Trimers, pairs and localisation on disordered random graphs

Abstract

Higher-order networks take group interactions as given. I ask what happens when a group interaction is made by pairwise dynamics on a graph, and follow it through the transition that decides whether it matters at large scale, Anderson localisation. An emergent hyperedge is a $k$-body bound state of a Hamiltonian whose terms live on the vertices and edges of the graph, Borromean when none of its $(k-1)$-body faces is bound, detected on a finite graph by a $k$-fold coincidence that survives the thermodynamic limit while the $(k-1)$-fold ones vanish. Sparse random graphs produce one from the simplest pairwise dynamics: three distinguishable particles with on-site attraction $U$ and hopping $t$ bind into a trimer before any pair binds. The thresholds are obtained on the Bethe lattice, in closed form for the pair, in both sectors that an expander graph has and a lattice lacks, the one a condensate sees and the one a Fermi liquid sees. On a disordered graph the emergent hyperedges nucleate on rare deep sites and localise before their parts: the compact trimer, with a narrow band and a threefold sensitivity to site energies, undergoes the Anderson transition at a disorder of $1.6t$, against $17t$ for a single particle and $4t$ for a pair, and the Fermi liquid of trimers is a glass above a line between $1.5t$ and $1.7t$, where a liquid of single particles at the same density needs $15t$ to $16t$. The transition is the single-particle cavity transition with the composite's hopping and effective disorder. A group interaction made by pairwise dynamics is an order of magnitude more fragile to heterogeneity than the network that makes it.