Multi-fractal geometry of finite networks of spins: Nonequilibrium dynamics beyond thermalization and many-body-localization

Quantum spin networks overcome the challenges of traditional charge-based electronics by encoding information into spin degrees of freedom. Although beneficial for transmitting information with minimal losses when compared to their charge-based counterparts, the mathematical formalization of the information propagation in a spin(tronic) network is challenging due to its complicated scaling properties. In this paper, we propose a fractal geometric approach for unraveling the information-theoretic phenomena of spin chains and rings by abstracting them as weighted graphs, where the vertices correspond to single spin excitation states and the edges represent the information theoretic distance between pair of nodes. The weighted graph exhibits a complex self-similar structure. To quantify this complex behavior, we develop a new box-counting-inspired algorithm which assesses the mono-fractal versus multi-fractal properties of quantum spin networks. Mono- and multi-fractal properties are in the same spirit as, but different from, Eigenstate Thermalization Hypothesis (ETH) and Many-Body Localization (MBL), respectively. To demonstrate criticality in finite size systems, we define a thermodynamics inspired framework for describing information propagation and show evidence that some spin chains and rings exhibit an informational phase transition phenomenon, akin to the MBL transition.