Cusp singularity in mean field Ising model

The diffusively coupled lattice of odd-symmetric chaotic maps introduced by Miller and Huse undergoes a continuous ordering phase transition, belonging to a universality class close but not identical to that of the two-dimensional Ising model. Here we consider a natural mean-field approach for this model, and find that it does not have a well-defined phase transition. We show how this is due to the coexistence of two attractors in its mean-field description, for the region of interest in the coupling. The behavior of the model in this limit then becomes dependent on initial conditions, as can be seen in direct simulations.

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Mean-field theory of the three-dimensional anisotropic Ising model as a four-dimensional mapping

Physical Review B ◽

10.1103/physrevb.27.6853 ◽

1983 ◽

Vol 27 (11) ◽

pp. 6853-6868 ◽

Cited By ~ 111

Author(s):

M. Høgh Jensen ◽

Per Bak

Keyword(s):

Field Theory ◽

Ising Model ◽

Mean Field Theory ◽

Mean Field ◽

Three Dimensional ◽

Dimensional Mapping

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Single-Crystal Magnetic Properties of [Co(NH3)5(OH2)] [Cr(CN)6] Between 2 and 300 K

Australian Journal of Chemistry ◽

10.1071/ch9921899 ◽

1992 ◽

Vol 45 (11) ◽

pp. 1899 ◽

Cited By ~ 1

Author(s):

PA Reynolds ◽

CD Delfs ◽

BN Figgis ◽

B Moubaraki ◽

KS Murray

Keyword(s):

Ising Model ◽

Field Model ◽

Mean Field ◽

Zero Field ◽

Spin Orbit Coupling ◽

Zero Field Splitting ◽

Magnetic Exchange Interaction ◽

Mean Field Model ◽

Magnetic Exchange ◽

The Mean

The magnetic susceptibilities along and perpendicular to the c axis (hexagonal setting) between 2.0 and 300 K at a magnetic field of 1.00 T, and the magnetizations at field strengths up to 5.00 T, are presented for single crystals of [Co(NH3)5(OH2)] [Cr(CN)6]. The results are interpreted in terms of zero-field splitting (2D) of the ground 4A2g term by spin-orbit coupling and of magnetic exchange interaction between the chromium atoms. The magnetic exchange is modelled as one of Ising or mean-field in type. The exchange is found to be quite small: J = -0.18(6) cm-1 if the Ising model is employed, and -0.03(1) cm-1 for the mean-field model. The model adopted for the exchange has a strong influence on the value of the parameter D obtained. When the Ising model is used D is deduced to be -0.28(9) cm-l; when the mean-field model is used D is -0.14(4) cm-l. The g-values deduced are in agreement with those from e.s.r. measurements at higher temperatures and do not depend on the exchange model. In any case, D is found to be sufficiently large that it must be considered in a polarized neutron diffraction experiment on the compound.

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