeycomb lattices with three different activities by using the number of its pure dimer coverings (perfect matchings). The explicit expression of the partition function and free energy per dimer for many types of plane honeycomb lattices with fixed shape of boundaries is obtained in this way (for a shape of plane honeycomb lattices, the procedure that the size goes to infinite, corresponds to a way that the honeycomb lattice goes to infinite). From these results, an interesting phenomena is observed. In the case of the regions of the plane honeycomb lattice has zero entropy per dimer—when its size goes to infinite—though in the thermodynamic limit, there is no freedom in placing a dimer at all, but if we distinguish three types of dimers with nonzero activities, then its free energy per dimer is nonzero. Furthermore, a sufficient condition for the plane honeycomb lattice with zero entropy per dimer (when the three activities are equal to 1) is obtained. Finally, the difference between the plane honeycomb lattices and the plane quadratic lattices is discussed and a related problem is proposed.

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. Baxter, R.J.: J. Math. Phys. 11, 784 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  2. Bhattacharjee, S.M., Nagle, J.F.: Phys. Rev. A 31, 3199 (1985)

    Article  ADS  Google Scholar 

  3. Chen, R.S., Cyvin, S.J., Cyvin, B.N., et al.: Advances in the Theory of Benzenoid Hydrocarbons. Topic in Current Chemistry, vol. 153. Springer, Berlin (1992)

    Google Scholar 

  4. Cohn, H., Elkies, N., Propp, J.: Duke Math. J. 85, 117 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cohn, H., Kenyon, R., Propp, J.: J. Am. Math. Soc. 14, 297 (2000)

    Article  MathSciNet  Google Scholar 

  6. Cyvin, B.N., Cyvin, B.N., Brunvoll, J.: Match 21, 291 (1986)

    MATH  MathSciNet  Google Scholar 

  7. Gordon, M., Davison, W.H.T.: J. Chem. Phys. 20, 428 (1952)

    Article  ADS  Google Scholar 

  8. Doslic, T.: Croat. Chem. Acta 78, 251 (2005)

    Google Scholar 

  9. Dresselhaus, M.S., Araujo, P.T.: ACS Nano 4, 6297 (2010)

    Article  Google Scholar 

  10. Eisenkolbl, T.: J. Comb. Theory, Ser. A 88, 368 (1999)

    Article  MathSciNet  Google Scholar 

  11. Elkies, N., Kuperberg, G., Larsen, M., Propp, J.: J. Algebr. Comb. 1, 111 and 219 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Elser, V.: J. Phys. A, Math. Gen. 17, 1509 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  13. Fisher, M.E.: Phys. Rev. 124, 1664 (1961)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Fowler, R.H., Rushbrooke, G.S.: Trans. Faraday Soc. 33, 1272 (1937)

    Article  Google Scholar 

  15. Grensing, D., Grensing, G.: J. Math. Phys. 24, 620 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  16. Gutman, I., Cyvin, S.J.: Kekulé Structures in Benzenoid Hydrocarbons. Springer, Berlin (1988)

    Google Scholar 

  17. Herndson, W.C.: J. Am. Chem. Soc. 95, 2404 (1973)

    Article  Google Scholar 

  18. Herndson, W.C.: Thermochem. Acta 8, 225 (1974)

    Article  Google Scholar 

  19. Huang, H.Y., Wu, F.Y., Kunz, H., Kim, D.: Physica A 228, 1 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  20. Kasteleyn, P.W.: Physica 27, 1209 (1961)

    Article  ADS  Google Scholar 

  21. Kasteleyn, P.W.: J. Math. Phys. 4, 287 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  22. Kenyon, R., Okounkov, A.: Not. Am. Math. Soc. 52, 342 (2005)

    MATH  MathSciNet  Google Scholar 

  23. Klein, D.J.: Int. J. Quantum Chem. Symp. 13, 293 (1979)

    Google Scholar 

  24. Klein, D.J., Schmalz, T.G.: Phys. Rev. B 41, 2244 (1990)

    Article  ADS  Google Scholar 

  25. Klein, D.J., Zhu, H.-Y.: Discrete Appl. Math. 67, 157 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Klein, D.J., Hite, G.E., Schmalz, T.G.: J. Comput. Chem. 7, 443 (1986)

    Article  Google Scholar 

  27. Klein, D.J., Hite, G.E., Seitz, W.A., Schmalz, T.G.: Theor. Chim. Acta 69, 409 (1986)

    Article  Google Scholar 

  28. Lieb, E.H.: J. Math. Phys. 8, 2339 (1967)

    Article  ADS  Google Scholar 

  29. Nagle, J.F.: Phys. Rev. Lett. 34, 1150 (1975)

    Article  ADS  Google Scholar 

  30. Nienhuis, B., Hishorst, H.J., Bloete, H.W.J.: J. Phys. A 17, 3559 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  31. Novoselov, K.S., Geiem, A.K., et al.: Science 306, 666 (2004)

    Article  ADS  Google Scholar 

  32. Percus, J.K.: J. Math. Phys. 10, 1881 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  33. Propp, J.: Int. J. Mod. Phys. B 11, 183 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Sachs, H.: Combinatorica 4, 89 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  35. Sachs, H., Zeritz, H.: Discrete Appl. Math. 51, 171 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  36. Sachs, H., Hansen, P., Zheng, M.: MATCH Commun. Math. Chem. 33, 169 (1996)

    MATH  MathSciNet  Google Scholar 

  37. Temperley, H.N.V., Fisher, M.E.: Philos. Mag. 6, 1061 (1961)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. Wannier, G.H.: Phys. Rev. 79, 357 (1950)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. Wu, F.Y.: Phys. Rev. 168, 539 (1968)

    Article  ADS  Google Scholar 

  40. Wu, F.Y., Wu, X.N., Blóte, H.W.J.: Phys. Rev. Lett. 62, 2773 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  41. Yan, W.G., Yeh, Y.-N., Zhang, F.J.: Theor. Comput. Sci. 349, 452 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  42. Yen, T.F.: Theor. Chem. Acta 20, 399 (1971)

    Article  Google Scholar 

  43. Yokoi, C.S., Nagle, J.F., Salinas, S.R.: J. Stat. Phys. 44, 729 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  44. Zhang, F.J., Chen, R.S., Guo, X.F., Gutman, I.: J. Serb. Chem. Soc. 51, 537 (1986)

    Google Scholar 

  45. Zhang, F.J., Guo, X.F., Chen, R.S.: Discrete Math. 72, 405 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  46. Zhu, H.-Y., Balaban, A.T., Klein, D.J., Zivkovic, T.P.: J. Chem. Phys. 101, 5281 (1994)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Sciences, Jimei University, Xiamen, 361021, China

    Weigen Yan

  2. Institute of Mathematics, Academia Sinica, Taipei, 11529, Taiwan

    Yeong-Nan Yeh

  3. School of Mathematical Science, Xiamen University, Xiamen, 361005, China

    Fuji Zhang

Authors
  1. Weigen Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yeong-Nan Yeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fuji Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fuji Zhang.

Additional information

W. Yan is partially supported by NSFC Grant (11171134).

Y.-N. Yeh is partially supported by NSC98-2115-M-001-010-MY3.

F. Zhang is partially supported by NSFC Grant (10831001).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yan, W., Yeh, YN. & Zhang, F. Dimers Belonging to Three Orientations on Plane Honeycomb Lattices. J Stat Phys 145, 1343–1356 (2011). https://doi.org/10.1007/s10955-011-0348-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-011-0348-y

Keywords

Search

Navigation