Branched critical multifractality and morphological phase transitions

It is shown, using results of recent numerical simulations of different authors, that multifractality of some quantum systems with a continuous spectrum has a branched critical nature. It is also shown, using multifractal analysis of the atomic spectral line series of cobalt and argon, that the clustering of lines in the series exhibits the branched critical multifractality as well. These results are compared with analogous results obtained in numerical simulations of turbulence. The well-known thermodynamic analogy of the multifractal description is used to explain a crossover from branch to branch observed in the numerical and experimental spectra of the generalized dimensions, Dq. In the thermodynamics terms this crossover appears to be a second-order phase transition. Then, the thermodynamic argument is used to calculate critical value of q. Self-organization processes related to the morphological phase transitions in some classic and quantum systems are also briefly discussed.