Field theoretical approach to spin models

Abstract

We developed a systematic non-perturbative method base on Dyson–Schwinger theory and the Φ-derivable theory for Ising model at broken phase. Based on these methods, we obtain critical temperature and spin spin correlation beyond mean field theory. The spectrum of Green function obtained from our methods become gapless at critical point, so the susceptibility become divergent at Tc. The critical temperature of Ising model obtained from this method is fairly good in comparison with other non-cluster methods. It is straightforward to extend this method to more complicate spin models for example with continue symmetry.

Graphical abstract

Keywords

Ising model / mean field theory / Dyson–Schwinger equations

References

[1] E. Ising, Beitrag zur theorie des ferromagnetismus, Z. Phys. 31(1), 253 (1925)

[2] L. Onsager, Crystal statistics (i): A two-dimensional model with an order-disorder transition, Phys. Rev. 65(3–4), 117 (1944)

[3] A. Kuzemsky, Statistical mechanics and the physics of many-particle model systems, Phys. Part. Nucl. 40(7), 949 (2009)

[4] P. Weiss and E. Stoner, Magnetism and atomic structure, J. Phys. 6, 667 (1907)

[5] G. Wysin and J. Kaplan, Correlated molecular-field theory for Ising models, Phys. Rev. E 61(6), 6399 (2000)

[6] H. A. Bethe, Statistical theory of superlattices, Proc. R. Soc. Lond. A 150(871), 552 (1935)

[7] R. Peierls, On Ising’s model of ferromagnetism, in: Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, Cambridge University Press, 1936, pp 477–481

[8] P. R. Weiss, The application of the Bethe–Peierls method to ferromagnetism, Phys. Rev. 74(10), 1493 (1948)

[9] K. K. Zhuravlev, Molecular-field theory method for evaluating critical points of the ising model, Phys. Rev. E 72(5), 056104 (2005)

[10] D. Yamamoto, Correlated cluster mean-field theory for spin systems, Phys. Rev. B 79(14), 144427 (2009)

[11] J. R. Viana, O. R. Salmon, J. R. de Sousa, M. A. Neto, and I. T. Padilha, An effective correlated mean-field theory applied in the spin-1/2 Ising ferromagnetic model, J. Magn. Magn. Mater. 369, 101 (2014)

[12] J. M. Luttinger and J. C. Ward, Ground-state energy of a many-fermion system (ii), Phys. Rev. 118(5), 1417 (1960)

[13] G. Baym and L. P. Kadanoff, Conservation laws and correlation functions, Phys. Rev. 124(2), 287 (1961)

[14] J. M. Cornwall, R. Jackiw, and E. Tomboulis, Effective action for composite operators, Phys. Rev. D 10(8), 2428 (1974)

[15] A. Kovner and B. Rosenstein, Covariant Gaussian approximation (i): Formalism, Phys. Rev. D 39(8), 2332 (1989)

[16] H. Van Hees and J. Knoll, Renormalization in selfconsistent approximation schemes at finite temperature (iii): Global symmetries, Phys. Rev. D 66(2), 025028 (2002)

[17] D. J. Amit and V. Martin-Mayor, Field Theory, the Renormalization Group, and Critical Phenomena: Graphs to Computers, World Scientific Publishing Company, 2005

[18] J. Wang, D. Li, H. Kao, and B. Rosenstein, Covariant Gaussian approximation in Ginzburg–Landau model, Ann. Phys. 380, 228 (2017)

[19] B. Rosenstein and A. Kovner, Covariant Gaussian approximation (ii): Scalar theories, Phys. Rev. D 40(2), 504 (1989)

[20] M. E. Fisher, The theory of equilibrium critical phenomena, Rep. Prog. Phys. 30(2), 615 (1967)

[21] N. W. Ashcroft, N. D. Mermin, , Solid state physics, Vol. 2005, Holt, Rinehart And Winston, New York, London, 1976

[22] H. Au-Yang and J. H. Perk, Correlation functions and susceptibility in the z-invariant Ising model, in: MathPhys Odyssey 2001, Springer, 2002, pp 23–48

[23] W. Orrick, B. Nickel, A. Guttmann, and J. Perk, The susceptibility of the square lattice Ising model: New developments, J. Stat. Phys. 102(3/4), 795 (2001) (for the complete set of series coefficients see https://blogs.unimelb.edu.au/tony-guttmann/)

[24] F. Ricci-Tersenghi, The Bethe approximation for solving the inverse Ising problem: A comparison with other inference methods, J. Stat. Mech. 2012(08), P08015 (2012)

RIGHTS & PERMISSIONS

2021 Higher Education Press