Time dependent Ginzburg-Landau model in the absence of translational invariance. Non Conserved Order Parameter domain growth

We have determined the static and dynamical properties of the Ginzburg-Landau model, with global coupling of the spherical type, on some non-translationally invariant lattices. Our solutions show that, in agreement with general theorems, fractal lattices with finite ramification do not display a finite temperature phase transition for any embedding dimension, d. On the other hand, the dynamical behaviour associated with the phase ordering dynamics of a non-conserved order parameter is non-trivial. Our analysis reveals that the domain size R grows in time as R(t) similar to t(z) and relates this exponent to the three exponents which characterize the static and dynamical properties of fractal structures, namely the fractal dimension of the lattice d(f), the random walk dimension d(w) and the spectral dimension d(s). We also present a brief renormalization group treatment of the model. Finally, we have considered lattices with infinite ramification numbers which have spectral dimensions larger that 2 and show a finite temperature phase transition