Describing the critical behavior of the Anderson transition in infinite dimension by random-matrix ensembles: Logarithmic multifractality and critical localization

Power-law random banded matrix ensembles have proven effective in predicting critical properties at finite-dimensional Anderson transitions. Here, the authors extend the use of random matrices to infinite dimensions by introducing ensembles specifically designed to capture infinite-dimensional Anderson transitions. Using both analytical arguments and advanced numerical simulations, they uncover two unique critical behaviors: logarithmic multifractality and critical localization. As shown in the image, logarithmic multifractality displays an algebraic dependence of the large moments of wavefunction amplitudes on the logarithm of system size, characterized by a spectrum of log-multifractal dimensions d0{0ex}q>0, while the conventional multifractal dimensions Dₐ-0. 22em, which describe scaling with system size, are zero. These insights provide a structured framework for understanding the pronounced finite-size effects and slow dynamics in these Anderson transitions, similar to the many-body localization transition.