Interaction geometry and ground-state properties of sparse quantum lattice models

We investigate how interaction geometry shapes the low-energy phases of sparse tunable long-range quantum models. We focus on a class of graphs whose degree grows logarithmically with system size, and show how symmetry and frustration in graph connectivity can drive, suppress, and reshape ground-state phase transitions. The central examples are power-of-p graphs, where even and odd values of p exhibit qualitatively distinct behaviour: even-p graphs inherit the rich phase structure of the power-of-two model, while odd-p graphs are governed by geometric frustration. Fibonacci graphs provide a contrasting case, lacking the discrete self-similarity of the power-of-p family but exhibiting a direct geometric mapping between the short- and long-range limits. Across our models, we find that phase structure and criticality are governed by the same effective-geometry principle, unifying our framework for experimentally motivated long-range quantum systems.