Modified Mean-Field Theory of One-Dimensional Spin Models with Anisotropy and Long-Range Dipolar Interactions


  • P. J. Camp School of Chemistry, University of Edinburgh, Department of Theoretical and Mathematical Physics, Institute of Natural Sciences and Mathematics, Ural Federal University
  • A. O. Ivanov Department of Theoretical and Mathematical Physics, Institute of Natural Sciences and Mathematics, Ural Federal University, M.N. Mikheev Institute of Metal Physics, UB of the RAS



Heisenberg model, Ising model, dipolar interactions, magnetization, magnetic susceptibility, modified mean-field theory, Monte Carlo simulations


The effects of interactions and anisotropy on the magnetic properties of linear chains of superparamagnetic nanoparticles are studied theoretically by mapping the problem onto spin models. With zero anisotropy, the magnetic dipole moments are free to rotate, and the system resembles a classical ferromagnetic Heisenberg model with long-range dipolar interactions. With strong anisotropy, they are constrained to align with the chain, and the system resembles the classical ferromagnetic Ising model with long-range interactions. Using a modified mean-field theory, expressions for the magnetization curve and initial magnetic susceptibility are derived from the response of a single particle subject to an effective field arising from the applied field and the interactions with the other particles. Various approximations for the effective field are tested against results from Monte Carlo simulations. It is shown that, for physically relevant interaction strengths, reliable theoretical predictions for both the zero-anisotropy and strong-anisotropy cases can be derived in a simple closed form.


R.E. Rosensweig. Ferrohydrodynamics (Dover, 1998).

J. Carrey, B. Mehdaoui, M. Respaud. Simple models for dynamic hysteresis loop calculations of magnetic single-domain nanoparticles: Application to magnetic hyperthermia optimization. J. Appl. Phys. 109, 083921 (2011).

A.L. Elrefai, T. Sasayama, T. Yoshida, K. Enpuku. Empirical expression for DC magnetization curve of immobilized magnetic nanoparticles for use in biomedical applications. AIP Advances 8, 056803 (2018).

E.A. Elfimova, A.O. Ivanov, P.J. Camp. Static magnetization of immobilized, weakly interacting, superparamagnetic nanoparticles. Nanoscale 11, 21834 (2019).

A.O. Ivanov, O.B. Kuznetsova. Magnetic properties of dense ferrofluids: An influence of interparticle correlations. Phys. Rev. E 64, 041405 (2001).

A.O. Ivanov, O.B. Kuznetsova. Magnetogranulometric analysis of ferrocolloids: Second-order modified mean field theory. Colloid J. 68, 430 (2006).

W.H. Keesom. On the deduction from Boltzmann's entropy principle of the second virial-coeficient for material particles (in the limit rigid spheres of central symmetry) which exert central forces upon each other and for rigid spheres of central symmetry containing an electric doublet at their centre. Comm. Phys. Lab. Leiden, Suppl. 24b, 23 (1912).

H.E. Stanley. Dependence of critical properties on dimensionality of spins. Phys. Rev. Lett. 20, 589 (1968).

M.E. Fisher. Magnetism in one-dimensional systems - the Heisenberg model for infinite spin. Am. J. Phys. 32, 343 (1964).

G.S. Joyce. Classical Heisenberg model. Phys. Rev. 155, 478 (1967).

R.J. Baxter. Exactly Solved Models in Statistical Mechanics (Academic Press, 1982).

N.D. Mermin, H. Wagner. Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1133 (1966).

J. Fr¨olich, R. Israel, E.H. Lieb, B. Simon. Phase transitions and reflection positivity. I. General theory and long range lattice models. Commun. Math. Phys. 62, 1 (1978).

P. Bruno. Absence of spontaneous magnetic order at nonzero temperature in one- and two-dimensional Heisenberg and XY systems with long-range interactions. Phys. Rev. Lett. 87, 137203 (2001).

D. Ruelle. Statistical mechanics of a one-dimensional lattice gas. Commun. Math. Phys. 9, 267 (1968).

F.J. Dyson. Existence of a phase transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12, 91 (1969).

F.J. Dyson. Non-existence of spontaneous magnetization in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12, 212 (1969).

F.J. Dyson. An Ising ferromagnet with discontinuous long-range order. Commun. Math. Phys. 21, 269 (1971).

J. Fr¨olich, T. Spencer. The phase transition in the one-dimensional Ising model with 1/r^2 interaction energy. Commun. Math. Phys. 84, 87 (1982).

T. Morita, T. Horiguchi. Classical one-dimensional Heisenberg model with an interaction of finite range. Physica A 83, 519 (1976).

J.-P. Hansen, I.R. McDonald. Theory of Simple Liquids (Academic Press, 2006).

P.Weiss. L'hypoth'ese du champ mol'eculaire et la propri'et'e ferromagn'etique. J. Phys. Theor. Appl. 6, 661 (1907).

M. Eisenbach, M. Dijkstra, B.L. Gy¨orffy. On the states of orientations along a magnetically inhomogeneous nanowire. J. Mag. Magn. Mater. 208, 137 (2000).

Y. Yamamura, H. Saitoh, M. Sumita, K. Saito. One-dimensional correlation in the dipolar Ising crystal tricyclohexyl-methanol: crystal structure revisited and heat capacity. J. Phys.: Condens. Matter 19, 176219 (2007).

J. K¨ofinger, G. Hummer, C. Dellago. Macroscopically ordered water in nanopores. Proc. Natl. Acad. Sci. U.S.A. 105, 13218 (2008).

J. K¨ofinger, G. Hummer, C. Dellago. A one-dimensional dipole lattice model for water in narrow nanopores. J. Chem. Phys. 130, 154110 (2009).

J. K¨ofinger, C. Dellago. Single-file water as a one-dimensional Ising model. New J. Phys. 12, 093044 (2010).

K. Binder, D.P. Landau. A Guide to Monte Carlo Simulations in Statistical Physics, 4th (Cambridge Univ. Press, 2014).

H.E. Stanley. Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press, 1971).




How to Cite

Camp, P. J., & Ivanov, A. O. (2020). Modified Mean-Field Theory of One-Dimensional Spin Models with Anisotropy and Long-Range Dipolar Interactions. Ukrainian Journal of Physics, 65(8), 691.



Physics of magnetic phenomena and physics of ferroics