Topological Edge States of Trimerized Spin Arrays in 1D Phononic Crystals

Topologically protected systems are characterized by robust boundary states immune to perturbations. Herein, we propose and analyze a tripartite scheme for simulating topological edge states with silicon‐vacancy (SiV) center arrays in 1D phononic crystals. We first investigate the pattern of trimer interactions extended from the standard Su‐Schrieffer‐Heeger (SSH) model. In momentum space, the results indicate that this structure exhibits unique topological properties, including specific symmetries and multiple types of boundary state configurations. Among them, depending on the hopping integrals, the system harbors both symmetry‐protected edge states and topologically trivial edge states. Moving to the realistic quantum system, we introduce a periodic driving protocol for mapping the trimerized spin interactions in 1D phononic crystals (PnCs). Based on the bandgap engineered spin‐phonon coupling, after adiabatic elimination of the phonon modes, we derive the highly tunable long‐range spin interactions. Thus, by means of a specially designed on‐site potential with periodic driving, one can selectively control and enhance the spin‐spin interactions, thereby achieving trimerized interaction in SiV center arrays. We show that, the effective Floquet Hamiltonian combines the inversion and time‐reversal (TR) symmetries, and then numerically simulate the topologically protected edge states with various periodic driving parameters. This work provides a tunable platform to engineer nontrivial topological states of matter in solid‐state quantum systems.