Inducing a metal-insulator transition through systematic alterations of local rewriting rules in a quantum graph

The Anderson localization transition in quantum graphs has garnered significant recent attention due to its relevance to many-body localization studies. Typically, graphs are constructed using top-down methods. Here, we explore a bottom-up approach, employing a simple local rewriting rule to construct the graph. Through the use of ratio statistics for the energy spectrum and Kullback-Leibler divergence correlations and multifractal analysis for the eigenstates, numerical analysis demonstrates that slight adjustments to the rewriting rule can induce a transition from a localized to an extended quantum phase. This extended state exhibits nonergodic behavior, akin to the nonergodic extended phase observed in the Porter-Rosenzweig model and suggested for many-body localization. Thus, by adapting straightforward local rewriting rules, it becomes feasible to assemble complex graphs from which desired global quantum phases emerge. This approach holds promise for numerical investigations and could be implemented in building optical realizations of complex networks using optical fibers and beam splitters.