Spectral properties of hyperbolic nanonetworks with tunable aggregation of simplexes

Cooperative self-assembly is a ubiquitous phenomenon found in natural systems which is used for designing nanostructured materials with new functional features. Its origin and mechanisms, leading to improved functionality of the assembly, have attracted much attention from researchers in many branches of science and engineering. These complex structures often come with hyperbolic geometry; however, the relation between the hyperbolicity and their spectral and dynamical properties remains unclear. Using the model of aggregation of simplexes introduced by Šuvakov et al. [ Sci. Rep. 8, 1987 (2018)], here we study topological and spectral properties of a large class of self-assembled structures or nanonetworks consisting of monodisperse building blocks (cliques of size $n=3,4,5,6$) which self-assemble via sharing the geometrical shapes of a lower order. The size of the shared substructure is tuned by varying the chemical affinity $\nu$ such that for significant positive $\nu$ sharing the largest face is the most probable, while for $\nu <0$, attaching via a single node dominates. Our results reveal that, while the parameter of hyperbolicity remains ${\delta }_{max}=1$ across the assemblies, their structure and spectral dimension $d_s$ vary with the size of cliques $n$ and the affinity when $\nu \ge 0$. In this range, we find that $d_s>4$ can be reached for $n\ge 5$ and sufficiently large $\nu$. For the aggregates of triangles and tetrahedra, the spectral dimension remains in the range $d_s\in [2,4)$, as well as for the higher cliques at vanishing affinity. On the other end, for $\nu <0$, we find $d_s\eqsim 1.57$ independently on $n$. Moreover, the spectral distribution of the normalized Laplacian eigenvalues has a characteristic shape with peaks and a pronounced minimum, representing the hierarchical architecture of the simplicial complexes. These findings show how the structures compatible with complex dynamical properties can be assembled by controlling the higher-order connectivity among the building blocks.