Tunable geometries from a sparse quantum spin network

Nonlocal light-mediated interactions between cold atoms coupled to the mode of an optical cavity present unique prospects for simulating the quantum dynamics of strongly-interacting many-body systems. In a recent publication, we introduced a tunable, nonlocal sparse spin network that can be engineered in near-term single-mode cavity QED platforms.1 In this companion paper, we study this spin network in detail and pedagogically review its basic dynamical properties, providing theoretical details and calculations that expand on the statements made in our original publication. We show that the network exhibits two distinct notions of emergent geometry - linear and treelike - that can be accessed using a single tunable parameter. In either of these two extreme limits, we find a succinct description of the resulting dynamics in terms of two distinct metrics on the network, encoding a notion of either linear or treelike distance between spins. We also show that the network can be mapped in these two extreme limits onto exactly solvable models: a linear Heisenberg spin chain in one limit, and a Dyson hierarchical model in the other. These observations highlight the essential role played by the geometry of the interaction structure in determining a system's dynamics, and raise prospects for novel studies of nonlocal and highly chaotic quantum dynamics in near-term experiments.