Phases and phase transitions on disordered Bethe lattice: analytical treatment by one-step replica symmetry breaking
Vladimir Kravtsov (ICTP, Trieste, Italy)
We apply the replica trick to the problem of localization on disordered Bethe lattice. We show that the formulation of the problem in terms of the one-step replica symmetry breaking leads naturally to existence of the multifractal phase and to phase transitions between this phase and the localized and fully extended (ergodic) delocalized phase. We prove the symmetry m→1-m of the problem with respect to the replica symmetry breaking parameter m and use this symmetry to derive exact expressions for the transition points in terms of the Lyapuniov exponents and the branching number K. We also suggest a simple approximation which allows to compute the fractal dimension D1 in the multifractal phase and the critical values of disorder at the phase transitions with the best known so far accuracy.
Reference :
-
B. L. Altshuler, L. B. Ioffe and V. E. Kravtsov, Multifractal states in self-consistent theory of localization: analytical solution, preprint arXiv:1610.00758