Quantum Physics in Highly Connected Worlds

Theoretical research into many-body quantum systems has mostly focused on either all-to-all models or regular structures which have a small, simple unit cell and where a vanishingly small number of pairs of the constituents directly interact. Motivated by advances in control over the pairwise interactions in many-body simulators, I will discuss the fate of many-body systems on more general, arbitrary graphs. In the spin case, I will present recent results which prove, when placing the minimum possible constraints on the underlying graph, such systems behave like a single collective spin in the thermodynamic limit. I will discuss the constraints necessary to violate these results and introduce hitherto unknown dense graphs where complex many-body physics emerges in the form of entanglement and highly non-uniform correlation functions.

The monogamy of entanglement makes matrix-product-states an accurate, controlled method for studying these highly connected systems. I will conclude by discussing how such insights are now paving the way to new tensor-networ based numerical methods for the study of random fermionic systems on high-dimensional structures.