Skip to main content
SpringerLink
Log in

Change in Density of States of 2D Ising Model when Next-Neighbor Interaction Is Introduced

  • Published:
Optical Memory and Neural Networks Aims and scope Submit manuscript

Abstract

In the present paper we analyzed a change in the density of states of a two-dimensional Ising model when a next-next-neighbor interaction is introduced. In other words, we examined two-dimensional lattices with diagonal connections. The same as in a three-dimensional model in this case each spin has 6 connections. Since the model is planar, we can calculate the free energy and other characteristics of the system using a polynomial algorithm. We performed computer simulations using the Kasteleyn–Fisher algorithm, which allowed us to study changes of critical values and the density of states when a long-range interaction is taken into account. From the obtained results it follows that the interactions of the type analyzed here result in quantitative changes of the system’s characteristics, but they did not change them qualitatively. In particular, we again obtained a logarithmic divergence of the heat capacity in the critical point.

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

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.
Fig. 12.

Similar content being viewed by others

REFERENCES

  1. Onsager, L., Crystal statistics. I. A two-dimensional model with an order–disorder transition, Phys. Rev., 1944, vol. 65, nos. 3–4, pp. 117–149.

    Article  MathSciNet  Google Scholar 

  2. Kasteleyn, P., Dimer statistics and phase transitions, J. Math. Phys., 1963, vol. 4, no. 2.

  3. Fisher, M., On the dimer solution of planar Ising models, J. Math. Phys., 1966, vol. 7, no. 11.

  4. Karandashev, Ya.M. and Malsagov, M.Yu., Polynomial algorithm for exact calculation of partition function for binary spin model on planar graphs, Opt. Mem. Neural Networks, 2017, vol. 26, no. 2. https://arxiv.org/abs/1611.00922.

  5. Schraudolph, N. and Kamenetsky, D., Efficient exact inference in planar Ising models, NIPS, 2008. https://arxiv.org/abs/0810.4401.

  6. Karandashev, I.M., Kryzhanovsky, B.V., and Malsagov, M.Yu., The analytical expressions for a finite-size 2D Ising model, Opt. Mem. Neural Networks, 2017, vol. 26, no. 3, pp. 165–171.

    Article  Google Scholar 

  7. Kryzhanovsky, B.V., Malsagov, M.Yu., and Karandashev, I.M., Dependence of critical parameters of 2D Ising model on lattice size, Opt. Mem. Neural Networks, 2018, vol. 27, no. 1, pp. 10–22.

    Article  Google Scholar 

  8. Lundow, P.H. and Markström, K., The discontinuity of the specific heat for the 5D Ising model, Nucl. Phys. B, 2015, vol. 895, pp. 305–318. https://doi.org/10.1016/j.nuclphysb.2015.04.013

    Article  MathSciNet  MATH  Google Scholar 

  9. Litinskii, L. and Kryzhanovsky, B., Spectral density and calculation of free energy, Phys. A: Stat. Mech. Its Appl., 2018, vol. 510, no. C, pp. 702–712.

    Article  MathSciNet  Google Scholar 

  10. Kryzhanovsky, B.V., Features of the spectral density of a spin system, Dokl. Math., 2018, vol. 97, no. 2, pp. 188–192.

    Article  MathSciNet  Google Scholar 

  11. Baxter, R.J., Exactly Solved Models in Statistical Mechanics, London: Academic Press, 1982.

    MATH  Google Scholar 

Download references

Funding

The work financially supported by State Program of SRISA RAS no. 0065-2019-0003 (AAA-A19-119011590090-2).

Author information

Authors and Affiliations

  1. Scientific Research Institute for System Analysis, Russian Academy of Sciences, 117218, Moscow, Russia

    I. M. Karandashev & B. V. Kryzhanovsky

Authors
  1. I. M. Karandashev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. V. Kryzhanovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. M. Karandashev.

Ethics declarations

The authors declare that they have no conflict of interest.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karandashev, I.M., Kryzhanovsky, B.V. Change in Density of States of 2D Ising Model when Next-Neighbor Interaction Is Introduced. Opt. Mem. Neural Networks 28, 165–174 (2019). https://doi.org/10.3103/S1060992X19030032

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1060992X19030032

Keywords:

Search

Navigation