A central part of Condensed Matter Physics is studying the properties of quantum systems
composed of many interacting particles. To pursue this goal, researchers typically consider the constituents of the system confined to some fixed geometry, such as discrete points on a one-dimensional grid or a triangular lattice, and try to understand how the properties of the system depend on microscopic parameters such as the strength of an external potential. The explicit geometry, however, is of fundamental import to the properties of the system hand, playing a decisive role in the emergence of exotic physical properties such as superconductivity and superfluidity . Moreover, in the state-of-the-art experimental platforms where such many-body systems can be realized, researchers are demonstrating increasing capability to tune the explicit couplings of the system and effectively control the geometry of the system by choosing which of its constituents directly interact [2-4].
These ideas motivated our work  which treats the geometry of the system as a parameter itself and tries to understand how the equilibrium properties vary as the geometry is varied and different constraints are placed upon it. We considered the geometry of the system to be specificied by a graph composed of vertices and edges. Each vertex of the graph corresponds to a quantum mechanical spin (such as a qubit) and each edge dictates whether there is a direct interaction between the spins. We then considered the question of how the Physics of the system explicitly depends on the structure of the graph.
Our first consideration was to place absolutely no constraints on the graph and select one at random with a prescribed number of sites and edges. Such a prescription is equivalent to having each possible edge appears independently with some probability p. By utilizing a number of statistical results we prove that, in equilibrium, for any finite p the physics of our system as tends to infinity is independent of and where the edges end up being placed and always equivalent to that of a single, large quantum spin. Moreover, we also proved that even when we split the spins into two separate groups and constrain the graph such that edges appearing between spins in the same group and different groups have separate probabilities p1 and p2, the physics of the system is just equivalent to two large spins.
These results tell us that one must put much stricter constraints on the graph in order to witness true, many-body physics and the inhomogeneity induced by the statistical properties of the random graphs we considered is not strong enough to prevent the system coalescing into a single spin. Importantly, the graph-theoretic approach of our work allows us to identify these constraints. Specifically, we identify two possibilities: one where the edge-density (ratio of edges to the total number of possible edges) of the system vanishes or one where there exists no way to split the spins into a finite number of groups where the spins in each group possess the same number of neighbors.
The first of these constraints leads us to the sparse, low-dimensional structures that are well-studied in Condensed Matter Physics and known to exhibit non-trivial strongly correlated Physics. Despite being a vanishingly small subset of all possible graphs, such structures are typical in the materials observed in the world around us. Our results suggest this ubiquity may be fundamental for the complex, rich behavior exhibited we observe in Nature.
Meanwhile, we term the graphs specified by the latter constraint as `irregular dense graphs’ and note that they have never before been studied in the context of quantum physics. Performing numerical simulations of a series of interacting qubits at zero temperature on such graphs, we show how the high inhomogeneity of this system sets it apart from the more generic `random graphs’, with the phase transition in the model we consider being of distinctly different order occurring compared to those more generic graphs.
The mathematical results in our paper all apply directly in the thermodynamic limit, i.e. the limit where the graph size (and thus number of spins) tends to infinity. We also performed numerical calculations at a finite-size system in order to observe the properties of our system tending towards those predicted by our mathematical results as we increased the system size. These numerical calculations involved the use of the Density Matrix Renormalization Group technique  where the degree of entanglement in the system can be chosen as a computational parameter and controls how difficult the simulation is. We found that whilst non-zero, the entanglement of the system, for all of the highly connected graphs we considered – whether random or highly irregular - was strongly bounded. This allowed us to reach systems sizes well beyond those available by conventional exact diagonalization and verify our mathematical proofs.
By placing the underlying graph as the central parameter in a many-body system we have been able to prove a number of significant results about many-body physics and identify the structures necessary to realize complexity in spin systems. We believe that this is just the beginning of such a graph-theoretic approach to many-body physics and there is the potential to unlock a whole new realm of phases of matter and address fundamental questions about the role of geometry in the many-body problem.
If you have any questions, please feel free to contact us.
 Törmä, P., Peotta, S. & Bernevig, B.A. “Superconductivity, superfluidity and quantum geometry in twisted multilayer systems”, Nat. Rev. Phys. 4, 528–542 (2022)
 Barredo, S. de Leseleuc, V. Lienhard, T. Lahaye, and A. Browaeys, “An atom-by-
atom assembler of defect-free arbitrary two-dimensional atomic arrays,” Science, vol. 354,
no. 6315, pp. 1021–1023 (2016)
 F. M. Gambetta, W. Li, F. Schmidt-Kaler, and I. Lesanovsky, “Engineering nonbinary
Rydberg interactions via phonons in an optical lattice,” Phys. Rev. Lett., vol. 124, p. 043402 (2020)
 S. Korenblit, D. Kafri, W. C. Campbell, R. Islam, E. E. Edwards, Z.-X. Gong, G.-D. Lin,
L. M. Duan, J. Kim, K. Kim, and C. Monroe, “Quantum simulation of spin models on
an arbitrary lattice with trapped ions,” New Journal of Physics, vol. 14, no. 9, p. 095024 (2012)
 J. Tindall, A. Searle, A . Alhajri and D. Jaksch, “Quantum Physics in Connected Worlds”, Nat. Comms. 13, 7445 (2022)
 Steven R. White, “Density matrix formulation for quantum renormalization groups”,
Phys. Rev. Lett. 69, 2863 (1992)