Topological Dynamics of Synthetic Molecules

TL;DR

This paper explores the topological dynamics of synthetic molecules generated through space transformations, using advanced algebraic tools to classify and interpolate their spectral properties.

Contribution

It introduces a mathematical framework linking molecule architectures to group algebra representations, enabling classification and topological analysis of their dynamics.

Findings

Dynamical matrices correspond to group algebra elements.

Identification of fundamental models for molecule dynamics.

Topological spectral flows arise from model interpolations.

Abstract

We study the dynamics of synthetic molecules whose architectures are generated by space transformations from a point group acting on seed resonators. We show that the dynamical matrix of any such molecule can be reproduced as the left regular representation of a self-adjoint element from the stabilized group's algebra. Furthermore, we use elements of representation theory and K-theory to rationalize the dynamical features supported by such synthetic molecules up to topological equivalences. These tools enable us to identify a set of fundamental models which generate by superposition all possible dynamical matrices up to homotopy equivalences. Interpolations between these fundamental models give rise to topological spectral flows.