Multifractality of wave functions on a Cayley tree: From root to leaves

Abstract

Disordered quantum systems undergo Anderson localization-delocalization transitions, which exhibit very rich physics. A remarkable feature of these transitions is the multifractality of critical wave functions. The eigenfunction multifractality in a $d$-dimensional disordered system holds only at the transition point and is characterized by universal critical exponents. The authors explore the evolution of wave-function statistics on a finite Bethe lattice from the central site (``root'') to the boundary (``leaves''). They show that eigenfunction moments exhibit generally a multifractal scaling with the volume $N$. The multifractality spectrum ${\ensuremath{\tau}}_{q}$ depends on the disorder strength and on the parameter $s$ characterizing the position of the observation point, $s$ = $r/R$, where $r$ is the distance to the root and $R$ is the ``radius'' of the lattice.