Landau–Ginzburg theory of phase transitions in fractal systems

Fracton Excitations in the Magnetic "Net Fractal" Systems

“…The ratio of α and β in Eq. (2) determines the fractional spectral (fracton) dimension, which in turn governs the thermodynamical behavior of the spin system [1,4]. Since there are many definitions of fractional derivatives in any approach, which involves the fractional calculus techniques, one should define, which definition of fractional pseudo--differential is used.…”

“…It can be proven [1] that, when presented in logarithmic scale, the family of mappings S (m,n,l) is isomorphic with a 3D crystal lattice. This means that the isomorphism…”

“…This property of fractals is called self-similarity, self-affinity or self-replicability. Although physical systems modeled by fractals are non-translation-invariant, it is a well-known fact that the self-similar fractals as well as the physical quantities on fractal systems show logperiodicities (see [1] and references therein). This opens a possibility to describe the symmetries of magnetic self-similar fractals in the way that is reminiscent of conventional formalism developed for crystalline systems.…”

RKKY-Reminiscent Interaction in Net Fractals

“…The very same refers to the placement of the characteristic building blocks of the "net fractal". This means that after transition to the logarithmic coordinates uniform distribution of spins and of electron density is restored [6].…”

“…It can be proven [6] that, when presented in a logarithmic scale, the family of mappings S (m,n,l) is isomorphic with a 3D crystal lattice. The isomorphic mapping is given by S (m,n,l) → (ma 1 , na 2 , la 3 ).…”