The study of topology in the context of electronic phases of matter is widespread1–6. Yet vibrational7 and dynamical8 topological phases of matter remain to be explored, especially in chemical systems such as polymers, self-assembled networks, metal-organic frameworks, and covalent organic frameworks, to mention a few. While topology in chemistry focuses on shapes in three-dimensional as well as in connectivity space8–15, topology in physics studies the topology of equations of motions, usually in the framework of matrix operators and band theory16. For a one-dimensional material like a linear polymer chain for instance, the space of vibrations of the material can be formally classified with topological tools such as holonomy groups by means of a winding number, for example8,17. This leads to unique global vibrational modes with rich dynamics, and as many vibrational phases with different global vibrational mode properties, as elements in the holonomy group of the vibrational space of the material class17. Regarding vibrations in a mechanical system, it is possible to study their topology in classical systems consisting of coupled oscillators18. Thus far, mostly mechanical models, such as Maxwell lattices19,20, have been employed to demonstrate vibrational topological properties21,22. These toy-models consisting of masses, springs, bars and plates offer a minimal framework for the design of mechanical systems with applications in acoustics23, robotics24, thermal diodes and waveguiding25, among others. In this context, two or more topological phases can be described in the space of vibrations. The trivial topology phase describes a system at the ‘atomic limit’26, wherein topological classification does not play a fundamental role describing the material’s vibrations. For nontrivial topological classifications, new vibrational phenomena can emerge, such as topological boundary modes (TBM) with intriguing properties such as robustness against local defects, nonreciprocity and unidirectionality of vibrational excitations20,27,28,18,29,30.

It is clear that the topological vibrational boundary modes which have been realized in mechanical systems consisting of spring chains20,28, can be realized in in polymer chains, molecular crystals and supramolecular systems which can be accurately described with springs, provided additional springs defining van der Waals interactions, dihedral angles, etc. are considered. The successful adaptation of topological mechanics in chemistry would lead to topological phononics at the molecular scale, with potential in polymerization dynamics31, and superconductivity enhancement32, for instance. However, molecular systems are usually described by unit cells of hundreds of atoms, making the determination of topological vibrational boundary modes very challenging. Self-assembly of molecular structures on a solid surface can simplify the study of polymers and crystals, by offering means to reduce the dimensionality of complex matter.33 Moreover, the versatility of supramolecular interactions can be employed to tailor effective spring constants of vibrational matter34, thereby leveraging topological design in semiconducting35,36, sensing37, switching38–40 and 3D applications41–43 of precision supramolecular lattices.

Here, we show by using molecular dynamics (MD) simulations that supramolecular self-assembled lattices under thermal fluctuations can host topological vibrational boundary modes. We investigate polymer chain models, and prototypical platforms for the formation of supramolecular axial coordination lattices, paving the way for creating 3D supramolecular architectures decoupling vibrations from surfaces. First, we introduce the phonon Su-Schrieffer-Heeger (pSSH) model for the study of topological vibrational modes in adsorbed polymers and supramolecular lattices. Second, we present the adsorbed SSH (aSSH) model on an implicit surface and depict the non-trivial boundary modes of its atomistic simulation under thermal fluctuations. We provide the trivial case counter-example in Supplementary Information. Third, we describe non-trivial boundary modes of the double-chain adsorbed SSH (daSSH) model and its supramolecular equivalent, followed by counter examples in the Supplementary Information. Finally, we deal with the extension of the daSSH model to identify topological boundary modes in a supramolecular ribbon, and the prospects for studying topological vibrational modes beyond the SSH model. Our results outline the robustness of phonon band topology under thermalized conditions and provide platforms for investigating topological chemical physics at the interface between organic chemistry and condensed matter physics.