Abstract
Using topology, fractal analysis and investigation of lattice formation process we find two types of equivalence transformations among Ising models: topological equivalence transformation and formation equivalence transformation. With the help of the transformations and the known data of the critical points of simple cubic (sc) lattice and planar square (sq) lattice we get directly the critical points for face-centered cubic (fcc) lattice, body-centered cubic (bcc) lattice and diamond (d) lattice. The transformation itself results no error in the calculation. Other than Monte Carlo method and series expansion approach the equivalence transformations help us simplify much more greatly the calculation of the critical points for the three-dimensional models and understand much more deeply the structural connection among Ising models.
Highlights
Fractal structure is a class of complex ordered structures in nature, which exhibits not a higher degree but an altogether different level of complexity
During the 1980s physicists tried to describe phenomena on fractal, they succeeded in calculating some of physical characteristics of fractals [1] [2]
Our approach combining the fractal analysis and the solvable Gaussian model has been succeeded in calculating the critical points for two-and-three-dimensional Ising models and analyzing the fluctuation structure of the details at the critical temperature [3]-[5]
Summary
Fractal structure is a class of complex ordered structures in nature, which exhibits not a higher degree but an altogether different level of complexity. Solving of three-dimensional Ising models has more far-reaching significance because the actual ferromagnetic elements have different crystal textures; for example iron is body-centered cubic (bcc) while nickel is facecentered cubic (fcc) The investigation of their structures will help us deeply understand general laws of ferromagnetic. There are two ways to research further: The first is to find out the fractal for the individual lattices one by one, except the sc we have solved, which requires us to divide correctly their sub-blocks; this may not be a normal way of affairs, especially for those composite lattices such as the d lattice Another way is to look for equivalence relations among the models, using which the fcc, bcc and d models can be described by the sc or the other such as the sq we have solved exactly.
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