Multifractality and dynamics at the Anderson transition: From finite dimension to infinite dimension

Multifractality is an exotic property that emerges at the Anderson transition. Meanwhile, the dynamics are highly influenced by the multifractality of the eigenstates. This presentation will focus on the emergence of multifractality and its dynamic signatures in random-matrix ensembles amenable to analytical treatment. Firstly, I will revisit random-matrix ensembles that capture multifractal properties in finite dimensions, emphasizing the scale-invariant properties of dynamics as a consequence of multifractality. Secondly, I will introduce new random-matrix ensembles featuring critical properties in infinite dimension, the upper critical dimension of the Anderson transition. Through analytical arguments, these models reveal two scenarios of critical properties: logarithmic multifractality and critical localization. These results will help to clarify some elusive problems of Anderson transitions in random graphs.

References: Physical Review E 108(5) , 054127 (2023); arXiv:2312.17481 (2023).